Comité local:

Pilar Carrasco Carrasco

Josefa María García Hernández

Esperanza López Centella

Evangelina Santos Aláez

Comité científico:

Juan Cuadra Díaz (UAL)

Ferrán Cedó Giné (UAB)

Esther García González (URJC)

José Gómez Torrecillas (UGR)

Ramón González Rodríguez (UVI)

Ángel Granja Barón (ULE)

Pedro A. Guil Asensio (UMU)

Pascual Jara Martínez (UGR)

Cándido Martín González (UMA)

Maribel Tocón Barroso (UCO)

Organiza:

Red Temática NcAlg.

Patrocinan y colaboran:

Universidad de Granada

Facultad de Ciencias UGR

Departamento de Álgebra UGR

GENIL. Granada Excellence Network of Innovation Laboratories

Real Sociedad Matemática Española RSME

Programa:

Jueves 31 de mayo. (15.00 - 19.00)

Lugar: Sala de Conferencias de Matemáticas

15.00-15.30 Inauguración de las Jornadas y sesiones de trabajo.

15.30-16.00 Cedó: Problemas sobre la ecuación de Yang-Baxter PDF

16.00-16.30 Anquela; Cortés; García; Gómez: Cocientes primos de Álgebras de Lie y sistemas de Jordan PDF

16.30-17.00 Andreu Juan: The Hochschild cohomology ring of preprojective algebras of type L_n PDF

17.00-17.30

17.30-18.00 Sánchez; Calderón: Weight modules over split Lie algebras PDF

18.00-18.30 Cortadellas; Jara; Lobillo: A graded criterion in the classification of cofinite homogeneous ideals PDF

18.30-19.00 Odabasi; Estrada: Kaplansky's Theorem for vector bundles PDF

Viernes 1 de junio. (9.00 - 13.00)

Lugar: Sala de Conferencias de Matemáticas

09.00-09.30 Brox; Fernández López: Ideales internos Jordan de los elementos asimétricos de un anillo simple con zócalo PDF

09.30-10.00 Rueda: Differential resultant formulas for linear od-polynomials PDF

10.00-10.30 Burgos Navarro: Strongly preserved problems PDF

10.30-11.00 Sánchez Ortega: An introduction to the diworld PDF

11.00-11.30

11.30-12.00 Soneira Calvo: Módulos Yetter-Drinfeld sobre álgebras de Hopf trenzadas débiles PDF

12.00-12.30 Siles Molina; Solanilla Hernández: Invariancia de Morita en anillos sin uno. Una aplicación a las álgebras de caminos de Leavitt PDF

12.30-13.00 Simón Pinero; Dokuchaev; Exel: Globalización de acciones parciales deformadas PDF

13.00

Viernes 1 de junio. (16.00 - 18.00)

Lugar: Sala de Conferencias de Matemáticas

16.00-16.30 Viruel Arbaizar; Costoya: Faithful actions on differential graded algebras determine the isomorphism type of finite groups PDF

16.30-17.00 Gómez Torrecillas; López Centella: Duality for groupoids and weak Hopf algebras PDF

17.00-17.30 Nicolás Zaragoza; Saorín: Generalized Tilting theory PDF

17.30-18.00 SESIÓN DE PROBLEMAS

18.00 REUNIÓN DE COMITÉS

21.00

Sábado 2 de junio. (9.00 - 13.30)

Lugar: Sala de Conferencias de Matemáticas

09.00-09.30 Antoine; Bosa; Perera: The Cuntz semigroup of Continuous Fields PDF

09.30-10.00 Martínez Verdú: Álgebras poliádicas. Formulación algebraica del Teorema de completitud de Gödel PDF

10.00-10.30 Cuadra Díaz: Sobre el grupo de Brauer del Álgebra de Hopf de Sweedler PDF

10.30-11.00 Caicedo; Margolis; del Río: Zassenhauss conjecture for torsion units PDF

11.00-11.30

11.30-12.00 Leamer: Groebner finite path algebras PDF

12.00-12.30 García Hernández; Marín: Basic module theory for general rings PDF

12.30-13.00 Lobillo Borrero; Cortadellas; Navarro: Prime fuzzy ideals over noncommutative rings PDF

13.00 CLAUSURA DE LAS JORNADAS

13.00

Resúmenes de las sesiones:

IDEALES INTERNOS JORDAN DE LOS ELEMENTOSANTISIMÉTRICOS DE UN ANILLO SIMPLE CON ZÓCALO

JOSE BROX, ANTONIO FERNÁNDEZ LÓPEZ

Abstract. In this work we characterize the Jordan inner ideals of the skew elementsof an associative, simple algebra with involution and nonzero socle, in terms of theirgeometric model as the continuous, finite-rank linear operators of a self-dual vector spacewith involution over a skew field, and describe them as the set of skew-symmetric tracesof the product of a right ideal R with a left ideal, which is either its conjugate R∗ or aminimal left ideal living inside R∗.

Introducción

Este trabajo forma parte de un proyecto, dirigido por el profesor Antonio FernándezLópez, cuyo objetivo es determinar los ideales internos abelianos del álgebra de Lie de loselementos antisimétricos de un álgebra asociativa prima centralmente cerrada con involución.En esta exposición nos ocuparemos del caso más clásico, el de un álgebra asociativa coninvolución que es simple y contiene ideales por la derecha minimales.

Sea A un anillo simple con zócalo no nulo, con involución ∗ y característica distinta de 2.Sea K := Skew(A, ∗) el conjunto de sus elementos antisimétricos (i.e., tales que a∗ = −a).Aunque K no es cerrado para el producto asociativo, sí que lo es para el operador cuadráticode Jordan Pab := aba, pues (aba)∗ = a∗b∗a∗ = −aba. Definimos un ideal interno Jordan deK como aquel conjunto I cuyos elementos absorben a K bajo dicho operador (Pa(K) ⊆ Ipara todo a ∈ I). En esta charla utilizaremos técnicas geométricas para describir todoideal interno Jordan I de K como el conjunto de las trazas antisimétricas de un ideal por laderecha de A multiplicado por un ideal por la izquierda, que es o bien su conjugado o bienun ideal minimal contenido en su conjugado (I = κ(RL) con L = R∗ o L ≤ R∗ minimal).

1. Modelo geométrico de los elementos antisimétricos

El anillo A puede modelarse geométricamente como el conjunto FX(X) de las aplica-ciones lineales y continuas de rango finito de un espacio vectorial con producto escalar nodegenerado sobre un anillo de división con involución (X,∆, < ,>), donde < ,> es o bienantihermítico o bien simétrico (en cuyo caso ∆ es un cuerpo y su involución la identidad); lainvolución de los elementos de A ∼= FX(X) viene dada por la transposición de aplicacioneslineales: recordemos que una aplicación a ∈ EndX es continua si existe otra aplicacióna∗ ∈ EndX (que resulta ser única) tal que ⟨ax, y⟩ = ⟨x, a∗y⟩ para todo par de vectores

Key words and phrases. Anillo simple, involución, ideal interno, elementos antisimétricos.1

2 J. BROX, A. FERNÁNDEZ LÓPEZ

x, y ∈ X. A su vez, FX(X) puede describirse completamente en términos del operadorx∗y : X → X,x∗y(v) := ⟨v, x⟩y, específicamente como el conjunto de sumas finitas de estetipo de aplicación lineal: FX(X) =

∑x∗i yi | xi, yi ∈ X ≡ X∗X. Si definimos la tra za

antisimétrica de una aplicación a como κ(a) := a− a∗, vemos que κ(a) ∈ K porque κ(a)∗ =a∗− a = −κ(a). En estos términos se obtiene que K = κ(X∗X) =

∑κ(x∗i yi) | xi, yi ∈ X.

2. Clasificación de los ideales internos Jordan

A la luz de este modelo geométrico podemos clasificar todos los ideales internos Jordande K, teniendo en cuenta si el producto escalar es antihermítico o simétrico. Asociado atodo ideal interno Jordan I encontramos su rango V (I), la unión de las imágenes de todaslas aplicaciones lineales de I sobre X (V (I) := ax | a ∈ I, x ∈ X), que resulta ser unsubespacio vectorial de X. Se demuestra lo siguiente:

(1) Si < ,> es antihermítico, entonces todo ideal interno Jordan I de K es de la formaκ(V ∗V ) = Skew(V ∗V ), donde se tiene que V ≤ X es V = V (I).

(2) Si < ,> es simétrico, entonces existen dos tipos de ideals internos Jordan:a) Ideales de la forma κ(V ∗V ) con V = V (I).b) Ideales de la forma κ(v∗V ), destacando un vector v ∈ V = V (I).

Se demuestra además que dados los subespacios V1, V2,W1,W2 ⊆ X se tiene que (V ∗1 W1)(V

∗2 W2) =

V ∗2 W1 siempre que ⟨V1,W2⟩ = 0, lo que lleva a que V ∗W = (X∗W )(V ∗X). Es cuestión de

cálculo directo ver que R := X∗W y L := V ∗X son, respectivamente, un ideal por la derechay otro por la izquierda de FX(X), y puesto que (x∗y)∗ = ±y∗x (en función de si el productoescalar es simétrico o antihermítico, respectivamente), se obtiene que W ∗X = R∗, por lo quesi I es de la forma Skew(V ∗V ) entonces I = Skew((X∗V )(V ∗X)) = Skew(RR∗), mientrasque si I es de la forma κ(v∗V ) entonces I = κ((X∗V )(v∗V )) = κ(RL) con L = v∗V ⊆ R∗

minimal.

Departamento de Álgebra, Geometría y Topología, Universidad de Málaga (Spain)E-mail address: brox@agt.cie.uma.es

ZASSENHAUS CONJECTURE FOR TORSION UNITS

MAURICIO CAICEDO, LEO MARGOLIS, AND ÁNGEL DEL RÍO

Abstract. Zassenhaus Conjecture for torsion units states that every augmentation onetorsion unit of the integral group ring of a finite group G is conjugate to an element of Gin the units of rational group algebra QG. This conjecture has been proved for nilpotentgroups, metacyclic groups and some other families of groups. We prove the conjecture forcyclic-by-abelian groups.

Introduction

Let G be a finite group and ZG denotes the integral group ring of G with coefficientsover the ring of integers Z. In the 1960s Hans Zassenhaus established a series of conjecturesabout the finite subgroups of units of augmentation one of ZG, this units are usually callednormalized units. Namely he conjectured that every finite group of normalized units of ZGis conjugate to a subgroup of G in the units of QG. These conjecture is usually denoted(ZC3), while the version of (ZC3) for the particular case of subgroups of normalized unitswith the same cardinality as G is usually denoted (ZC2). These conjectures have importantconsequences. For example, a positive solution of (ZC2) implies a positive solution for theIsomorphism and Automorphism Problems (see [Seh93] for details). The most celebratedpositive result for Zassenhaus Conjectures is due to Weiss [Wei91] who proved (ZC3) fornilpotent groups. However Roggenkamp and Scott founded a counterexample to the Au-tomorphism Problem, and henceforth to (ZC2) (see [Rog91] and [Kli91]). Later Hertweck[Her01] provided a counterexample to the Isomorphism Problem.

The only conjecture of Zassenhaus that is still up is the version for cyclic subgroupsnamely:

Zassenhaus Conjecture for Torsion Units (ZC1). If G is a finite groupthen every normalized torsion unit of ZG is conjugate in QG to an elementof G.

Besides the family of nilpotent groups, (ZC1) has been proved for some concrete groups[BH08, BHK04, HK06, LP89, LT91, Her08b], for groups having a Sylow subgroup withan abelian complement [Her06], for some families of cyclic-by-abelian groups [LB83, LT90,LS98, MRSW87, PMS84, PMRS86, dRS06, RS83] and some classes of metabelian groups notnecessarily cyclic-by-abelian [MRSW87, SW86]. Other results on Zassenhaus Conjecturescan be found in [Seh93, Seh01] and [Seh03, Section 8]

The latest and most general result for (ZC1) on the class of cyclic-by-abelian groups isdue to Hertweck [Her08a] who proved (ZC1) for finite groups of the form G = AX with Aa cyclic normal subgroup of G and X an abelian subgroup of G. This includes the class ofmetacyclic groups that was not covered in previous results.

We prove (ZC1) for arbitrary cyclic-by-abelian groups.1

2 MAURICIO CAICEDO, LEO MARGOLIS, AND ÁNGEL DEL RÍO

Theorem. Let G be a finite cyclic-by-abelian group. Then every normalizedtorsion unit of ZG is conjugate in QG to an element of G.

References

[BH08] V. Bovdi and M. Hertweck, Zassenhaus conjecture for central extensions of S5, J. Group Theory11 (2008), no. 1, 63–74. MR 2381018 (2009a:20010)

[BHK04] V. Bovdi, C. Höfert, and W. Kimmerle, On the first Zassenhaus conjecture for integral grouprings, Publ. Math. Debrecen 65 (2004), no. 3-4, 291–303. MR 2107948 (2006f:20009)

[dRS06] Á. del Río and S.K. Sehgal, Zassenhaus conjecture (ZC1) on torsion units of integral grouprings for some metabelian groups, Arch. Math. (Basel) 86 (2006), no. 5, 392–397. MR 2229354(2007c:16064)

[Her01] M. Hertweck, A counterexample to the isomorphism problem for integral group rings, Ann. ofMath. 154 (2001), 115–138.

[Her06] , On the torsion units of some integral group rings, Algebra Colloq. 13 (2006), no. 2,329–348. MR 2208368 (2006k:16049)

[Her08a] , Torsion units in integral group rings of certain metabelian groups, Proc. Edinb. Math.Soc. (2) 51 (2008), no. 2, 363–385. MR 2465913 (2009j:16027)

[Her08b] , Zassenhaus conjecture for A6, Proc. Indian Acad. Sci. Math. Sci. 118 (2008), no. 2,189–195. MR 2423231 (2009c:20010)

[HK06] C. Höfert and W. Kimmerle, On torsion units of integral group rings of groups of small order,Groups, rings and group rings, Lect. Notes Pure Appl. Math., vol. 248, Chapman & Hall/CRC,Boca Raton, FL, 2006, pp. 243–252. MR 2226199 (2007d:16077)

[Kli91] L. Klingler, Construction of a counterexample to a conjecture of Zassenhaus, Comm. Algebra19 (1991), no. 8, 2303–2330. MR 1123126 (92i:20004)

[LB83] I.S. Luthar and A.K. Bhandari, Torsion units of integral group rings of metacyclic groups, J.Number Theory 17 (1983), no. 2, 270–283. MR 716946 (85c:20004)

[LP89] I.S. Luthar and I.B.S. Passi, Zassenhaus conjecture for A5, Proc. Indian Acad. Sci. Math. Sci.99 (1989), no. 1, 1–5. MR 1004634 (90g:20007)

[LS98] I.S. Luthar and P. Sehgal, Torsion units in integral group rings of some metacyclic groups, Res.Bull. Panjab Univ. Sci. 48 (1998), no. 1-4, 137–153 (1999). MR 1773990 (2001f:16065)

[LT90] I.S. Luthar and P. Trama, Zassenhaus conjecture for certain integral group rings, J. Indian Math.Soc. (N.S.) 55 (1990), no. 1-4, 199–212. MR 1088139 (92b:20008)

[LT91] , Zassenhaus conjecture for S5, Comm. Algebra 19 (1991), no. 8, 2353–2362. MR 1123128(92g:20003)

[MRSW87] Z. Marciniak, J. Ritter, S. K. Sehgal, and A. Weiss, Torsion units in integral group rings of somemetabelian groups. II, J. Number Theory 25 (1987), no. 3, 340–352. MR 880467 (88k:20019)

[PMRS86] C. Polcino Milies, J. Ritter, and S.K. Sehgal, On a conjecture of Zassenhaus on torsion unitsin integral group rings. II, Proc. Amer. Math. Soc. 97 (1986), no. 2, 201–206. MR 835865(87i:16013)

[PMS84] C. Polcino Milies and S.K. Sehgal, Torsion units in integral group rings of metacyclic groups, J.Number Theory 19 (1984), no. 1, 103–114. MR 751167 (86i:16009)

[Rog91] K.W. Roggenkamp, Observations on a conjecture of Hans Zassenhaus, Groups—St. Andrews1989, Vol. 2, London Math. Soc. Lecture Note Ser., vol. 160, Cambridge Univ. Press, Cambridge,1991, pp. 427–444. MR 1123997 (92g:20004)

[RS83] J. Ritter and S.K. Sehgal, On a conjecture of Zassenhaus on torsion units in integral group rings,Math. Ann. 264 (1983), no. 2, 257–270. MR 711882 (85e:16014)

[Seh93] S.K. Sehgal, Units in integral group rings, Pitman Monographs and Surveys in Pure and AppliedMathematics, vol. 69, Longman Scientific & Technical, Harlow, 1993, With an appendix by AlWeiss. MR 1242557 (94m:16039)

[Seh01] , Zassenhaus conjecture, Encyclopaedia of mathematics. Supplement. Vol. III(M. Hazewinkel, ed.), Kluwer Academic Publishers, Dordrecht, 2001, pp. 453–454. MR 1935796(2003j:00009)

ZASSENHAUS CONJECTURE FOR TORSION UNITS 3

[Seh03] , Group rings, Handbook of algebra, Vol. 3, North-Holland, Amsterdam, 2003, pp. 455–541. MR 2035104 (2005d:16044)

[SW86] S.K. Sehgal and A. Weiss, Torsion units in integral group rings of some metabelian groups, J.Algebra 103 (1986), no. 2, 490–499. MR 864426 (88f:20015)

[Wei91] A. Weiss, Torsion units in integral group rings, J. Reine Angew. Math. 415 (1991), 175–187.MR 1096905 (92c:20009)

Departamento de Matemáticas, Universidad de Murcia, 30100 Murcia, SpainE-mail address: mauriciojc02@hotmail.com

Fachbereich Mathematik, Universitaet Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart,Germany

E-mail address: leo.imsueden@yahoo.com

Departamento de Matemáticas, Universidad de Murcia, 30100 Murcia, SpainE-mail address: adelrio@um.es

PROBLEMAS SOBRE LA ECUACIÓN DE YANG-BAXTER

FERRAN CEDÓ

Abstract. This is a survey about some open problems on set-theoretic solutions of theYang-Baxter equation and related algebraic structures.

Introducción

Sean X un conjunto no vacío y r : X2 → X2 una aplicación, denotando r(x, y) porr(x, y) = (σx(y), γy(x)). Se dice que (X, r) es una solución conjuntista, involutiva, nodegenerada por la derecha, de la ecuación de Yang-Baxter si se cumplen las propiedadessiguientes:

(1) r2 = idX2 ,(2) σx ∈ SymX , para todo x ∈ X,(3) r1r2r1 = r2r1r2, donde r1 = r × idX : X3 → X3 y r2 = idX × r : X3 → X3.Si además se cumple:(2’) γx ∈ SymX , para todo x ∈ X,

decimos que (X, r) es no degenerada (por los dos lados).Sea (X, r) una solución conjuntista, involutiva, no degenerada, de la ecuación de Yang-

Baxter. El grupo G(X, r) presentado con conjunto de generadores X y relaciones:

xy = σx(y)γy(x),

para todo x, y ∈ X, se llama grupo de estructura de (X, r), y Etingof, Shedler y Soloviev en[3] demostraron que es isomorfo a un subgrupo de ZX o SymX de la forma

(a, φ(a)) | a ∈ ZX,donde ZX denota el grupo abeliano libre con base X y φ es una aplicación de ZX a SymX

tal que φ(x) = σx para todo x ∈ X.El subgrupo de permutaciones G(X, r) ⊆ SymX generado por σx | x ∈ X claramente

es una imagen homomórfica de G(X, r). En [3] se demuestra que si X es finito, entoncesG(X, r) es resoluble, y por tanto G(X, r) es resoluble y finito. Además en [6] se demuestraque si X es finito entonces G(X, r) es libre de torsión.

En [3] se introduce la relación binaria ∼ de retracción sobre X con respecto a r definidapor

x ∼ y si y sólo si σx = σy,

para x, y ∈ X. Es claro que ∼ es de equivalencia e induce una solución conjuntista, involu-tiva, no degenerada, de la ecuación de Yang-Baxter (X/ ∼, r), con

r([x], [y]) = ([σx(y)], [γy(x)]),1

2 FERRAN CEDÓ

donde [x] denota la clase de x ∈ X en el conjunto cociente X/ ∼. Esta solución se diceque es la retracción de (X, r) y la denotamos por Ret(X, r). Definimos Ret0(X, r) = (X, r)y, para n > 0, Retn(X, r) = Ret(Retn−1(X, r)). Si existe un entero no negativo m talque Retm(X, r) es la solución trivial (sobre un conjunto de un elemento), entonces se diceque (X, r) es una solución multipermutación. En este caso, si m es el menor posible conRetm(X, r) trivial, decimos que (X, r) tiene nivel de multipermutación m y escribimosmpl(X, r) = m.

Se dice que la solución (X, r) es libre de cuadrados si σx(x) = x para todo x ∈ X.Un avance importante en el estudio de esta tipo de soluciones es el siguiente resultado de

Rump.

Teorema 0.1. (Rump [8, Theorem 1]) Sea X un conjunto finito con |X| > 1. Si (X, r) esuna solución conjuntista, involutiva, no degenerada y libre de cuadrados de la ecuación deYang-Baxter, entonces X tiene al menos dos órbitas bajo la acción de G(X, r).

Una de las consecuencias de este resultado es el siguiente:

Corolario 0.2. ([7, Proposition 4.2]) Sea X un conjunto finito no vacío. Si (X, r) esuna solución conjuntista, involutiva, no degenerada y libre de cuadrados de la ecuación deYang-Baxter, entonces G(X, r) es poli-Z.

1. Problemas y Conjeturas

Sea (X, r) una solución conjuntista, involutiva, no degenerada de la ecuación de Yang-Baxter.

Conjetura 1. ([4]) Si X es finito y (X, r) es libre de cuadrados, entonces (X, r) es unasolución multipermutación.

En [1] se demuestra que la conjetura es cierta si G(X, r) es abeliano.Esta conjetura es equivalente a la siguiente, que tiene un enunciado muy elemental.

Conjetura 1’. Sean X = 1, 2, . . . , n y σ1, σ2, . . . , σn ∈ Sn tales que(i) σx(x) = x para todo x ∈ X,(ii) σxσσ−1

x (y) = σyσσ−1y (x) para todo x, y ∈ X.

Entonces existen dos elementos distintos x, y ∈ X tales que σx = σy.

En un artículo muy reciente [5] Gateva-Ivanova y Cameron obtienen la siguiente relaciónentre la longitud de la serie derivada de G(X, r) y mpl(G, r) cuando (X, r) es una soluciónmultipermutación libre de cuadrados.

Teorema 1.1. ([5, Theorem 6.10]) Si (X, r) es una solución multipermutación libre decuadrados, entonces G(X, r) es resoluble y la longitud de su serie derivada es menor o iguala mpl(G, r).

PROBLEMAS SOBRE LA ECUACIÓN DE YANG-BAXTER 3

En el mismo artículo plantean el siguiente problema.

Problema 1. ([5]) Dado un entero positivo m, ¿existe alguna solución multipermutación(X, r) de nivel m libre de cuadrados, con X finito y G(X, r) abeliano?

Recientemente junto con Eric Jespers y Jan Okniński hemos contestado afirmativamenteesta pregunta.

En [5] hay otros muchos problemas abiertos interesantes.Hemos visto antes que si X es finito entonces G(X, r) es resoluble. En [2] planteamos el

problema siguiente.

Problema 2. ([2]) Sea G un grupo finito resoluble. ¿Existe alguna solución conjuntista,involutiva, no degenerada (X, r) de la ecuación de Yang-Baxter tal que G ∼= G(X, r)?

En [2] se obtienen algunos resultados que apoyan una respuesta afirmativa de esta pre-gunta. Por ejemplo, se demuestra que existe una solución conjuntista, involutiva, no de-generada (X, r) de la ecuación de Yang-Baxter tal que X es finito y G es isomorfo a unsubgrupo de G(X, r).

Para grupos abelianos la respuesta es afirmativa. De hecho la respuesta es afirmativa siG es nilpotente de clase ≤ 2.

No se conoce la respuesta para grupos nilpotentes de clase 3 (o mayor).

References

[1] F. Cedó, E. Jespers and J. Okniński, Retractability of the set theoretic solutions of the Yang-Baxterequation, Adv. Math. 224 (2010), 2472–2484.

[2] F. Cedó, E. Jespers and Á. del Río, Involutive Yang-Baxter groups, Trans. Amer. Math. Soc. 362 (2010),2541–2558.

[3] P. Etingof, T. Schedler and A. Soloviev, Set-theoretical solutions to the quantum Yang-Baxter equation,Duke Math. J. 100 (1999), 169–209.

[4] T. Gateva-Ivanova, A combinatorial approach to the set-theoretic solutions of the Yang-Baxter equation,J. Math. Phys. 45 (2004), 3828–3858.

[5] T. Gateva-Ivanova and P. Cameron, Multipermutation solutions of the Yang-Baxter equation, Comm.Math. Phys. 309 (2012), 583–621.

[6] T. Gateva-Ivanova and M. Van den Bergh, Semigroups of I-type, J. Algebra 206 (1998), 97–112.[7] E. Jespers and J. Okniński, Monoids and Group of I-Type, Algebr. Represent. Theory 8 (2005), 709-729.[8] W. Rump, A decomposition theorem for square-free unitary solutions of the quantum Yang-Baxter

equation, Adv. Math. 193 (2005), 40–55.

Universitat Autònoma de BarcelonaE-mail address: cedo@mat.uab.cat

SOBRE EL GRUPO DE BRAUER DEL ÁLGEBRA DE HOPF DESWEEDLER

JUAN CUADRA DÍAZ

Abstract. In 1994 Caenepeel, Van Oystaeyen and Zhang defined the Brauer group of aHopf algebra with bijective antipode by considering Yetter-Drinfeld module algebras [1].They extended so a previous construction by Long for commutative and cocommutativeHopf algebras [6]. Since then it was a main goal to compute the Brauer group of the smallestnoncommutative noncocommutative Hopf algebra, namely, Sweedler four dimensional Hopfalgebra. In this talk we will report about the current state of knowledge on this problem.The results to be presented are based on the joint work with Giovanna Carnovale [5].

Resumen

En 1994 Caenepeel, Van Oystaeyen y Zhang definieron el grupo de Brauer de un álgebrade Hopf con antípoda biyectiva mediante la estructura de módulo-álgebra de Yetter-Drinfeld[1]. Extendieron así una construcción previa de Long para álgebras de Hopf conmutativas ycoconmutativas [6]. Desde entonces el principal objetivo ha sido calcular el grupo de Brauerdel álgebra de Hopf de Sweedler, que tiene dimensión 4 y es el álgebra de Hopf no conmutativay no coconmutativa de dimensión más pequeña. En esta charla discutiremos cuál es el estadoactual de conocimiento sobre este problema. Los resultados que se presentarán aparecen enel trabajo conjunto con Giovanna Carnovale [5].

References

[1] S. Caenepeel, F. Van Oystaeyen y Y.H. Zhang, Quantum Yang-Baxter Module Algebras. K-theory 8 no.3 (1994), 231-255.

[2] , The Brauer group of Yetter-Drinfeld module algebras. Trans. Amer. Math. Soc. 349 no. 9(1997), 3737-3771.

[3] S. Caenepeel, Brauer groups, Hopf algebras and Galois Theory. K-Monographs in Mathematics 4.Kluwer Academic Publishers, Dordrecht, 1998.

[4] G. Carnovale, Some isomorphisms for the Brauer groups of a Hopf algebra. Comm. Algebra 29 no. 11(2001), 5291-5305.

[5] G. Carnovale y J. Cuadra, On the subgroup structure of the full Brauer group of Sweedler Hopf algebra.Israel J. Math 183 (2011), 61-92.

[6] F.W. Long, The Brauer group of dimodule algebras. J. Algebra 31 (1974), 559-601.[7] F. Van Oystaeyen y Y.H. Zhang, The Brauer group of Sweedler’s Hopf algebra H4. Proc. Amer. Math.

Soc. 129 no. 2 (2001), 371-380.

Universidad de Almería, Dpto. Álgebra y Análisis Matemático, E-04120 AlmeríaE-mail address: jcdiaz@ual.es

1

BASIC MODULE THEORY FOR GENERAL RINGS

J. L. GARCÍA AND L. MARÍN

Abstract.

Introduction

We consider here associative rings which do not necessarily have an identity element, andwe call them general rings. Our purpose is to construct a theory of modules over generalrings which is reasonably simple and, at the same time, is a direct extension of the usualmodule theory for rings with identity. We show that this is indeed feasible, so that most ofthe elementary results of the module theory of rings with identity are but particularizationsof the results of the theory of modules over general rings; the noteworthy exceptions beingthose results depending upon the existence of enough finitely presented objects, somethingthat cannot be guaranteed in our setting for general rings.

The article containing the results we shall present has appeared in Communications inAlgebra, 40 (2012), 291-314.

References

University of Murcia . . .E-mail address: jlgarcia@um.es

1

COCIENTES PRIMOS DE ÁLGEBRAS DE LIE Y SISTEMAS DE

JORDAN

JOSÉ A. ANQUELA, TERESA CORTÉS, ESTHER GARCÍA, AND MIGUEL A. GÓMEZ LOZANO

Abstract. We show that, unlike alternative algebras, prime quotients of a nondegenerateJordan system or a Lie algebra need not be nondegenerate, even if the original Jordansystem is primitive, or the Lie algebra is strongly prime, both with nonzero simple hearts.For Jordan systems and Lie algebras directly linked to associative systems, we prove thatsemiprime quotients are necessarily nondegenerate.

Introducción

Un resultado clásico en teoría de anillos asociativos es: para un anillo asociativo R lassiguientes condiciones son equivalentes:

(1) R es un anillo semiprimo.(2) R no posee ideales por la izquierda no nulos nilpotentes.(3) R no posee ideales por la derecha no nulos nilpotentes.(4) Si x ∈ R verifica xRx = 0, entonces x = 0.

No obstante, cuando salimos del mundo asociativo, incluso en el mundo alternativo, estasequivalencias no son ciertas. Así, si A es un álgebra alternativa, el carácter semiprimo,carecer de ideales nilpotentes no nulos, no es equivalente a ser no degenerada, es decir,no poseer divisores absolutos de cero no nulos (elementos x ∈ A tales que xAx = 0). Noobstante, en [5], Beidar, Mikhalev y Shestakov demostraron que si A es un álgebra alternativano degenerada e I es un ideal de A tal que el cociente A/I es semiprimo, entonces A/I es nodegenerado, lo cual es una extensión del resultado de Kleinfeld [14, Ch. 9, Sect. 2, Th. 5].Obsérvese que este resultado demuestra que existen álgebras alternativa libres degeneradas.

En este trabajo estudiamos si un resultado análogo al de [5] es también cierto en elcontexto de las álgebras de Lie o de los sistemas (álgebras, pares y sistemas triples) deJordan.

1. Preliminares

1.1. Vamos a trabajar con sistemas asociativos, sistemas de Jordan (álgebras, pares y sis-temas triples) y álgebras de Lie sobre un anillo de escalares arbitrario Φ (cf. [8, 9, 10, 11, 12]).

— Dada un álgebra de Jordan J , sus productos serán denotados por x2, Uxy, para x, y ∈ J .Estos productos son cuadráticos en x y lineales en y con linealizaciones x y, Ux,zy =x, y, z = Vx,yz, respectivamente.

— Dado un par de Jordan V = (V +, V −), escribiremos sus productos Qxy ∈ V ε, x ∈ V ε,y ∈ V −ε, ε = ±, con linealizaciones Qx,zy = x, y, z = Dx,yz.

1

2 JOSÉ A. ANQUELA, TERESA CORTÉS, ESTHER GARCÍA, AND MIGUEL A. GÓMEZ LOZANO

— Dado un sistema triple de Jordan T , escribiremos sus productos Pxy, x, y ∈ T , conlinealizaciones Px,zy = x, y, z = Lx,yz.

— Dada un álgebra de Lie L, el producto (bilineal) de dos elementos x, y ∈ L serádenotado por [x, y]. La aplicación adjunta adx : L −→ L está definida por adx(y) = [x, y].

1.2. Un álgebra de Jordan da lugar a un sistema triple de Jordan simplemente olvidando laoperación cuadrado, y tomando P := U . Si duplicamos cualquier sistema triple de JordanT , V (T ) := (T, T ) con producto Q := P es un par de Jordan. Por último, si partimos deun par de Jordan V = (V +, V −), T (V ) := V + ⊕ V − con producto Px+⊕x−(y+ ⊕ y−) =Qx+y− ⊕Qx−y+ es un sistema triple de Jordan (cf. [11, 1.13, 1.14]).

1.3. Un sistema asociativo R da lugar a un sistema de Jordan R(+), llamado su simetrización:con la misma estructura de Φ-módulo, definimos el mismo cuadrado y producto cuadráticoUxy = xyx, con x, y ∈ R en el caso de álgebras, Pxy = xyx en el caso de sistemas triples,y Qxσy−σ = xσy−σxσ, σ = ± en el caso de pares, en donde la yuxtaposición denota elproducto asociativo de R.

Dado un álgebra asociativa R, podemos construir un álgebra de Lie R(−), llamada suantisimetrización, quedándonos con la misma estructura de Φ-módulo y definiendo el nuevoproducto por [x, y] = xy − yx.

1.4. Un divisor absoluto de cero en un sistema de Jordan J es un x ∈ J tal que Ux = 0(resp. Px = 0 o Qx = 0), mientras que en un álgebra de Lie L es un elemento x ∈ L tal quead2x = 0. Un sistema de Jordan o álgebra de Lie se dice que es no degenerado si el únicodivisor absoluto de cero que posee es el cero. Un sistema de Jordan o álgebra de Lie se diceque es semiprima (resp. prima) si el único ideal nilpotente es el nulo (resp. no contieneideales ortogonales no nulos).

2. Contraejemplos

2.1. Si consideramos la Φ-álgebra asociativa libre FAssalg[X] sobre un conjunto X, la Φ-subálgebra de Jordan de FAssalg[X] generada por X es la Φ-álgebra de Jordan libre especialsobre X, denotada FSJalg[X]. Es más, cuando Φ es un cuerpo, FSJalg[X] es fuertementeprima. Por tanto, el ejemplo dado en [13], de un álgebra de Jordan especial prima y dege-nerada sobre un cuerpo de característica cero, produce un primer contraejemplo al análogobuscado, ya que es cociente del álgebra de Jordan libre especial fuertemente prima FSJalg[X].Los funtores dados en (1.2), junto con la transferencias de regularidades demostradas en [1],producen ejemplos de un par de Jordan o un sistema triple de Jordan fuertemente primosobre un cuerpo de característica cero con cocientes primos y degenerados.

2.2. Siguiendo esta misma filosofía, en el mundo Lie podemos encontrar contraejemplos alresultado dado en [5] para el mundo alternativo. Sea X un conjunto y FLiealg[X] el álgebrade Lie libre sobre X. Como consecuencia del teorema de Poincare-Birkhoff-Witt (cf. [8, Cor.17.3 B], [9, Cor. 1, p. 160]), si estamos trabajando en álgebras sobre cuerpos, FLiealg [X]es isomorfa a la subálgebra de Lie generada por X en FAssalg[X], lo que implica que, siX es infinito, es un álgebra de Lie fuertemente prima. Por tanto, si partimos del álgebrade Jordan prima y degenerada dada en [13] sobre un cuerpo de característica cero, el parde Jordan V = (J, J) es primo y degenerado, por lo que el álgebra de Lie TKK(V ) es un

COCIENTES PRIMOS DE ÁLGEBRAS DE LIE Y SISTEMAS DE JORDAN 3

álgebra de Lie prima y degenerada sobre un cuerpo de característica cero (cf. [7, 1.2, 2.2,2.6]), que es cociente del álgebra de Lie libre fuertemente prima FLiealg[X].

2.3. Es más, en [4] podemos encontrar un álgebra (par o triple) de Jordan primitivo concorazón no nulo que posee cocientes primos y degenerados, mientras que en [2] se muestranejemplos de álgebras de Lie fuertemente primas con corazón no nulo y cocientes primosdegenerados.

3. Sistemas de Jordan o Lie asociados con sistemas asociativos

En esta sección vamos a ver que, a pesar de lo anterior, un resultado análogo al dado en[5] para álgebras alternativas es cierto para estructuras Jordan o Lie que están próximas alas asociativas.

Definición 3.1. Sea R un sistema asociativo (álgebra, pair, o sistema triple) con involución∗, H0 := H0(R, ∗) un subespacio amplio de R, y B un ∗-ideal de R.

— Si R es un álgebra, definimos

K(B,H0) =

b+ b∗ +∑

i

λibib∗

i +∑

j

bjhjb∗

j | b, bi, bj ∈ B,hj ∈ H0, λi ∈ Φ

,

que es un ideal de H0 contenido en B ∩H0 [3, 2.2].— Si R = (R+, R−) es un par, definimos

K(Bσ,H0) =

b+ b∗ +∑

i

bihib∗

i | b, bi ∈ Bσ, hi ∈ H−σ0

,

que es un semi-ideal de H0 contenido en Bσ ∩ Hσ0 , σ = ± [3, 3.1] Denotaremos por

K(B,H0) = (K(B+,H0),K(B−,H0)).— Si R es un sistema triple, definimos

K(B,H0) =

b+ b∗ +∑

i

bihib∗

i | b, bi,∈ B,hi ∈ H0

,

que es un semi-ideal de H0 contenido en B ∩H0 [3, 2.2].

Teorema 3.2. Si R es un sistema asociativo con involución ∗, H0 := H0(R, ∗) es unsubespacio amplio de R, y P es un ideal semiprimo (resp. primo) de H0, entonces existe un∗- ideal semiprimo (resp. primo) I de R tal que P = I ∩H0. Es más, I es el mayor ∗-idealde R tal que K(I,H0) ⊆ P . Más aún, P es un ideal no degenerado (resp. fuertementeprimo) de H0.

Corolario 3.3. Si R es un sistema asociativo y P es un ideal semiprimo (resp. primo) de

R(+), entonces P es un ideal semiprimo (resp. primo) de R y, por tanto, P es un ideal no

degenerado de R(+).

Observemos que con estos dos resultados no sólo hemos demostrado que para los sistemasde Jordan no degenerados R(+) y H0(R, ∗) cualquier cociente semiprimo es no degenerado,sino que hemos caracterizado los ideales semiprimos de estos sistemas a partir de idealessemiprimos del sistema asociativo.

4 JOSÉ A. ANQUELA, TERESA CORTÉS, ESTHER GARCÍA, AND MIGUEL A. GÓMEZ LOZANO

Teorema 3.4. Sea R un álgebra asociativa y P un ideal semiprimo de R(−) y sea I unideal de R maximal entre los contenidos en P . Entonces I es un ideal semiprimo de R,P = Z(R/I) y, por tanto, P es un ideal no degenerado de R(−).

En el mundo Lie no es cierto, en general, que los ideales semiprimos provengan de idealesasociativos semiprimos: si consideramos R el álgebra de matrices 4 por 4 con la involucióntrasposición, tenemos que L = K(R, ∗) es un álgebra de Lie que es suma directa de dosálgebras simples, por lo que cualesquiera de estas álgebras es un ideal fuertemente primode L que no tiene relación posible con los ideales de R, ya que ésta es simple (cf. [6]). Noobstante si que podemos decir algo en este caso.

Teorema 3.5. Sea (R, ∗) un álgebra asociativa con involución ∗-prima tal que R no es unorden central en un álgebra de matrices Mn(F) con F un cuerpo y n ≤ 4. Entonces todoideal semiprimo (resp. primo) de K(R, ∗) es no degenerado (resp. fuertemente primo).

References

[1] J. A. Anquela, T. Cortés, “Local and Subquotient Inheritance of Simplicity in Jordan Systems", J.

Algebra 240 (2001) 680-704.[2] J. A. Anquela, T. Cortés, “Imbedding Lie Algebras in Strongly Prime Algebras", Comm. Algebra (en

prensa).[3] J. A. Anquela, T. Cortés, E. García, “Herstein’s Theorems and Simplicity of Hermitian Jordan Systems",

J. Algebra, 246 (2001) 193-214.[4] J. A. Anquela, T. Cortés, K. McCrimmon, “Imbedding Jordan Systems in Primitive Systems" (en

preparación).[5] K. I. Beidar, A. V. Mikhalev, “The Structure of Nondegenerate Alternative Algebras", J. Soviet. Math.

47 (1989) 2525-2536.[6] T. S. Erickson, The Lie Structure in Prime Rings with involution Journal of Algebra 21, (1972) 523-534.[7] E. García, “Tits-Kantor-Koecher Algebras of Strongly Prime Hermitian Jordan Pairs", J. Algebra 277

(2004) 559-571.[8] J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer-Verlag, New York,

1980.[9] N. Jacobson, Lie Algebras, Dover Publications, Inc., New York, 1979.

[10] N. Jacobson, Structure and Representations of Jordan Algebras, Amer. Math. Soc. Colloq. Publ., vol.39, Providence, 1968.

[11] O. Loos, Jordan Pairs, Lecture Notes in Mathematics, Vol. 460, Springer-Verlag, New York, 1975.[12] K. McCrimmon, A Taste of Jordan Algebras, Springer-Verlag, New York, 2004.[13] S. V. Pchelintsev, “Prime Algebras and Absolute Zero Divisors" Math. USSR Izvestiya 28 (1) (1987)

79-98.[14] Rings that are nearly associative, by K. A. Zhevlakov, A. M. Slin’ko, I. P. Shestakov and A. I. Shirshov,

translated by Harry F. Smith, Academic Press, New York, 1982, xi + 371 pp.

Dpto. de Matemáticas, Universidad de Oviedo, C/ Calvo Sotelo s/n, 33007 Oviedo, SpainE-mail address: anque@orion.ciencias.uniovi.es;cortes@orion.ciencias.uniovi.es

Dpto. de Matemática Aplicada, Universidad Rey Juan Carlos, 28933 Móstoles, Madrid, SpainE-mail address: esther.garcia@urjc.es

Dpto. de Álgebra, Geometría y Topología, Universidad de Málaga, 29071 MálagaE-mail address: miggl@uma.es

DUALITY FOR GROUPOIDS AND WEAK HOPF ALGEBRAS

JOSÉ GÓMEZ-TORRECILLAS AND ESPERANZA LÓPEZ-CENTELLA

Abstract. We prove that the category of finite grupoids is anti-equivalent to the categoryof commutative semisimple finite-dimensional weak Hopf algebras over an algebraicallyclosed field of characteristic 0. This result is obtained as a consequence of our main result,which shows that there is an adjoint pair of functors between the category of grupoidswith finitely many objects and the category of weak Hopf algebras.

Introduction

The category of finite groups is anti-equivalent to the category of commutative semisimplefinite-dimensional Hopf algebras over an algebraically closed field of characteristic 0 (see [1,Theorem 3.4.2]). This basic result could be understood as the simplest discrete version ofTannaka’s duality theorem. Weak Hopf algebras were introduced in [2, 4] in the frameworkof the theory of quantum groupoids, so they have been treated basically as genuine noncom-mutative since their inception. In particular, the extension of the aforementioned dualityfor finite grupoids seems have not been explored. The aim of this communication is to studythis topic and, more generally, to explain a precise relationship between the category of (settheoretical) grupoids and that of weak Hopf algebras.

1. The groupoid weak Hopf algebra

We work over a field K of characteristic 0, and algebras and coalgebras are meant over K.The tensor product of vector spaces over K is denoted by ⊗. For an excelent introductionto the theory of Hopf algebras, we refer to [1]. A systematic account of the fundamentals ofthe theory of weak Hopf algebras is [3].

Definition 1.1. A weak bialgebra is a quintuple (H,µ, η,∆, ε), where (H,µ, η) is an as-sociative unital algebra with multiplication µ : H ⊗ H → H and unit η : K → H, and(H,∆, ε) is a coassociative counital coalgebra with comultiplication ∆ : H → H ⊗ H andcounit ε : H → K such that the following compatibility conditions hold.(a) The comultiplication ∆ is multiplicative (equivalently, the multiplication µ is comulti-

plicative.(b) For all x, y, z ∈ H

ε(xyz) = ε(xy(1))ε(y(2)z)

ε(xyz) = ε(xy(2))ε(y(1)z)

SUPPORT FROM THE MINISTERIO DE CIENCIA E INNOVACIÓN AND FEDER, GRANT MTM2010-20940-C02-01 IS GRATEFULLY ACKNOWLEDGED.

1

2 JOSÉ GÓMEZ-TORRECILLAS AND ESPERANZA LÓPEZ-CENTELLA

(c)

(∆⊗H) ∆(1) = (∆(1)⊗ 1)(1⊗∆(1))

(∆⊗H) ∆(1) = (1⊗∆(1))(∆(1)⊗ 1)

Example 1.2. Consider a small category C, with a finite set of objects C0. Let C1 denotethe set of arrows, and s, t : C1 → C0 the source and target maps. By KC we denote the weakHopf algebra built on the k–vector space with basis C1. The coalgebra structure on KC isdetermined by ∆(g) = g⊗g and ε(g) = 1 for every g ∈ C1. The multiplication is given on C1by gh = g h if s(g) = t(h), and gh = 0 if s(g) 6= t(h), and it is extended to KC by linearity.The unit for this product is given by

∑x∈C0 1x, where 1x denotes the identity morphism at

the object x. Some straightfoward computations show that KC is a weak bialgebra.

Definition 1.3. A weak bialgebra (H,µ, η,∆, ε, S) is a weak Hopf algebra if there exists alinear map S : H → H, called the antipode, satisfying that for all x, y ∈ H

x1S(x(2)) = ε(1(1)x)1(2),

S(x(1))x(2) = 1(1)ε(x1(2)),

S(x(1))x(2)S(x(3)) = S(x)

Example 1.4. Let G be a grupoid over a finite set, that is, a category G with a finite setof objects G0 such that every arrow is an isomorphism. The weak bialgebra KG defined inExample 1.2 is a weak Hopf algebra in this case because the linear map S : KG → KGdefined by S(g) = g−1 for every g ∈ G1 is an antipode. We refer to KG as the groupoidweak Hopf algebra of the groupoid G.

Attached to any weak Hopf algebra H we have two relevant maps uL,uR : H → Hdefined by

uL(x) = ε(1(1)x)1(2), uR(x) = 1(1)ε(x1(2)),

Their images HL = uL(H), HR = uR(H) are separable K–subalgebras of H (see [3, Propo-sition 2.11]).

2. An adjoint pair

Let Gpd denote the category whose objects are the grupoids with finitely many objects,and with functors between them as morphisms. The construction of Example 1.4 give theobjects part of a functor K(−) : Gpd→WHAK , where WHAK is the category of weak Hopfalgebras over K. Morphisms in WHAK are multiplicative homomorphisms of coalgebrasf : H → H ′ between weak Hopf algebras H,H ′ such that fuL = uLf and fuR = uRf . Ourmain result is the following.

Theorem 2.1. The functor K(−) : Gpd → WHAK has a right adjoint G(−) : WHAK →Gpd.

The proof of Theorem 2.1 requires the construction of the groupoid G(H) for any weakHopf algebra H. The arrows of G(H) are defined as

G(H)1 = g ∈ H : ∆(g) = g ⊗ g, ε(g) = 1, S2(g) = g

DUALITY FOR GROUPOIDS AND WEAK HOPF ALGEBRAS1 3

The objects are then given by

G(H)0 = uL(G(H)1) = uR(G(H)1),

and the source and target maps, used to specify which morphisms of G(H) will be com-posable, are the restrictions to G(H)1 of uR and uL, respectively, to G(H)0. The resultingcategory G(H) is a groupoid, as the inverse of each g ∈ G(H)1 is given by S(g).

Theorem 2.1 allows to identify Gpd as a subcategory of WHAK , according to the followingcorollary.

Corollary 2.2. The category Gpd is equivalent to the full subcategory of WHAK of allcosemisimple pointed weak Hopf algebras.

When K is algebraically closed, the subcategory of WHAK equivalent to Gpd has analternative description:

Corollary 2.3. If K is algebraically closed, then the category Gpd is equivalent to the fullsubcategory of WHAK of all cosemisimple cocommutative weak Hopf algebras.

3. Duality

The dual H∗ = HomK(H,K) of each finite-dimensional Hopf algebra H is a weak Hopfalgebra (see [3]). A consequence of Corollary 2.3 is the following duality theorem for finitegrupoids.

Theorem 3.1. Let Gpdf denote the category of finite grupoids, and consider the categoryCWHf

K of finite-dimensional commutative semisimple weak Hopf algebras. If K is alge-braically closed, then these categories are anti-equivalent by the contravariant functors:

Φ : Gpdf → CWHfK , G 7→ (KG)∗

Ψ : CWHfK → Gpdf , H 7→ G(H∗)

We consider Theorem 3.1 as a starting point to study duality between grupoids andweak Hopf algebras from an algebraic point of view. To deal with non finite grupoids, thefirst step is to consider the finite dual Ho of a weak Hopf algebra H of any dimensionas the pertinent generalization of H∗. This should be helpful to study duality for algebraicgroupoids, for instance, and as a guideline to extend it to genuine quantum grupoids withoutstrict finiteness conditions. This is left for future work.

Aknowledgement: We thank to Gabriella Böhm for helpful discussions.

References

[1] E. Abe, Hopf Algebras, Cambridge University Press, (1980). ISBN 0 521 22240 0.[2] G. Böhm, K. Szlachányi, A Coassociative C∗-Quantum Group with Nonintegral Dimensions, Lett.

Math. Phys. 35, 437 (1996).[3] G. Böhm, F. Nill, K. Szlachányi, Weak Hopf Algebras I. Integral Theory and C∗-Structure. J. Algebra

221 (1999), 385-438.

4 JOSÉ GÓMEZ-TORRECILLAS AND ESPERANZA LÓPEZ-CENTELLA

[4] K. Szlachányi Weak Hopf Algebras, in Operator Algebras and Quantum Field Theory, eds.: S. Doplicher,R. Longo, J.E. Roberts, and L. Zsidó, International Press (1996).

Departamento de Álgebra, Universidad de Granada, Granada, E18071 SpainE-mail address: gomezj@ugr.es

Departamento de Álgebra, Universidad de Granada, Granada, E18071 SpainE-mail address: esperanza@ugr.es

PRIME FUZZY IDEALS OVER NONCOMMUTATIVE RINGS

OSCAR CORTADELLAS, F. J. LOBILLO, AND GABRIEL NAVARRO

Abstract. In this paper we introduce prime fuzzy ideals over a noncommutative ring.This notion of primeness is equivalent to level cuts being crisp prime ideals. It also gen-eralizes the one provided by Kumbhojkar and Bapat in 1993, which lacks this equivalencein a noncommutative setting. Semiprime fuzzy ideals over a noncommutative ring arealso defined and characterized as intersection of primes. This allows us to introduce thefuzzy prime radical and contribute to establish the basis of a Fuzzy Noncommutative RingTheory. This work is going to appear in [17].

Introduction

Since the well-known paper of Rosenfeld [19] dealing with fuzzy sets of a group, manyresearchers have focused on giving an algebraic structure to the universe space, definingalgebraic topics on a fuzzy environment and studying their properties. See, for instance,[7, 8, 10, 5, 13, 20, 26, 11, 18, 12, 1, 21].

Focusing on the structure of ring, the early paper of Liu [10] defining fuzzy ideals initiatedthe investigation of rings by expanding the class of ideals with these fuzzy objects. Someyears later, during the late eighties and nineties, many works of different authors werepublished in order to develop a Fuzzy Ring Theory. Most of these authors limit theirattention to commutative rings or, even, fail to mention that this requirement is necessary.Nevertheless, the commutativity condition becomes too restrictive when we realize thatnoncommutative rings can be found in a wide range of knowledge areas as Particle Physics,Quantum Field Theory, Gauge Theory, Cryptography or Coding Theory.

In this work we study the notion of primeness on fuzzy ideals clarifying relationshipsbetween various definitions appearing in the literature, and we propose a new definition ofprimeness over arbitrary rings. It is generally accepted that the concept of fuzzy primenessconsidered in [5, 6] is the most appropriate since, in commutative rings, it is equivalent tothe level cuts being crisp prime ideals. Nevertheless, when working over arbitrary rings, thisis no longer valid in previous papers. Our definition fills in this gap. It generalizes the onegiven in [5, 6] and verifies the aforementioned property.

Throughout R will be an arbitrary ring with unity. We recall from [10] that a (two-sided)fuzzy ideal over R is a fuzzy set I : R→ [0, 1] satisfying the following properties:

(1) I(x− y) ≥ I(x) ∧ I(y) for any x, y ∈ R, i.e., it is an additive fuzzy subgroup.(2) I(xy) ≥ I(x) ∨ I(y) for any x, y ∈ R.For any α ≤ I(0), we may consider the α-cut, Iα, as the subset Iα = x ∈ R such that I(x) ≥ α.

It is easy to prove that I is a fuzzy ideal if and only if Iα is an ideal of R for anyI(1) < α ≤ I(0). For any fuzzy set F , the fuzzy ideal generated by F will be the least

1

2 OSCAR CORTADELLAS, F. J. LOBILLO, AND GABRIEL NAVARRO

ideal containing F , i.e., the intersection of all fuzzy ideals I satisfying that F ≤ I. We shalldenote it by 〈F 〉.

1. A little survey about fuzzy primeness

Accordingly with crisp Ring Theory, the notion of prime ideal was one of the first conceptsunder consideration in its fuzzy version. In [20] the product of fuzzy ideals is defined as:

IJ(x) =∨

x=∑

i aibi

∧i

(I(ai) ∧ J(bi)),

and Zahedi proposes the following definition in [24].

Definition 1.1 (D1). A non-constant fuzzy ideal P : R → [0, 1] is said to be prime if,whenever IJ ≤ P for some fuzzy ideals I and J , it satisfies that I ≤ P or J ≤ P .

Malik and Mordeson [13], completing the work of Mukherjee and Sen [16], and Swamyand Swamy [20] for L-fuzzy ideals, give a nice characterization of all D1-prime fuzzy ideals.

Lemma 1.2. [13, 20] A fuzzy ideal P : R→ [0, 1] is D1-prime if and only if P has the form

P (x) =

1 if x ∈ Q,t otherwise,

where Q is a crisp prime ideal of R and 0 ≤ t < 1.

Albeit the former result makes fuzzy D1-primeness clear and transparent, the reader mayfigure out a subtle problem about its usefulness: there is no plenty more D1-prime fuzzyideals than crisp ones. Hence, the additional information about the ring provided by D1-prime fuzzy ideals is quite reduced. Additionally, as pointed out in [4], the notion seems tobe too strong since there are fuzzy ideals whose level cuts are prime, despite of they are notD1-prime.

Although the first inconvenient becomes much more difficult to resolve, an evident solutionto the second one is to define primeness as the property that it should verify.

Definition 1.3 (D2). A non-constant fuzzy ideal P : R→ [0, 1] is said to be prime if Pα isprime for any P (0) ≥ α > P (1).

As the reader may think in reading Definition 1.3, it only translates our problem sincenow we need a characterization which makes operational D2-primeness.

In [5], Kumbhojkar and Bapat deal with prime fuzzy ideals from an element-like per-spective and they state primeness looking forward similarities with standard commutativealgebra.

Definition 1.4 (D3). A non-constant fuzzy ideal P : R → [0, 1] is said to be prime if, forany x, y ∈ R, whenever P (xy) = P (0), then P (x) = P (0) or P (y) = P (0).

D2-primeness implies D3-primeness, although the converse does not hold. For this reason,the authors give a stronger notion which they call strongly primeness.

Definition 1.5 (D4). A non-constant fuzzy ideal P : R → [0, 1] is said to be prime if, forany x, y ∈ R, P (xy) = P (x) or P (xy) = P (y).

PRIME FUZZY IDEALS OVER NONCOMMUTATIVE RINGS 3

Obviously, D3-primeness is weaker than D4-primeness, but we are interested in the rela-tion between D2-primeness and D4-primeness.

Lemma 1.6. A fuzzy ideal P is D4-prime if and only if each level cut Pα is completelyprime for all P (0) ≥ α > P (1).

Hence, D4-primeness implies D2-primeness. Although the converse result does not holdfor arbitrary rings.

In order to obtain an element-like definition, in [24], Zahedi develops two notions involvingthe so-called singletons. We recall the reader that, given an element x ∈ R and t ∈ (0, 1],the singleton xt is the fuzzy set defined by xt(x) = t and zero otherwise.

Definition 1.7 (D0). A non-constant fuzzy ideal P : R → [0, 1] is said to be prime if,whenever xtys ≤ P for some singletons xt and ys, then xt ≤ P or ys ≤ P . This is calledcompletely prime by Zahedi in [24, Definition 2.7].

Definition 1.8 (D0′). A non-constant fuzzy ideal P : R → [0, 1] is said to be prime if,whenever 〈xt〉〈ys〉 ≤ P for some singletons xt and ys, then xt ≤ P or ys ≤ P .

In [24, Theorem 4.9], D0′-primeness is proven to be equivalent to D1-primeness. In [6,Theorem 3.5], D0-primeness is proven to be equivalent to D1-primeness, see also [15, Theo-rem 2.6]. Nevertheless, the proof is no longer valid when R is an arbitrary non-commutativering. As the reader may see, in a general setting, there is no characterization of primenessconsistent with the level cuts.

2. Fuzzy primeness over noncommutative rings

Definition 2.1. Let R be an arbitrary ring with unity. A non-constant fuzzy ideal P : R→[0, 1] is said to be prime if, for any x, y ∈ R,

∧P (xRy) = P (x) ∨ P (y).

Proposition 2.2. Let R be an arbitrary ring with unity and P : R→ [0, 1] be a non-constantfuzzy ideal of R. The following conditions are equivalent:

(1) P is prime.(2) Pα is prime for all P (0) ≥ α > P (1).

Moreover, if R is commutative, any of these statements is equivalent to P being D4-prime.

Definition 2.3. Let R be an arbitrary ring with unity. A non-constant fuzzy ideal P : R→[0, 1] is said to be semiprime if

∧P (xRx) = P (x) for all x ∈ R.

Proposition 2.4. Let R be an arbitrary ring with unity and P : R→ [0, 1] be a non-constantfuzzy ideal of R. The following conditions are equivalent:

(1) P is semiprime.(2) Pα is semiprime for all P (0) ≥ α > P (1).

Theorem 2.5. A fuzzy ideal is semiprime if and only if it is the intersection of prime fuzzyideals.

Corollary 2.6. Let R be a ring with unity and I be a non-constant fuzzy ideal over R. Thefollowing fuzzy ideals coincide:

(1) The intersection F1 of all semiprime fuzzy ideals containing I.

4 OSCAR CORTADELLAS, F. J. LOBILLO, AND GABRIEL NAVARRO

(2) The intersection F2 of all prime fuzzy ideals containing I.(3) The fuzzy ideal F3 given by F3(x) =

∨t ∈ [0, 1] such that x ∈ Rad(It).

For any non-constant fuzzy ideal I, we define the fuzzy prime radical of I, FRad(I), asany of the fuzzy ideals described in Corollary 2.6. Hence, the following result is immediate.

Corollary 2.7. P is a semiprime fuzzy ideal if and only if FRad(P ) = P .

References

[1] W. Chen, Fuzzy subcoalgebras and duality, Bull. Malays. Math. Sci. Soc. (2) 32 (2009) 283–294.[2] V.N. Dixit, R. Kumar, N. Ajmal, On fuzzy rings, Fuzzy Sets Syst. 49 (1992) 205–213.[3] K.C. Gupta, M.K. Kantroo, The nil radical of a fuzzy ideal, Fuzzy Sets Syst. 59 (1993) 87–93.[4] R. Kumar, Certain fuzzy ideals of rings redefined, Fuzzy Sets Syst. 46 (1992) 251–260.[5] H.V. Kumbhojkar, M.S. Bapat, Not-so-fuzzy fuzzy ideals, Fuzzy Sets Syst. 37 (1990) 237–243.[6] H.V. Kumbhojkar, M.S. Bapat, On prime and primary fuzzy ideals and their radicals, Fuzzy Sets Syst.

53 (1993) 203–216.[7] N. Kuroki, Fuzzy semiprime ideals in semigroups, Fuzzy Sets Syst. 8 (1982) 71–79.[8] N. Kuroki, On fuzzy semigroups, Inform. Sci. 53 (1991) 203–236.[9] J. Levitzki, Prime ideals and the lower radical, Amer. J. Math. 73 (1951) 25–29.

[10] W.J. Liu, Fuzzy invariant subgroups and fuzzy ideals, Fuzzy Sets Syst. 8 (1982) 133–139.[11] S.R. López-Permouth, D.S. Malik, On categories of fuzzy modules, Inform. Sci. 52 (1990) 211–220.[12] D.S. Malik, J.N. Mordeson, Fuzzy subfields, Fuzzy Sets Syst. 37 (1990) 383–388.[13] D.S. Malik, J.N. Mordeson, Fuzzy prime ideals of a ring, Fuzzy Sets Syst. 37 (1990) 93–98.[14] D.S. Malik, J.N. Mordeson, Fuzzy maximal, radical, and primary ideals of a ring, Inform. Sci. 53 (1991)

237–250.[15] D.S. Malik, J.N. Mordeson, Radicals of fuzzy ideals, Inform. Sci. 65 (1992) 239–252.[16] T.K. Mukherjee, M.K. Sen, On fuzzy ideals of a ring. I, Fuzzy Sets Syst. 21 (1987) 99–104.[17] G. Navarro, O. Cortadellas and F.J. Lobillo, Prime fuzzy ideals over noncommutative rings, Fuzzy Sets

and Systems, in press, 2012. doi:10.1016/j.fss.2011.11.002[18] F.Z. Pan, Fuzzy finitely generated modules, Fuzzy Sets Syst. 21 (1987) 105–113.[19] A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl. 35 (1971) 512–517.[20] U.M. Swamy, K.L.N. Swamy, Fuzzy prime ideals of rings, J. Math. Anal. Appl. 134 (1988) 94–103.[21] S.E.-B. Yehia, Fuzzy ideals and fuzzy subalgebras of Lie algebras, Fuzzy Sets Syst. 80 (1996) 237–244.[22] L.A. Zadeh, Fuzzy sets, Inform. Control 8 (1965) 338–353.[23] L.A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning. I, Inform.

Sci. 8 (1975) 199–249.[24] M.M. Zahedi, A characterization of L-fuzzy prime ideals, Fuzzy Sets Syst. 44 (1991) 147–160.[25] M.M. Zahedi, A note on L-fuzzy primary and semiprime ideals, Fuzzy Sets Syst. 51 (1992) 243–247.[26] Y. Zhang, Prime L-fuzzy ideals and primary L-fuzzy ideals, Fuzzy Sets Syst. 27 (1988) 345–350.

Department of Algebra, Faculty of Sciences, University of Granada, Avda. Fuentenuevas/n, E-18071, Granada, Spain

E-mail address: ocortad@ugr.es

Department of Algebra, ETSIIT, University of Granada, c/ Periodista Daniel SaucedoAranda s/n, E-18071, Granada, Spain

E-mail address: jlobillo@ugr.es

Department of Computer Sciences and AI, University of Granada, C/ El Greco s/n, E-51002, Ceuta, Spain

E-mail address: gnavarro@ugr.es

ALGEBRAS POLIÁDICAS. FORMULACIÓN ALGEBRAICA DEL

TEOREMA DE COMPLETITUD DE GÖDEL

DOMINGO MARTÍNEZ VERDÚ

Abstract. Between 1954 and 1959, P.R. Halmos published, in various mathematicaljournals, a number of articles on algebraic logic developing the basics of monadic andpolyadic Boolean algebras in order to make a algebraization of first-order logic. Specifically,the most significant result of polyadic algebras is that they provide an purely algebraicformulation of Gödel’s Completeness Theorem. The main idea is based on the equivalenceof the semisimplicity of polyadics algebras and the esemantically completeness. Likewise,and in the same line, Halmos also started, but left unfinished, the algebraic version ofGödel’s Incompleteness theorems. Like Hamos pointed, these works were perhaps for himthe best scientific work done over all his years.

Resumen

Entre 1954 y 1959, P.R. Halmos publicó, en diferentes revistas matemáticas, una seriede artículos sobre lógica algebraica, en los que definió, estudió y desarrolló los sistemasalgebraicos de las álgebras Booleanas monádicas y poliádicas con el objetivo de realizar unaalgebraización de la lógica de primer orden. En concreto, el resultado más significativo de lasálgebras poliádicas ha sido la formulación puramente algebraica del Teorema de Completitudde Gödel. La idea principal se basa en la equivalencia entre la semisimplicidad de las álgebraspoliádicas y la completitud semántica. Así mismo, y en la misma línea, Halmos tambiéninició, aunque dejo inconclusa, la versión algebraica de los Teoremas de Incompletitud deGödel. Como el mismo Halmos apuntó, estos trabajos fueron para él quizá la mejor laborcientífica que realizó a lo largo de sus años.

Bibliográfica básica: relación reducida

References

[1] P.R. Halmos, Algebraic logic, Chelsea publishing Company, (1962), New York. (Papers collected to-gether appeared 1954-1959)

[2] P.R. Halmos, Algebraic logic, I, Monadic Boolean algebras, Compositio Mathematica, vol. 12 (1955),217-249.

[3] P.R. Halmos, Algebraic logic, II, Homogeneous locally finite polyadic Boolean algebras of infinite degree,Fundamenta Mathematicae, vol. 43 (1956), 255-325.

Domingo Martínez Verdú (Universidad de Murcia)E-mail address: dverdu@ono.com

Pedro Antonio Guil Asensio (Director del trabajo) (Universidad de Murcia)E-mail address: paguil@um.es

1

GENERALIZED TILTING THEORY

PEDRO NICOLÁS AND MANUEL SAORÍN

Abstract. We study the relationship between (generalized) tilting modules and triangleequivalences between (quotients of) derived categories.

Introduction

The fundamental theorem of Morita theory gives a list of necessary and sucient con-ditions for an equivalence between two module categories to exist (see for example [6]).Equivalences between derived categories of rings generalize Morita theory, and in this sense,derived Morita theory is a generalization of Morita theory. The fundamental theorem ofderived Morita theory gives a list of necessary and sucient conditions for a triangle equiv-alence between two derived categories of rings to exist (see [4, 5]). This is at the basis ofclassical tilting theory. Indeed, if A is a ring, a right A-module T is classical tilting if andonly if the adjoint pair of derived functors

DA

RHomA(T,?)

DB

?⊗LBT

OO

induces a triangle equivalence between the derived category DA of A and the derived cate-gory DB of its endomorphisms ring B = EndA(T ). In particular, a classical tilting moduleis nitely generated. The notion of classical tilting module has evolved to the notion of gen-eralized tilting module, which includes other interesting modules which are not necessarilynitely generated. Recently, Bazzoni-Mantese-Tonolo [2] proved that a good (generalized)tilting A-module T induces a triangle equivalence

DA

RHomA(T,?)

DB/ ker(?⊗LB T)

?⊗LBT

OO

between the derived category of A and the quotient of the derived category of B = EndA(T )by the kernel of the derived tensor functor ? ⊗L

B T . This equivalence exists if and only if(see [3]) the derived Hom-functor

RHomA(T, ?) : DA → DB

is fully faithful. A detailed inspection reveals that if T is a good tilting module, this functoris not only fully faithful, but it also preserves compact objects.

Date: February 18, 2012.1

2 PEDRO NICOLÁS AND MANUEL SAORÍN

We summarize the dierent notions of tilting modules we have been refering:

Denition 0.1. Let A be an ordinary algebra, and let T be a right A-module. Considerthe following conditions:

a) There is an exact sequence 0 → Pn → Pn−1 → · · · → P1 → P0 → T → 0 in Mod Awhere the modules Pi are projective.

a') There is an exact sequence 0 → Pn → Pn−1 → · · · → P1 → P0 → T → 0 in Mod Awhere the modules Pi are nitely generated projective.

b) There is an exact sequence 0 → A → T0 → T1 → · · · → Tm → 0 in Mod A where themodules Ti are direct summands of coproducts of copies of T .

b') There is an exact sequence 0 → A → T0 → T1 → · · · → Tm → 0 in Mod A where themodules Ti are direct summands of nite coproducts of copies of T .

c) ExtiA(T, T (α)) = 0 for each i > 0 and each cardinal α.

We say that T is a generalized n-tilting module if it satises a), b) and c). We say that itis a classical n-tilting module if it satises a'), b') and c) (for α = 1). We say that T is agood n-tilting module if it satises a), b') and c). We say that a module is generalized tilting(resp. classical tilting, good tilting) if it is generalized n-tilting (resp. classical n-tilting,good n-tilting) for some n ≥ 1.

However, it is was still unclear which is the precise relationship between generalized tiltingmodules and triangle equivalences between (quotients of) derived categories.

Let A be a dierential graded(=dg) algebra, T an arbitrary right dg A-module and letB = REndA(T ). In this work we contribute to the enlightenment of this relationship with:

1) A theorem which characterizes those A-modules T such that RHomA(T, ?) : DA →DB is fully faithful and preserves compact objects ( 1). For this we use a derivedversion of the classical notion of faithfully balanced bimodule.

2) A theorem which studies the link between generalized tilting modules and the mod-ules appearing in the theorem mentioned above ( 2).

1. Special localizations

Let thickDA(T ) be the smallest full subcategory of DA containing T and closed undershifts, extensions and direct summands.

Theorem 1.1. The following statements are equivalent:

1) RHomA(T, ?) : DA → DB is fully faithful and preserves compact objects.2) A belongs to thickDA(T ).3) The natural map A → RHomBop(T, T ) is a quasi-isomorphism and T is compact

regarded as an object of D(Bop).

Remark 1.2. a) Notice that a good tilting module T satises 2), and so part 1) willalways be true. Hence, the former theorem gives an alternative proof of the mainresult of [2].

b) The theorem is a particular case of a more general result stated in terms of dgcategories and arbitrary bimodules.

GENERALIZED TILTING THEORY 3

2. Tilting

Let now A be an ordinary algebra, T a right A-module and EndA(T ) = B.

Proposition 2.1. If ExtiA(T, T ) = 0 for each i > 0, the following conditions are equivalent:

1) A ∈ thickDA(T ),2) A admits a coresolution

0 → A → T0 → T1 → · · · → Tn → 0

in Mod A, where each Ti is a direct summand of a nite coproduct of copies of T .

Corollary 2.2. If a generalized tilting module T is such that RHomA(T, ?) : DA → DB isfully faithful and preserves compact objects, then T is a good tilting module.

Theorem 2.3. The following assertions hold:

1) There are 1-tilting modules T such that RHomA(T, ?) : DA → DB is not fullyfaithful.

2) There are right A-modules T satisfying the following conditions:a) pdA T ≤ 1,b) the functor RHomA(T, ?) : DA → DB is fully faithful and preserves compact

objects,c) T does not satisfy condition c) of Denition 0.1, and so it is not a tilting module.

References

[1] F. K. Anderson, K. R. Fuller, Rings and Categories of Modules, Springer-Verlag Graduate Texts inMathematics, (1992).

[2] S. Bazzoni, F. Mantese and A. Tonolo, Derived Equivalence induced by n-tilting modules,arXiv:0905.3696v1 [math.RA].

[3] P. Gabriel, M. Zisman, Calculus of Factions and Homotopy Theory, Springer-Verlag, (1967).[4] B. Keller, Deriving DG categories, Ann. Scient. Ec. Norm. Sup., 27(1) (1994), 63102.[5] B. Keller, On the construction of triangle equivalences, in: Derived equivalences for group rings, Springer

Lecture Notes in Mathematics, (1998).[6] B. Stenström, Rings of quotients, Springer-Verlag, (1975).

Universidad de Murcia, Departamendo de Didáctica de las CC. Matemáticas y SocialesE-mail address: pedronz@um.es

Universidad de Murcia, Departamento de MatemáticasE-mail address: saorinc@um.es

A LAZARD-LIKE THEOREM FOR QUASI-COHERENT SHEAVES

SERGIO ESTRADA, PEDRO A. GUIL ASENSIO, AND SINEM ODABASI

Abstract. We study ltration of quasi-coherent sheaves. We prove a version of Kaplan-sky Theorem for quasi-coherent sheaves, by using Drinfeld's notion and the Hill Lemma.We also show a Lazard-like theorem for at quasi-coherent sheaves for quasi-compact andsemi-separated schemes which satisfy the resolution property.

Introduction

Let X = SpecR be an ane scheme for a commutative ring R with unity. It is known thatthe category of all quasi-coherent sheaves on X is equivalent to R-Mod. In this equivalence,nite dimensional vector bundles on X correspond to nitely generated projective modules.Then a (classical) vector bundle on an arbitrary scheme X corresponds to a quasicoherentsheaf F such that, for each ane open subset U = SpecR, the corresponding R-module ofsections Γ(U,F) is nitely generated and free.

[Drinfeld, 2006] asks the following key problem: `Is there a reasonable notion of notnecessarily nite dimensional vector bundles on a scheme?'. In the same paper he proposesseveral possible answers to this question. Each one of these involves dierent classes ofmodules. In this work, we focus on the projective R-modules and at Mittag-Leer for acommutative ring R with unity. Then we center on the category of quasi-coherent sheavesover some special projective schemes and the several `new' notions of (innite dimensional)vector bundles attained to that class as proposed by Drinfeld. We prove structural resultsrelative to the dierent generalization of vector bundles in terms of certain ltrations oflocally countably generated quasi-coherent sheaves. In the case in which the vector bundlesare built from the class of projective R-modules, our structural theorem yields a versionof Kaplansky's Theorem for innite dimensional vector bundles on these special projectiveschemes.

1. Qco(X) As a Category of Representation

The aim of this chapter is to give a new and simpler category that is equivalent toQco(X).So, it allow us to work in Qco(X) more easily.

Let Q be a quiver. A representation R of the quiver Q in the category of rings meansthat for each vertex v ∈ V we have a ring R(v) and a ring homomorphism

R(a) : R(v) −→ R(w),

for each edge a : v → w. An R-module M is given by an R(v)-moduleM(v), for each vertexv ∈ V , and an R(v)-linear morphism. The category of quasi-coherent R-modules for a xedquiver Q and a xed representation R of the quiver Q is dened as the full subcategory

1

2 SERGIO ESTRADA, PEDRO A. GUIL ASENSIO, AND SINEM ODABASI

of the category R-Mod that contains all quasi-coherent R-modules. We will denote it byRQco-Mod.

Theorem 1.1. Let X be a scheme. For a suitable quiver Q, the category of quasi-coherent

R-modules for this quiver Q is equivalent to Qco(X).

Proof. Consider the category of quasi-coherent sheaves on a scheme (X,OX), denoted byQco(X). By the denition of a scheme, the scheme X has a family B of ane open subsetswhich is a base for X such that this family uniquely determines the scheme (X,OX). Now,dene a quiver Q having the family B as the set of vertices, and an edge between two aneopen subsets U, V ∈ B as the only one arrow U → V provided that V ( U . Fix thisquiver. Take the representation R as R(U) = OX(U) for each U ∈ B and the restrictionmap ρUV : OX(U)→ OX(V ) for the edge U → V . Then the functor

Φ : Qco(X) 7−→ RQco-Mod,

which was dened by above argument, is well-dened.

2. Filtration in Qco(PnR)

It is known that there is a bijection between the class of the vector bundles in the sense ofclassical algebraic geometry and the class of the locally free coherent OX -modules of niterank. So, in Sheaf Theory, the denition of vector bundle is taken as locally free coherentOX -module of nite rank. But in our study, we will drop the conditions niteness andfreeness. This leads to Drinfeld's denition of innite-dimensional vector bundles.

Denition 2.1. [Drinfeld, 2006, Section 2] Let (X,OX) be a scheme. A quasi-coherent OX -module F is said to be a vector bundle (in the sense [Drinfeld, 2006]) if F(U) is a projectiveOX(U)-module for every ane open subset U of X.

Denition 2.2. [Drinfeld, 2006, Section 2] Let (X,OX) be a scheme. A quasi-coherentOX -module F is said to be a locally at Mittag-Leer if F(U) is a at Mittag-LeerOX(U)-module for every ane open subset U of X.

In Drinfeld's paper, it is stated that the notion in 2.2 is a local property. That is,conversely, if M is an R-module such thatM(u) is a projective R(u)-module for each vertexu, then there exists a unique vector bundleM on the scheme X. As in the case of projectivemodules, the property of the at Mittag-Leer module is also local.

Theorem 2.3. Every vector bundle on PnR is a ltration of locally countably generated vector

bundles.

Theorem 2.4. Every locally at Mittag-Lefer quasi-coherent sheaf on X is a direct union

of locally countably generated vector bundles.

Theorem 2.5. Let X be a scheme having enough locally countably generated vector bundles.

Let F be a at quasi-coherent sheaf on X. Then F = lim−→Fi, where Fi is locally countably

generated and at with V dimFi ≤ 1.

A LAZARD-LIKE THEOREM FOR QUASI-COHERENT SHEAVES 3

References

[Drinfeld, 2006] Drinfeld, V. (2006). Innite-dimensional vector bundles in algebraic geometry. P. Etingof,R. Vilademir, & I.M. Singer, (Ed), The unity of mathematics(263-304). Boston: Birkäuser.

[Eklof, 1977] Eklof, P.C. (1977). Homological algebra and set theory. Transactions of the American Mathe-

matical Society, 227, 207-225.[Enochs & Estrada, 2005] Enochs, E., & Estrada, S. (2005). Relative homological algebra in the category of

quasi-coherent sheaves. Advances in Mathematics, 194, 284-295.[Enochs, Estrada, García Rozas & Oyonarte, 2004a] Enochs, E., Estrada, S., García Rozas, J.R., & Oy-

onarte, L. (2004a). Flat covers in the category of quasi-coherent sheaves over the projective Line.Communications in Algebra, 32, 1497-1508.

[Enochs, Estrada, García Rozas & Oyonarte, 2004b] Enochs, E., Estrada, S., García Rozas, J.R., & Oy-onarte, L. (2004b). Flat and cotorsion quasi-coherent sheaves, Applications. Algebras and Representation

Theory, 7, 441-456.[Enochs, Estrada, García Rozas & Oyonarte, 2003] Enochs, E., Estrada, S., García Rozas, J.R., & Oy-

onarte, L. (2003). Generalized quasi-coherent sheaves. Journal of Algebra and Its Applications, 2 (1),1-21.

[Enochs, Estrada & Torrecillas, 2006] Enochs, E., Estrada, S., & Torrecillas, B. (2006). An elementary proofof Grothendieck's Theorem. P. Goeters, & O. M. G. Jenda, (Ed), Abelian groups, rings, modules, and

homological algebra(151-157). Boca Raton: Chapman & Hall/CRC[Estrada, Asensio, Prest & Trlifaj, 2009] Estrada, S., Asensio, P.G., Prest, M., & Trlifaj, J. (2009). Model

category structures arising from Drinfeld vector bundles. Preprint. Retrieved June 10, 2010, fromhttp://arxiv.org/archive/math, arXiv:0906.5213v1

[Estrada, Asensio, Odabasi, 2011] Estrada, S., Asensio, P.G., Odabasi, S., (2011). A Lazard-like theorem for

quasi-coherent sheaves Preprint. from http://arxiv.org/archive/math, arXiv:1109.0439[Göbel & Trlifaj, 2006] Göbel, R., & Trlifaj, J. (2006). Approximations and endomorphism algebras of mod-

ules. Berlin: Walter de Gruyter.[Grothendieck, 1957] Grothendieck, A. (1957). Sur la classication des bres holomorphes sur la sphere de

Riemann. American Journal of Mathematics, 79, 121-138.[Serre, 1955] Serre, J.P. (1955). Faisceaux algebriques coherents. Annals of Mathematics, 61, 197-278.[Serre, 1958] Serre, J.P. (1958). Modules projectifs et espaces brés à bre vectorielle. Séminaire P. Dubreil,

M.-L. Dubreil-Jacotin et C. Pisot 11, 1957-1958, Paris: Exposè 23.

Murcia UniversityE-mail address: sestrada@um.es

Murcia UniversityE-mail address: paguil@um.es

Murcia UniversityE-mail address: sinem.odabasi@um.es

DIFFERENTIAL RESULTANT FORMULAS FOR LINEAR

OD-POLYNOMIALS

SONIA L. RUEDA

Abstract. The sparse dierential resultant ∂Res(P) of an overdetermined system P ofgeneric nonhomogeneous ordinary dierential polynomials, was formally dened recentlyby Li, Gao and Yuan (2011). In this note, a dierential resultant formula ∂FRes(P)is dened for linear systems. Under some conditions on the supports of the dierentialoperators determining P, the formula is proved to be always nonzero and, up to a constant,equal to ∂Res(P).

Introduction

The implicitization problem of unirational algebraic varieties has been widely studied,and the results on the computation of the implicit equation of a system of algebraic rationalparametric equations by algebraic resultants are well known. The generalization of theseresults to the dierential (and more generally noncommutative) case is a eld of research atan initial stage of development, where many interesting problems arise.

Let P be an system of n generic sparse nonhomogeneous ordinary dierential polynomialsin n−1 dierential variables. It would be useful to represent the sparse dierential resultant∂Res(P), dened in [4], as the quotient of two determinants, as done for the algebraic casein [3]. In the dierential case, so called Macaulay style formulas do not exist, even in thesimplest situation. The matrices used in the algebraic case to dene Macaulay style formulas[3], are coecient matrices of sets of polynomials obtained by multiplying the original onesby appropriate sets of monomials, [1]. In the dierential case, in addition, derivatives ofthe original polynomials should be considered. The dierential resultant formula denedby Carrà-Ferro in [2], is the algebraic resultant of Macaulay [5], of a set of derivatives ofthe ordinary dierential polynomials in P. Already for linear dierential polynomials theseformulas vanish often, giving no information about the dierential resultant ∂Res(P). Thelinear case can be seen as a previous stage to get ready to approach the nonlinear case,considering only the problem of taking the appropriate set of derivatives of the elements inP for the moment.

Let us assume that P is a system of linear dierential polynomials. In [8], the linearcomplete dierential resultant ∂CRes(P) was dened, as an improvement, in the linear case(non necessarily generic), of the dierential resultant formula given by Carrà-Ferro. Still,∂CRes(P) is the determinant of a matrix having zero columns in many cases. The lineardierential polynomials in P can be described via dierential operators. We use appropriatebounds of the supports of those dierential operators to decide on a convenient set ps(P) ofderivatives of P, such that its coecient matrix M(P) is squared and has no zero columns.

1

2 SONIA L. RUEDA

Furthermore, we can guarantee that the linear sparse dierential resultant ∂Res(P) canalways be computed (up to a constant) as the determinant of a matrix M(P∗), for a superessential subsystem P∗ of P, as dened in Section 2. A key fact is that not every polynomialin P is involved in the computation of ∂Res(P), only those in a super essential subsystemP∗ of P are, and P∗ is proved to exist in all cases. An extended version of the resultspresented can be found in [7].

1. Sparse linear differential resultant

Let us suppose that the eld Q of rational numbers is a eld of constants of a derivation ∂.Let us consider the set U = u1, . . . , un−1 of dierential indeterminates over Q. By N0 wemean the natural numbers including 0. For k ∈ N0, we denote by uj,k the k-th derivative ofuj and for uj,0 we simply write uj . We denote by U the set of derivatives of the elementsof U .

For i = 1, . . . , n and j = 1, . . . , n− 1, let us consider subsets Si,j of Z to be the supports ofgeneric dierential operators

Gi,j :=

∑k∈Si,j

ci,j,k∂k ,Si,j = ∅,

0 ,Si,j = ∅.

Let us consider the set of dierential indeterminates over Q

C = c1, . . . , cn and C := ∪ni=1 ∪n−1

j=1 ci,j,k | k ∈ Si,j.

Let K = Q⟨C⟩, a dierential eld extension of Q, and D = KC, a dierential domain.Consider the set P = F1, . . . ,Fn of generic sparse linear dierential polynomials in DU asfollows

Fi := ci −n−1∑j=1

Gi,j(uj) = ci −n−1∑j=1

∑k∈Si,j

ci,j,kuj,k, i = 1, . . . , n.

Let xi,j , i = 1, . . . , n, j = 1, . . . , n−1 be algebraic indeterminates over Q. Let X(P) = (Xi,j)be the n× (n− 1) matrix, such that

Xi,j :=

xi,j ,Gi,j = 0,0 ,Gi,j = 0.

The system P is said to be dierentially essential if rank(X(P)) = n− 1.

Let [P] be the dierential ideal generated byP in DU. By [4], Corollary 3.4, the dimensionof the elimination ideal

ID(P) = [P] ∩ Dis n − 1 if and only if P is a dierentially essential system. In such case, ID(P) = sat(R),the saturation ideal of a dierential polynomial R in D, we can assume that R ∈ QC,Cis irreducible. By [4], Denition 3.5, R is the sparse dierential resultant of P, we will denoteit by ∂Res(P).

DIFFERENTIAL RESULTANT FORMULAS FOR LINEAR OD-POLYNOMIALS 3

2. Sparse differential resultant formula for supper essential systems

Let us assume that the order of Fi is oi ≥ 0, i = 1, . . . , n. We dene positive integers, toconstruct convenient intervals bounding the supports of the dierential operators Gi,j . Letldeg(Gi,j) := min Si,j and deg(Gi,j) := max Si,j . For j = 1, . . . , n− 1,

γj(P) := minoi − deg(Gi,j) | Gi,j = 0, i = 1, . . . , n,γj(P) := minldeg(Gi,j) | Gi,j = 0, i = 1, . . . , n,

γj(P) := γj(P) + γj(P).

Therefore, for Gi,j = 0 the next set of lattice points contains Si,j ,

Ii,j(P) := [γj(P), oi − γj(P)] ∩ Z.

Finally,

γ(P) :=

n−1∑j=1

γj(P).

We denote by Xi(P), i = 1, . . . , n, the submatrix of X(P) obtained by removing its ithrow. The system P is said to be supper essential if det(Xi(P)) = 0, i = 1, . . . , n. GivenN :=

∑ni=1 oi, let

Li := N − oi − γ(P), i = 1, . . . , n.

If P is super essential then Li ≥ 0, i = 1, . . . , n and we can construct the set

ps(P) := ∂kFi | k ∈ [0, Li] ∩ Z, i = 1, . . . , n,containing L :=

∑ni=1(Li + 1) dierential polynomials, in the set V of L − 1 dierential

indeterminates

V := uj,k | k ∈ [γj(P), N − γj(P)− γ(P)] ∩ Z, j = 1, . . . , n− 1.

The coecient matrix M(P) of the dierential polynomials in ps(P) as polynomials inD[V] is an L×L matrix. We dene a linear dierential resultant formula for P, denoted by∂FRes(P), and equal to:

∂FRes(P) := det(M(P)).

3. Main results

The implicitization of linear DPPEs (dierential polynomial parametric equations) bydierential resultant formulas was studied in [8] and [6]. In [7], some of the results in [6] areextended and used to obtain the next conclusions.

Given a dierentially essential system P, ID(P) = [∂Res(P)]D, the dierential ideal gen-erated by ∂Res(P) in D. Furthermore, R = ∂Res(P) is a linear dierential polynomialverifying:

(1) R =∑n

i=1 Li(ci), Li ∈ K[∂] and a greatest common left divisor of L1, . . . ,Ln belongsto K, that is R is ID-primitive.

(2) R belongs to (ps(P)) ∩ D, where (ps(P)) is the algebraic ideal generated by ps(P)in D[V].

4 SONIA L. RUEDA

(3) The highest positive integer c such that ∂cR ∈ (ps(P) is

c = |ps(P)| − 1− rank(M(V)),where M(V) is the submatrix of M(P) whose L − 1 columns are indexed by theelements in V.

Using these properties we can prove the next result.

Theorem 3.1. Let P be a system of generic sparse linear dierential polynomials. If P is

super essential then ∂FRes(P) = 0.

Furthermore, if P is not super essential, we can prove the existence of a super essentialsubsystem P∗ of P and provide a computation method, [7], Section 4. Furthermore, P isdierentially essential if and only it has a unique super essential subsytem.

Theorem 3.2. Let us consider a dierentially essential system P, of generic sparse linear

dierential polynomials, and the super essential subsystem P∗ of P. There exists a nonzero

constant α ∈ K such that ∂Res(P) = α∂FRes(P∗).

The previous results, allow us to give a bound of the order of ∂Res(P) in the dierentialindeterminates C. Namely, given I∗ := i | Fi ∈ P∗ and i ∈ 1, . . . , n

ord(∂Res(P), ci) = −1 if i /∈ I∗,

ord(∂Res(P), ci) = N∗ − oi − γ(P∗) if i ∈ I∗,

with N∗ =∑

i∈I∗ oi and equality holds for some i ∈ I∗.

References

[1] J. Canny, I. Emiris, A subdivision based algorithm for the sparse resultant, J. ACM 47 (2000), 417-451.[2] G. Carrà-Ferro, A resultant theory for ordinary algebraic dierential equations. Lecture Notes in Com-

puter Science, 1255. Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. Proceedings(1997).

[3] D'Andrea, C., Macaulay Style Formulas for Sparse Resultants. Trans. of AMS, 354(7) (2002), 2595-2629.[4] W. Li, X.S. Gao, C.M. Yuan, Sparse Dierential Resultant, Proceedings of the ISSAC'2011 (2011),

225-232.[5] F.S. Macaulay, The Algebraic Theory of Modular Systems. Proc. Cambridge Univ. Press., Cambridge,

(1916).[6] S.L. Rueda, A perturbed dierential resultant based implicitization algorithm for linear DPPEs. Journal

of Symbolic Computation, 46 (2011), 977-996.[7] S.L. Rueda, Linear sparse dierential resultant formulas. arXiv:1112.3921 (2011).[8] S.L. Rueda and J.F. Sendra, Linear complete dierential resultants and the implicitization of linear

DPPEs. Journal of Symbolic Computation, 45 (2010), 324-341.

Dpto de Matemática Aplicada, E.T.S. Arquitectura, Universidad Politécnica de Madrid. Memberof the Research Group ASYNACS. Partially supported by the Ministerio de Ciencia e Innovación"under the Project MTM2008-04699-C03-01 and by the "Ministerio de Economía y Competitividad"under the project MTM2011-25816-C02-01.

E-mail address: sonialuisa.rueda@upm.es

Weight modules over split Lie algebras

Antonio J. Calderon Martın ∗

Departamento de Matematicas.Universidad de Cadiz. 11510 Puerto Real, Cadiz, Spain.

e-mail: ajesus.calderon@uca.es

Jose M. Sanchez-Delgado

Departamento de Algebra, Geometrıa y Topologıa.Universidad de Malaga. 29080 Malaga, Spain.

e-mail: txema.sanchez@uma.es

Abstract

We study the structure of arbitrary weight modules V , (with no re-strictions neither on the dimension nor on the base field), over split Liealgebras L. We show that if L is perfect and V satisfies LV = V andZ(V ) = 0, then

L =⊕i∈I

Ii and V =⊕j∈J

Vj

with any Ii an ideal of L satisfying [Ii, Ik] = 0 if i 6= k, and any Vj a(weight) submodule of V in such a way that for any j ∈ J there exists aunique i ∈ I such that IiVj 6= 0, being Vj a weight module over Ii. Undercertain conditions, it is shown that the above decomposition of V is bymeans of the family of its minimal submodules, each one being a simple(weight) submodule.

Keywords: Infinite dimensional Lie module, infinite dimensional splitLie algebra, structure theory.

2010 MSC: 17B10, 17B65, 17B05.

1 Introduction and previous definitions

Throughout this paper, weight modules V and split Lie algebras L are consid-ered of arbitrary dimensions and over an arbitrary base field K. It is worth tomention that, unless otherwise stated, there is not any restriction on dim Vγ ,dim Lα or the products LαVγ where Vγ denotes the weight space associated tothe weight γ of V and Lα the root space associated to the root α of L.

∗Supported by the PCI of the UCA ‘Teorıa de Lie y Teorıa de Espacios de Banach’, by thePAI with project numbers FQM298, FQM2467, FQM3737 and by the project of the SpanishMinisterio de Educacion y Ciencia MTM2007-60333.

1

Given an element x of a Lie algebra L, we denote by adx the adjoint mappingadx defined as adx(y) := [x, y] for any y ∈ L. A splitting Cartan subalgebra Hof L is defined as a maximal abelian subalgebra of L, satisfying that the adjointmappings adh, for h ∈ H, are simultaneously diagonalizable. If L containsa splitting Cartan subalgebra H, then L is called a split Lie algebra, (see forinstance [5]). This means that we have a root spaces decomposition

L = H ⊕ (⊕α∈Λ

Lα)

where Lα = vα ∈ L : [vα, h] = α(h)vα for any h ∈ H for a linear functionalα ∈ H∗ and Λ := α ∈ H∗\0 : Lα 6= 0. The subspaces Lα for α ∈ H∗

are called root spaces of L, (respect to H), and the elements α ∈ Λ ∪ 0 arecalled roots of L, (respect to H). Clearly L0 = H and, as consequence ofJacobi identity, [Lα, Lβ ] ⊂ Lα+β for any α, β ∈ Λ ∪ 0. We also say that Λ issymmetric if for any α ∈ Λ we have that −α ∈ Λ.

Definition 1.1. Let V be a module over a Lie algebra L with splitting Cartansubalgebra H. For a linear functional γ : H −→ K, the weight space of V ,(respect to H), associated to γ is the subspace

Vγ = vγ ∈ V : hvγ = γ(h)vγ for any h ∈ H.

The elements γ ∈ H∗ satisfying Vγ 6= 0 are called weights of V respect to H andwe denote P := γ ∈ H∗\0 : Vγ 6= 0. We say that V is a weight module,respect to H, if

V = V0 ⊕ (⊕γ∈P

Vγ).

We also say that P is the weight system of V .

The weight system P is called symmetric if for any γ ∈ P we have that−γ ∈ P.

Split Lie algebras are examples of weight modules over themselves, whereP = Λ and Vγ = Lγ for γ ∈ P ∪ 0. Since the even part L0 of the standardembedding of a split Lie triple system T and of a split twisted inner derivationtriple system M is a split Lie algebra, the natural actions of L0 over T and Mmake of T and M weight modules over the split Lie algebra L0. So the presentpaper extend the results in [1, 2, 3].

2 Connections of weights. Decompositions of V

In the following, V = V0 ⊕ (⊕

γ∈PVγ) denotes a weight module with a symmetric

weight system P, respect to a split Lie algebra L = H ⊕ (⊕

α∈Λ

Lα) with a sym-

metric root system Λ. We begin by developing connections of weights techniquesin this framework.

2

Definition 2.1. Let γ and δ be two nonzero weights. We say that γ is connectedto δ if there exist α1, ..., αn ∈ Λ such that

1. γ + α1, γ + α1 + α2, ..., γ + α1 + α2 + · · ·+ αn−1 ⊂ P,

2. γ + α1 + α2 + · · ·+ αn ∈ δ,−δ,

where the sums are considered in H∗.We also say that γ, α1, ..., αn is a connection from γ to δ.

Proposition 2.1. The relation ∼ in P defined by γ ∼ δ if and only if γ isconnected to δ is of equivalence.

Given γ ∈ P, we denote by

Pγ := δ ∈ P : δ ∼ γ.

Clearly if δ ∈ Pγ then −δ ∈ Pγ and, by Proposition 2.1, if η /∈ Pγ thenPγ ∩ Pη = ∅.

Our next goal is to associate an (adequate) weight submodule VPγ to anyPγ . For Pγ , γ ∈ P, we define the following linear subspace of V :

VPγ:= (

∑α∈Λ∩Pγ

L−αVα)⊕ (⊕δ∈Pγ

Vδ).

We also denote by V0,Pγ :=∑

α∈Λ∩Pγ

L−αVα ⊂ V0.

Lemma 2.1. The following assertions hold:

1. For any α ∈ Λ and γ ∈ P with α 6= −γ, if LαVγ 6= 0 then α + γ ∼ γ.

2. For any α, β ∈ Λ ∩ P, if Lβ(L−αVα) 6= 0 then α ∼ β.

We recall that a Lie module V is said to be simple if its only submodulesare 0 and V .

Theorem 2.1. Let γ ∈ P. Then the following assertions hold.

1. VPγ is a weight submodule of V .

2. If V is simple, then there exists a connection from γ to δ for any γ, δ ∈ Pand V0 =

∑α∈Λ∩Pγ

L−αVα.

Theorem 2.1-1 let us assert that for any γ ∈ P, VPγ is a weight submoduleof V that we call the submodule of V associated to Pγ .

Proposition 2.2. For a linear complement U of spanKL−αVα : α ∈ Λ ∩ Pin V0, we have

V = U + (∑

[γ]∈P/∼

V[γ]),

where any V[γ] is one of the weight submodules described in Theorem 2.1-1.

3

Recall that the center of V is defined as the set Z(V ) = v ∈ V : Lv = 0.

Theorem 2.2. If LV = V and Z(V ) = 0, then V is the direct sum of the idealsgiven in Proposition 2.2,

V =⊕

[γ]∈P/∼

V[γ].

3 Connections of roots. Decompositions of L

We begin this section by introducing a concept of connections of roots for L in aslightly different way to the one of connection of weights for P developed in theprevious section. To do that, we will connect the nonzero roots of L throughnonzero roots of L and nonzero weights of P considered both as elements in H∗.

Definition 3.1. Let α, β be two nonzero roots of L. We say that α is connectedto β if there exist ζ1, ..., ζn ∈ Λ ∪ P such that

1. α = ζ1,

1. ζ1, ζ1 + ζ2, ..., ζ1 + · · ·+ ζn−1 ⊂ Λ ∪ P,

2. ζ1 + · · ·+ ζn ∈ β,−β,

where the sums are considered in H∗.We also say that ζ1, ..., ζn is a connection from α to β.

Proposition 3.1. The relation ≈ in Λ defined by α ≈ β if and only if α isconnected to β is of equivalence.

Given α ∈ Λ, we denote by

Λα := β ∈ Λ : β ≈ α.

We also have that if β ∈ Λα then −β ∈ Λα and, by Proposition 3.1, if µ /∈ Λα

then Λα ∩ Λµ = ∅.Our next aim is to associate an adequate ideal of L to any Λα. For Λα, α ∈ Λ,

we defineHΛα := spanK[Lβ , L−β ] : β ∈ Λα ⊂ H,

andVΛα

:=⊕

β∈Λα

Lβ .

We denote by LΛαthe following linear subspace of L,

LΛα:= HΛα

⊕ VΛα.

Proposition 3.2. Let α ∈ Λ. Then the following assertions hold.

1. [LΛα, LΛα

] ⊂ LΛα.

4

2. If µ /∈ Λα then [LΛα , LΛµ ] = 0.

By Proposition 3.2-1 we can assert that for any α ∈ Λ, LΛα is a subalgebraof L that we call the subalgebra of L associated to Λα.

Theorem 3.1. The following assertions hold.

1. For any α ∈ Λ, the subalgebra

LΛα = HΛα ⊕ VΛα

of L associated to Λα is an ideal of L.

2. If L is simple, then there exists a connection from α to β for any α, β ∈ Λand H =

∑α∈Λ

[Lα, L−α].

Proposition 3.3. For a linear complement U of spanK[Lα, L−α] : α ∈ Λ inH, we have

L = U +∑

[α]∈Λ/≈

I[α],

where any I[α] is one of the ideals LΛαof L described in Theorem 3.1-1, satis-

fying [I[α], I[β]] = 0 if [α] 6= [β].

Let us denote by Z(L) = e ∈ L : [e, L] = 0 the center of L.

Theorem 3.2. If Z(L) = 0 and H =∑

α∈Λ

[Lα, L−α], then L is the direct sum

of the ideals given in Theorem 3.1,

L =⊕

[α]∈Λ/≈

I[α].

4 Relating the decompositions of V and L

Theorem 4.1. Let V be weight module respect to a perfect split Lie algebra Lsuch that LV = V and Z(V ) = 0. Then

L =⊕i∈I

Ii and V =⊕j∈J

Vj

with any Ii a nonzero ideal of L satisfying [Ii, Ik] = 0 if i 6= k, and any Vj anonzero weight submodule of V in such a way that for any j ∈ J there exists aunique i ∈ I such that

IiVj 6= 0.

Furthermore Vj is a weight module over Ii.

5

5 The simple components

In this section we are showing that, under certain conditions, the decompositionof V given in Theorem 4.1 can be given by means of the family of its minimalsubmodules, each one being a simple (weight) submodule.

Lemma 5.1. Suppose Z(V ) = 0. If W is a submodule of V such that W ⊂ V0,then W = 0.

Let us introduce the concepts of weight-multiplicativity and maximal lengthin the framework of weight modules over spit Lie algebras, in a similar way to theanalogous ones for split Lie algebras, split Lie triple systems and split twistedinner derivation triple systems, (see [1, 2, 3] for these notions and examples).

Definition 5.1. We say that a weight module V is of maximal length if dim Vγ =1 for any γ ∈ P.

Let us note that weight modules of maximal length appears in a natural wayin several contexts. See for instance [4], [6] and [7] for the cases over Virasoro,generalized Virasolo and Witt algebras respectively.

Given any submodule W of V , it is well known that any submodule of aweight module is again a weight module. So we have

W = (W ∩ V0)⊕ (⊕γ∈P

(W ∩ Vγ)).

Observe that if V is of maximal length then we can write

W = (W ∩ V0)⊕ (⊕

γ∈PW

Vγ),

wherePW := γ ∈ P : W ∩ Vγ 6= 0.

Definition 5.2. We say that a weight module V over a split Lie algebra L isweight-multiplicative if given α ∈ Λ and γ ∈ P such that α + γ ∈ P, thenLαVγ 6= 0.

Here we note that if V satisfies V0 =∑

β∈Λ∩PL−βVβ we will understand the

weight-multiplicativity of V by supposing also that if Lβ(L−βVβ) 6= 0 thenL−β(LβV−β) 6= 0.

Theorem 5.1. Let V be a weight module of maximal length, weight-multiplicativeand with Z(V ) = 0 over a split Lie algebra L. If V has all its nonzero weightsconnected and V0 =

∑α∈Λ∩P

L−αVα then either V is simple or V = W ⊕W ′ with

W and W ′ simple (weight) submodules of V .

6

Theorem 5.2. Let V be a weight module of maximal length, weight-multiplicativeand with LV = V , Z(V ) = 0 over a split Lie algebra L. Then L =

⊕i∈I

Ii with

any Ii a nonzero ideal of L satisfying [Ii, Ij ] = 0 if i 6= j, and V =⊕

k∈K

Vk is

the direct sum of the family of its minimal submodules, each one being a simpleweight submodule of V in such a way that for any k ∈ K there exists a uniquei ∈ I such that IiVk 6= 0. Furthermore Vk is a weight module over Ii.

References

[1] Calderon, A.J.: On split Lie algebras with symmetric root systems. Proc.Indian. Acad. Sci, Math. Sci. 118, 351-356, (2008).

[2] Calderon, A.J.: On split Lie triple systems II. Proc. Indian. Acad. Sci,Math. Sci. 120, 185-198, (2010).

[3] Calderon, A.J., Forero M.: Split Twisted inner derivation triple systems.Comm. Alg. 38, n.1, 28-45, (2010).

[4] Kaplansky I.: Virasolo algebras. Comm. Math. Phys. 86, 49-54, (1982).

[5] Stumme N.: The structure of locally finite split Lie algebras. J. Algebra220, 664-693, (1999).

[6] Su, Y.C., Zhao K.: Generalized Virasolo and super-Virasolo algebras andmodules of intermediate series. J. Algebra 252, 1-19, (2002).

[7] Zhao K.: Weight modules over generalized Witt algebras with 1-dimensional weight spaces. Forum Math. 16, no. 5, 725748, (2004).

7

AN INTRODUCTION TO THE “DIWORLD”

JUANA SÁNCHEZ ORTEGA

Abstract. We present a generalization of associative algebras, called associative dialge-bras. We give a simplified statement of the KP algorithm introduced by Kolesnikov andPozhidaev for extending polynomial identities for algebras to corresponding identities fordialgebras. Applications to the KP algorithm are given.

Introduction

Dialgebras were introduced by Loday [10] to provide a natural setting for Leibniz alge-bras, a “noncommutative” version of Lie algebras. To be more precise, a (right) Leibnizalgebra (see [6, 9] for details) is a vector space L, together with a bilinear map L×L→ L,denoted (a, b) 7→ 〈a, b〉, satisfying the (right) Leibniz identity, which says that rightmultiplications are derivations:

〈〈a, b〉, c〉 ≡ 〈〈a, c〉, b〉+ 〈a, 〈b, c〉〉.If 〈a, a〉 ≡ 0 then the Leibniz identity is the Jacobi identity and L is a Lie algebra.

It is well known that every associative algebra becomes a Lie algebra if the product ab isreplaced by the Lie bracket ab − ba. Loday’s goal was to introduce a new structure whichgives a Leibniz algebra by a similar procedure. His idea was to replace the product ab andits opposite ba by two distinct operations a a b and b ` a. As this way, the Leibniz bracketa a b− b ` a is not necessarily skew-symmetric, and we obtain the notion of an associativedialgebra.

An associative dialgebra is a vector space A with two bilinear maps A × A → A,denoted a and ` and called the left and right products, satisfying the left and right baridentities, and left, right and inner associativity:

(a a b) ` c ≡ (a ` b) ` c, a a (b a c) ≡ a a (b ` c),

(a a b) a c ≡ a a (b a c), (a ` b) ` c ≡ a ` (b ` c), (a ` b) a c ≡ a ` (b a c).

The Leibniz bracket in an associative dialgebra satisfies the Leibniz identity.Dialgebras have become an active research area, attracting the attention of numerous

authors who have considered other varieties of nonassociative dialgebras, which have beenstudied by Velásquez and Felipe [14, 15], Gubarev and Kolesnikov [8, 7], Pozhidaev [12, 13],Voronin [16], and Bremner, Felipe, Peresi and J.S.O [1, 2, 4, 3, 5], among many others.

The purpose of the talk is to present the distinct varieties of nonassociative dialgebras,and their related systems (here by a system, we will understand a pair of a triple).

Section 1 recalls basic definitions for free dialgebras. Section 2 presents a simplifiedstatement of the general Kolesnikov-Pozhidaev (KP) algorithm for converting an arbitrary

Date: 20 February, 2012.1

2 JUANA SÁNCHEZ ORTEGA

variety of multioperator algebras into a variety of dialgebras. Sections 3 and 4 are devotedto the application of the KP algorithm to the Jordan and Lie setting, respectively.

1. Free dialgebras

Loday has determined a basis for the free dialgebra.

Definition 1.1. A dialgebra monomial on a set X is a product x = a1a2 · · · an wherea1, . . . , an ∈ X with some placement of parentheses and some choice of operations. Thecenter of x is defined inductively: if n = 1 then c(x) = x; if n ≥ 2 then x = y a z orx = y ` z and we set c(y a z) = c(y) or c(y ` z) = c(z).

Lemma 1.2. (Loday [11]) If x = a1a2 · · · an is a monomial with c(x) = ai then x isdetermined by the order of its factors and the position of its center:

x = (a1 ` · · · ` ai−1) ` ai a (ai+1 a · · · a an).

Definition 1.3. The right side of the last equation is the normal form of x and is abbre-viated by the hat notation a1 · · · ai−1aiai+1 · · · an.

Lemma 1.4. (Loday [11]) The set of monomials a1 · · · ai−1aiai+1 · · · an in normal form with1 ≤ i ≤ n and a1, . . . , an ∈ X forms a basis of the free dialgebra on X.

2. The Kolesnikov-Pozhidaev algorithm

This algorithm, introduced by Kolesnikov [8] and Pozhidaev [13], converts a multilinearpolynomial identity of degree d for an n-ary operation into d multilinear identities of degreed for n new n-ary operations.

Definition 2.1. KP Algorithm.Part 1: We consider a multilinear n-ary operation, denoted by the symbol

(1) −,−, . . . ,− (n arguments).

Given a multilinear polynomial identity of degree d in this operation, we describe the appli-cation of the algorithm to one monomial in the identity, and from this the application to thecomplete identity follows by linearity. Let a1a2 . . . ad be a multilinear monomial of degreed, where the bar denotes some placement of n-ary operation symbols. We introduce n newn-ary operations, denoted by the same symbol but distinguished by subscripts:

(2) −,−, . . . ,−1, −,−, . . . ,−2, . . . , −,−, . . . ,−n.For each i ∈ 1, 2, . . . , d we convert the monomial a1a2 . . . ad in the original n-ary operation(1) into a new monomial of the same degree d in the n new n-ary operations (2), accordingto the following rule which is based on the position of ai. For each occurrence of the originaloperation symbol in the monomial, either ai occurs within one of the n arguments or not,and we have the following cases:

• If ai occurs within the j-th argument then we convert the original operation symbol. . . to the j-th new operation symbol . . . j .• If ai does not occur within any of the n arguments, then either

– ai occurs to the left of the original operation symbol, in which case we convert. . . to the first new operation symbol . . . 1, or

AN INTRODUCTION TO THE “DIWORLD” 3

– ai occurs to the right of the original operation symbol, in which case we convert. . . to the last new operation symbol . . . n.

In this process, we call ai the central argument of the monomial.Part 2: In addition to the identities constructed in Part 1, we also include the following

identities for all i, j ∈ 1, 2, . . . , n with i 6= j and all k, ` ∈ 1, 2, . . . , n:a1, . . . , ai−1, b1, · · · , bnk, ai+1, . . . , anj ≡a1, . . . , ai−1, b1, · · · , bn`, ai+1, . . . , anj .

This identity says that the n new operations are interchangeable in the i-th argument of thej-th new operation when i 6= j.

Example 2.2. The defining identities for associative dialgebras can be obtained by applyingthe KP algorithm to the associativity identity, which we write in the form a, b, c ≡a, b, c. The original operation produces two new operations −,−1 and −,−2. Sinceassociativity has degree 3, Part 1 produces three new identities of degree 3 by making a, b,c in turn the central argument:

a, b1, c1≡a, b, c11, a, b2, c1≡a, b, c12, a, b2, c2≡a, b, c22,and Part 2 produces these two identities:

a, b, c11 ≡ a, b, c21, a, b1, c2 ≡ a, b2, c2.If we revert to the standard notation by writing a a b = a, b1 and a ` b = a, b2, thenthese five identities are the defining identities for associative dialgebras.

3. Jordan dialgebras and Jordan triple disystems

We apply the KP algorithm to the defining identities for Jordan algebras to obtain thevariety of Jordan dialgebras. The most recent results related to Jordan dialgebras will bestablished.

4. Leibniz triple systems

The KP algorithm will be applied to Lie triple systems to get a new variety of triplesystems; we call these structures Leibniz triple systems. To conclude, we verify thatLeibniz triple systems are the natural analogues of Lie triple systems in the context ofdialgebras.

References

[1] M. R. Bremner: On the definition of quasi-Jordan algebra. Communications in Algebra 38 (2010)4695–4704.

[2] M. R. Bremner, L. A. Peresi: Special identities for quasi-Jordan algebras. Communications inAlgebra 39:7 (2011) 2313-2337.

[3] M. R. Bremner, R. Felipe and J. Sánchez-Ortega: Jordan triple disystems. Computers andMathematics with Applications (to appear).

[4] M. R. Bremner, L. A. Peresi, and J. Sánchez-Ortega: Malcev dialgebras. Linear and MultilinearAlgebra (to appear).

[5] M. R. Bremner and J. Sánchez-Ortega: Leibniz triple systems (Preprint)[6] C. Cuvier: Algèbres de Leibnitz: définitions, propriétés. Annales Scientifiques de l’École Normale

Supérieure (4) 27, 1 (1994) 1–45.

4 JUANA SÁNCHEZ ORTEGA

[7] V. Y. Gubarev, P. S. Kolesnikov: The Tits-Kantor-Koecher construction for Jordan dialgebras.Communications in Algebra 39 (2011) 497–520.

[8] P. S. Kolesnikov: Varieties of dialgebras and conformal algebras. Sibirskiı Matematicheskiı Zhurnal49 (2008) 322–339; translation in Siberian Mathematical Journal 49 (2008) 257–272.

[9] J.-L. Loday: Une version non commutative des algèbres de Lie: les algèbres de Leibniz. L’EnseignementMathématique 39, 3-4 (1993) 269–293.

[10] J.-L. Loday: Algèbres ayant deux opérations associatives: les digèbres. Comptes Rendus de l’Académiedes Sciences. Série I. Mathématique 321, 2 (1995) 141–146.

[11] J. L. Loday: Dialgebras. Dialgebras and Related Operads, pages 7–66. Lectures Notes in Mathematics,1763. Springer-Verlag, 2001.

[12] A. P. Pozhidaev: Dialgebras and related triple systems. Sibirskiı Matematicheskiı Zhurnal 49 (2008)870–885; translation in Siberian Mathematical Journal 49 (2008) 696Ð708.

[13] A. P. Pozhidaev: 0-dialgebras with bar-unity, Rota-Baxter and 3-Leibniz algebras. ContemporaryMathematics 499, 245–256. American Mathematical Society, 2009.

[14] R. Velásquez, R. Felipe: Quasi-Jordan algebras. Communications in Algebra 36 (2008) 1580–1602.[15] R. Velásquez, R. Felipe: Split dialgebras, split quasi-Jordan algebras and regular elements. Journal

of Algebra and its Applications 8 (2009) 191–218.[16] V. Voronin: Special and exceptional Jordan dialgebras. arXiv:1011.3683v1 [math.RA]

Dpto. de Matemática Aplicada, E. T. S. I. de Telecomunicación, Campus de Teatinos, Universidadde Málaga

E-mail address: jsanchezo@uma.es

GLOBALIZACIÓN DE ACCIONES PARCIALES DEFORMADAS

M. DOKUCHAEV, R. EXEL, AND J. J. SIMÓN.

Abstract. Let A be a unital ring which is a product of possibly infinitely many indecom-posable rings. We establish a criteria for the existence of a globalization for a given twistedpartial action of a group on A. If the globalization exists, it is unique up to a certain equiv-alence relation and, moreover, the crossed product corresponding to the twisted partialaction is Morita equivalent to that corresponding to its globalization. For arbitrary unitalrings the globalization problem is reduced to an extendibility property of the multipliersinvolved in the twisted partial action.

Introducción

Las acciones parciales deformadas de grupos localmente compactos en C∗-álgebras fueronintroducidas en [6] como herramienta para la construcción general de los productos cruzadosde C∗-álgebras. El alcance de este concepto le permite a uno probar (véase [6, Theorem7.3]) que algunos fibrados vectoriales C∗-algebraicos (C∗-algebraic vector bundles) puedenser obtenidos mediante esta construcción. Las acciones parciales deformadas de gruposen álgebras abstractas y sus correspondientes productos cruzados fueron introducidos en[4], donde un análogo algebraico de la estabilización que se construyó en C∗-álgebras fueobtenido. En [2, 7] pueden encontrarse aplicaciones y otros desarrollos de estos conceptos.

El estudio de la globalización de acciones parciales ha producido mucha literatura enmuy diversas ramas de matemáticas (véase, por ejemplo, [1, 9, 10]). El problema de laglobalización ha resultado esencial para un estudio mayor del producto cruzado general yotros temas relacionados (véase [3, 8]). Los resultados de este resumen se encuentran en elartículo de los autores, Globalization of twisted partial actions, Trans. Amer. Math. Soc.362 (2010), 4137-4160.

1. Preliminares

A lo largo de todo el resumen los anillos A serán siempre asociativos y no necesaria-mente con uno. Denotaremos con U(A) el grupo de las unidades y por 1A la identidad,cuando tenga. Recordemos que el anillo de multiplicadores M(A) de un anillo asociativo,no necesariamente con identidad A es el conjunto

M(A) = (R,L) ∈ End(AA)× End(AA) : (aR)b = a(Lb) para todo a, b ∈ Ajunto con la suma y multiplicación componente a componente. Aquí nosotros usamos laescritura a la derecha para los homomorfismos de A-módulos por la izquierda, mientras quela acción a la izquierda se usa para módulos por la derecha.

Así, dado R :A A →A A, L : AA → AA and a ∈ A escribimos a 7→ aR y a 7→ La. Paraun multiplicador w = (R,L) ∈ M(A) y un elemento a ∈ A hacemos aw = aR y wa = La,así que siempre tenemos (aw)b = a(wb) (a, b ∈ A).

1

2 M. DOKUCHAEV, R. EXEL, AND J. J. SIMÓN.

Definición 1.1. Una acción parcial de un grupo G en A es una terna

α = (Dgg∈G, αgg∈G, wg,h(g,h)∈G×G),

donde para cada g ∈ G, se tiene que Dg es un ideal bilátero en A, αg es un isomorfismo deanillos Dg−1 → Dg, y para cada (g, h) ∈ G×G, wg,h es un elemento invertible deM(Dg ·Dgh).Además, se satisfacen los siguientes axiomas, para g, h y t ∈ G.

(i) D2g = Dg, Dg · Dh = Dh · Dg.

(ii) D1 = A y α1 es la identidad en A.(iii) αg(Dg−1 · Dh) = Dg · Dgh.(iv) αg αh(a) = wg,hαgh(a)w−1

g,h, ∀a ∈ Dh−1 · Dh−1g−1 .(v) w1,g = wg,1 = 1.(vi) αg(awh,t)wg,ht = αg(a)wg,hwgh,t, ∀a ∈ Dg−1 · Dh · Dht.

Denotaremos también a los wg,h como w[g, h]. Como se ha visto en [4], de (i) se sigue queun producto finito de ideales idempotentes Dg · · · Dh es idempotente y

αg(Dg−1 · Dh · Df ) = Dg · Dgh · Dgfpara todo g, h, f ∈ G, por (iii). Así, tiene sentido aplicar todos los multiplicadores de (vi).

Decimos que α es global si Dg = A (g ∈ G). Nótese que en una acción global deformada

(1) β = (B, βgg∈Gug,h(g,h)∈G×G)

de un grupo G sobre un anillo B, no necesariamente unitario, uno puede restringir β aun ideal bilátero A de B, de tal manera que A, tenga 1A, como sigue. Hacemos Dg =A ∩ βg(A) = A · βg(A) vemos que cada Dg tiene 1; a saber, 1Aβg(1A). Entonces haciendoαg = βg |Dg−1 , los axiomas (i), (ii) y (iii) de la definición anterior se satisfacen. Aún más, sidefinimos wg,h = ug,h1Aβg(1A)βgh(1A) podemos comprobar que el resto de los axiomas sesatisface; así que, de hecho, hemos obtenido una acción parcial deformada en A.

Definición 1.2. Una acción parcial deformada β, como en (1), de un grupo G sobre unanillo (posiblemente sin 1) B, decimos que es una globalización (o una acción envolvente)para la acción parcial α de G en A si existe un monomorfismo ϕ : A → B tal que:

(i) ϕ(A) es un ideal de B.(ii) B =

∑g∈G βg(ϕ(A)).

(iii) ϕ(Dg) = ϕ(A) ∩ βg(ϕ(A)) para todo g ∈ G.(iv) ϕ αg = βg ϕ sobre Dg−1 para todo g ∈ G.(v) ϕ(awg,h) = ϕ(a)ug,h, ϕ(wg,ha) = ug,hϕ(a) para cualesquier g, h ∈ G y a ∈ DgDgh.

En este caso, decimos que α es globalizable.En caso de que A sea anillo con uno, si B también tiene identidad decimos que β es una

globalización unitaria y que α es unitariamente globalizable.

2. Resultados

2.1. Equivalencia de Morita de productos cruzados. Dada una acción parcial α de Gsobre A, el producto cruzado A∗αG es la suma directa ⊕g∈GDgδg, donde los δg son símboloscon la regla de multiplicar

(agδg) · (bhδh) = αg(α−1g (ag)bh)wg,hδgh,

GLOBALIZACIÓN DE ACCIONES PARCIALES DEFORMADAS 3

donde wg,h actúa como multiplicador por la derecha de αg(α−1g (ag)bh) ∈ αg(Dg−1 · Dh) =

Dg · Dgh. La asociatividad del producto se probó en [4]. Cuando α es globalizable, losproductos cruzados parcial y global son equivalentes de Morita.

Teorema 2.1. Sea α una acción parcial deformada globalizable de G sobre un anillo s-unitario por la izquierda A y β la globalización sobre B. Entonces los productos cruzadosA ∗α G and B ∗β G son equivalentes de Morita.

2.2. Reducción del problema a la extensión de los multiplicadores. Con la notaciónde la Definición 1.1. Si α es una acción parcial deformada de G sobre A, de tal manera quetantoA, como los ideales Dg tienen identidad (que denotamos 1g = 1Dg) entonces se verificanlas siguientes propiedades.

(1) DgDh es unitario con identidad 1g1h. Como consecuencia,(2) M(DgDgh) ∼= DgDgh, así que wg,h es invertible.(3) Finalmente, αg(1g−11h) = 1g1gh. Recuérdese que αg(Dg−1 · Dh) = Dg · Dgh.Esto nos permite probar el siguiente teorema, que nos muestra que la globalización de-

pende exclusivamente de que podamos extender las unidades de los multiplicadores.

Teorema 2.2. Con la notación de la Definición 1.1. Sea α una acción parcial deformadade G sobre A, donde los Dg tienen identidad. Entonces, α admite globalización si y sólo sipara todo (g, h) ∈ G×G existe wg,h ∈ U(A), tal que wg,h1g1gh = wg,h y además

αg(wh,t 1g−1) wg,ht = 1g wg,h wgh,t, (g, h, t ∈ G)

2.3. Acciones sobre anillos que son producto arbitrario de indescomponibles. Deahora en adelante, supondremos que nuestro anillo se descompone como A =

∏λ∈ΛRλ, con

Rλ anillo con uno, indescomponible. En este caso, es fácil ver que si α es una acción parcialdeformada de G sobre A, los ideales Dg son producto de algunos de los Rλ. De hecho, αpermuta la descomposición en bloques. De aquí surge la idea de las acciones transitivas.

Definición 2.3. Sea A =∏λ∈ΛRλ, producto de indescomponibles y α una acción parcial

de G sobre A. Decimos que α es transitiva si para cualesquiera λ, λ′ ∈ Λ existe g ∈ G detal manera que Rλ ⊆ Dg−1 y αg(Rλ) = Rλ′ .

2.4. Existencia de la globalización. Sea A un anillo de la forma A =∏λ∈ΛRλ, con

Rλ anillo con uno, indescomponible. Utilizando una adaptación al caso no conmutativoy parcial de la idea de la correstricción en el álgebra homológica podemos, partiendo decualquier acción parcial deformada, crear “acciones tipo” que son equivalentes a las origi-nales (en el sentido que veremos más adelante, en el Párrafo 2.5, por cuestiones de espacio)pero más manejables. De hecho, se puede partir de una acción parcial deformada no transi-tiva y obtener una equivalente que sea transitiva. A partir de éstas nuevas acciones parcialesdeformadas transitivas más simples obtenemos nuestro resultado principal sobre la global-ización.

Teorema 2.4. Sea A un anillo de la forma A =∏λ∈ΛRλ, con Rλ anillo con uno, inde-

scomponible. Una acción parcial deformada

α = (Dxx∈G, αxx∈G, w[x, y](x,y)∈G×G),

de un grupo G sobre A es globalizable si y sólo si cada Dg (g ∈ G) es anillo con uno.

4 M. DOKUCHAEV, R. EXEL, AND J. J. SIMÓN.

2.5. Unicidad. Como hemos comentado, al hacer la globalización, la acción envolvente nocontiene a la original sino a una equivalente, luego es normal que la globalización se obtengasalvo equivalencia. Vamos a ver esos conceptos. Sean A1 y A2 anillos y

αi = (Digg∈G, αi,gg∈G, wi[g, h](g,h)∈G×G), (i = 1, 2)

acciones parciales deformadas. Decimos que α1 y α2 son equivalentes si D1g = D2

g (g ∈ G)y existe un conjunto de unidades εgg∈G ⊆ U(Dg) tal que α′g(a) = εgαg(a)ε−1

x y tal quew′g,h = εg αg(εh1g−1)wg,h ε−1

gh . Diremos que α1 y α2 son isomorfas si existe un isomorfismode anillos φ : A1 → A2 tal que φ(D1

g) = D2g (g ∈ G); φ α1,g φ−1(a) = α2,g(a) (g ∈ G

y a ∈ D2g−1) y φg,hw1[g, h]φ−1

g,h = w2[g, h], como multiplicadores de D2gD2

gh, donde φg,h =φ|D1

gD1gh. Diremos que α1 y α2 son isomorfo-equivalentes si α1 es isomorfa a una acción

equivalente a α2. Si β1 y β2 son globalizaciones de α1 y α2, respectivamente, con inclusionesϕ1 : A → B1 y ϕ2 : A → B2, diremos que son globalizaciones equivalentes si α1 y α2 sonisomorfo-equivalentes y existe un isomorfismo de anillos φ : B1 → B2 tal que φ ϕ1 = ϕ2.

Teorema 2.5. Sea A un anillo de la forma A =∏λ∈ΛRλ, con Rλ anillo con uno, inde-

scomponible y sea α una acción parcial deformada globalizablemente unitaria, de G sobre A.Entonces cualesquiera dos globalizaciones son equivalentes.

References

[1] F. Abadie, Enveloping Actions and Takai Duality for Partial Actions, J. Funct. Analysis 197 (2003),no. 1, 14–67.

[2] D. Bagio, W. Cortes, M. Ferrero, A. Paques, Actions of inverse semigroups on algebras, Commun.Algebra 35 (2007), no. 12, 3865–3874.

[3] D. Bagio, J. Lazzarin, A. Paques, Crossed products by twisted partial actions: separability, semisim-plicity and Frobenius properties, Commun. Algebra, 38(2), 2010

[4] M. Dokuchaev, R. Exel, J. J. Simón, Crossed products by twisted partial actions and graded algebras,J. Algebra, 320, (2008), no. 8, 3278–3310.

[5] M. Dokuchaev, M. Ferrero, A. Paques, Partial Actions and Galois Theory, J. Pure Appl. Algebra 208(2007), no. 1, 77–87.

[6] R. Exel, Twisted partial actions: a classification of regular C∗-algebraic bundles, Proc. London Math.Soc. 74 (1997), no. 3, 417–443.

[7] R. Exel, M. Laca, J. Quigg, Partial dynamical systems and C∗-algebras generated by partial isometries,J. Operator Theory 47 (2002), no. 1, 169–186.

[8] M. Ferrero, Partial actions of groups on semiprime rings, Groups, rings and group rings, Lect. NotesPure Appl. Math. 248, Chapman & Hall/CRC, Boca Raton, FL, (2006), 155–162.

[9] J. Kellendonk, M. V. Lawson, Partial actions of groups, Internat. J. Algebra Comput. 14 (2004), no. 1,87 – 114.

[10] M. V. Lawson, Inverse semigroups. The theory of partial symmetries, World Scientific Publishing Co.,Inc., River Edge, NJ, 1998.

Universidad de São Paulo, BrasilE-mail address: dokucha@ime.usp.br

Universidade Federal de Santa CatarinaE-mail address: exel@mtm.ufsc.br

Universidad de MurciaE-mail address: jsimon@um.es

INVARIANCIA DE MORITA EN ANILLOS SIN UNO. UNAAPLICACIÓN A LAS ÁLGEBRAS DE CAMINOS DE LEAVITT

JOSÉ FÉLIX SOLANILLA HERNÁNDEZ AND MERCEDES SILES MOLINA . . .

Abstract. En esta charla nos proponemos contar algunos resultados relacionados conla invariancia de Morita de ciertas propiedades para anillos no necesariamente unitarios,el uso de locales en elementos y su aplicación a las álgebras de caminos de Leavitt. Separtirá de la noción de anillos Morita equivalentes en el caso sin uno, y se ilustrarán ciertobloque de propiedades Morita invariantes que se mantienen para estos anillos. Luegose utilizará la noción de anillo local en elementos para enunciar el resultado principal decaracterización de propiedades Morita invariantes. Señalaremos que aún se mantiene comoproblema abierto si el ser un I-anillo es una propiedad Morita invariante (en particularla propiedad de elevación de idempotentes módulo el radical de Jacobson). Por últimodaremos una aplicación de lo anterior a las álgebras de Caminos de Leavitt y probaremosque la Condición (L) es una propiedad Morita invariante.

In this talk we stablish some results concerning to Morita invariance about theoreticproperties for nonunital rings, using local at element rings and how to be applied to LeavithPath Algebras. We begin from Equivalents Morita rings notion for rings without unit andwill show some Morita’s properties block which are remained into this rings. Moreover we’lluse the local at elements sense to be declared the main theorem about its caracterizationon invariance Morita’s properties. Additionally, we note even is holding as open questionif been I-rings are Morita invariance property ( particularly, lifting idempotents moduloJacobson radical property). Finally, we give an interesting aplication above to LeavithPath Algebras and will be shown that Condition L is a Morita invariant property.

Introducción

La teoría de Morita, su invariabilidad y el estudio de su equivalencias, ha sido un terrenoampliamente abonado y objeto, por tanto de numerosos trabajos. En [1],[4],[8],[9],[10],[11]hay fuentes de excelente referencia sobre equivalencias de Morita para anillos no necesari-amente unitarios.También en [5], desde perspectivas diversas, se puede tener una valiosafuente de consulta sobre Morita invariabilidad, asi como contextos de Morita, tanto paraanillos unitarios como para no unitarios. Los contextos de Morita que trataremos serán paraanillos idempotentes. Se hará necesario partir entonces, recordando algunas nociones sobrecontextos y equivalencia de Morita para estos anillos.

References

[1] G. Abrams, Morita equivalence for rings with local units, Comm. in Algebra 11(8) (1983) 801-837.[2] G. D. Abrams, G. Aranda Pino, The Leavitt path algebra of a graph, J. Algebra 293 (2) (2005),

319–334.[3] G. Abrams, P. Ara, M. Siles Molina, Leavitt path algebras, Springer Lecture Notes in Mathematics (to

appear).1

2 JOSÉ FÉLIX SOLANILLA HERNÁNDEZ AND MERCEDES SILES MOLINA . . .

[4] G. Abrams, M. Tomforde, Isomorphism and Morita equivalence of graph algebras, (Pre impresion).[5] G. Aranda Pino, Propiedades Morita invariantes y contextos de Morita, Tesis de Licenciatura. Univer-

sidad de Málaga, 2004.[6] G. Aranda Pino, K. R. Goodearl, F. Perera, M. Siles Molina, Socle theory for Leavitt path

algebras of arbitrary graphs, Rev. Mat. Iberoam. (to appear).[7] G. Aranda Pino; D. Martín Barquero, C. Martín González, M. Siles Molina, Non-simple

purely infinite rings (submitted).[8] Ara, P, Rings without identity which are Morita equivalent to regular rings, Algebra Colloq. 11 (4)

(2004), 533–540.[9] P. N. Ánh, L. Márki, Morita equivalence for rings without identity, Tsukuba J. Math. 11 (1) (1987),

1–16.[10] Brown, L. G.; Green, P.; Rieffel, M. A., Stable isomorphism and strong Morita equivalence of C∗-

algebras, Pacific J. Math. 71 (2) (1977), 349–363.[11] J. L. García, J.J. Simón, Morita equivalence for idempotent rings. J. Pure Appl. Algebra 76 (1991),

39–56.[12] M. Gómez Lozano, M. Siles Molina, Quotient rings and Fountain-Gould left orders by the local

approach, Acta Math. Hungar. 97 (2002), 287–301.[13] S. Kyuno, Equivalence of module categories. Math. J. Okayama Univ. 28 (1974), 147–150.[14] T. Y. Lam, A First Course in Noncommutative Rings. Graduate texts in Mathematics 131, Springer-

Verlag, New York (2001).[15] W.K. Nicholson.,I-rings Transactions of the american mathematical society. Vol 207, 1975.

Universidad de Málaga.. . .E-mail address: msiles@uma.es

Universidad de Málaga.. . .E-mail address: jose_solanilla@uma.es

MÓDULOS YETTER-DRINFELD SOBRE ÁLGEBRAS DE HOPF

TRENZADAS DÉBILES

C. SONEIRA CALVO

Abstract. In this talk we introduce the notion of weak operator and the theory of Yetter-Drinfeld modules over a weak braided Hopf algebra with invertible antipode in a strictmonoidal category. We prove that the class of such objects constitute a non strict monoidalcategory and provide a generic example of Yetter-Drinfeld module in this context via theadjoint action (coaction) associated to the weak braided Hopf algebra.

Introducción

Los módulos de Yetter-Drinfeld sobre un álgebra de Hopf fueron considerados por primeravez por Radford en [7] para estudiar las proyecciones de álgebras de Hopf, si bien su deniciónformal la establece, aunque con el nombre de módulo cruzado, Yetter en [10].

En los años noventa, para estudiar problemas relacionados con la teoría cuántica decampos en dimensión baja, Bohm, Nill y Schalanyi [5] introducen las álgebras de Hopfdébiles como una generalización de las álgebras de Hopf. Recientemente, para probar elteorema de de Radford para proyecciones de álgebras de Hopf débiles, Alonso, Fernándezy González introducen las álgebras trenzadas débiles [1], concepto que engloba como casosparticulares a las álgebras de Hopf débiles y las álgebras de Hopf trenzadas de Takeuchi [9].

En [3] los autores prueban que muchas propiedades relevantes sobre proyecciones asociadasa álgebras de Hopf trenzadas débiles pueden obtenerse sin recurrir a una trenza global enla categoría de partida. Este hecho plantea la cuestión del posible desarrollo de una teoríageneral de módulos Yetter-Drinfeld sobre un álgebra de Hopf trenzada débil en una categoríamonoidal estricta.

En esta charla mostraremos que que es posible dar una respuesta armativa y ademásrecuperar la teoría clásica desarrollada en [8]. Se dan también dos colecciones de ejemplosen este contexto general basadas en el uso de la (co)acción (co)adjunta.

1. Operadores débiles

En esta charla, salvo indicación expresa de lo contrario, denotamos por C una categoríamonoidal estricta con idempotentes escindidos.

Denición 1.1. Sea D una biálgebra trenzada débil en C (Ver [4]) y M un objeto en C.Un operador débil entre M y D, denotado por (M,D)-OD, se dene como una cuádrupla(rM , r

′M , sM , s

′M ) formada por cuatro morsmos en C:

rM : M⊗D → D⊗M, r′M : D⊗M →M⊗D, sM : D⊗M →M⊗D, s′M : M⊗D → D⊗M,

tales que1

2 C. SONEIRA CALVO

(e1) Se cumple:(e1-1) (D ⊗ rM ) (rM ⊗D) (M ⊗ tD,D) = (tD,D ⊗M) (D ⊗ rM ) (rM ⊗D)(e1-2) (r′M ⊗D) (D ⊗ r′M ) (tD,D ⊗M) = (M ⊗ tD,D) (r′M ⊗D) (D ⊗ r′M )(e1-3) (sM ⊗D) (D ⊗ sM ) (tD,D ⊗M) = (M ⊗ tD,D) (sM ⊗D) (D ⊗ sM )(e1-4) (D ⊗ s′M ) (s′M ⊗D) (M ⊗ tD,D) = (tD,D ⊗M) (D ⊗ s′M ) (s′M ⊗D)

y las igualdades análogas sustituyendo tD,D por t′D,D.

(e2) Se cumple:(e2-1) (r′M ⊗D) (D ⊗ sM ) (tD,D ⊗M) = (M ⊗ tD,D) (sM ⊗D) (D ⊗ r′M )(e2-2) (sM ⊗D) (D ⊗ r′M ) (t′D,D ⊗M) = (M ⊗ t′D,D) (r′M ⊗D) (D ⊗ sM )

(e2-3) (D ⊗ s′M ) (rM ⊗D) (M ⊗ tD,D) = (tD,D ⊗M) (D ⊗ rM ) (s′M ⊗D)(e2-4) (D ⊗ rM ) (s′M ⊗D) (M ⊗ t′D,D) = (t′D,D ⊗M) (D ⊗ s′M ) (rM ⊗D).

(e3) Si

∇rM := r′M rM : M ⊗D →M ⊗D, ∇r′M:= rM r′M : D ⊗M → D ⊗M,

∇sM := s′M sM : D ⊗M → D ⊗M, ∇s′M:= sM s′M : M ⊗D →M ⊗D,

se tiene que:(e3-1) ∇rM = (((εD⊗M) rM )⊗D) (M ⊗ δD) = (M ⊗µD) (((r′M (ηD⊗M))⊗D)(e3-2) ∇r′M

= (D⊗ ((M ⊗ εD) r′M )) (δD⊗M) = (µD⊗M) (D⊗ (rM (M ⊗ ηD)))

(e3-3) ∇sM = (D⊗ ((M ⊗ εD) sM )) (δD⊗M) = (µD⊗M) (D⊗ (s′M (M ⊗ ηD)))(e3-4) ∇s′M

= (((εD⊗M)s′M )⊗D) (M ⊗δD) = (M ⊗µD) (((sM (ηD⊗M))⊗D).

(e4) Se cumple:(e4-1) rM (M ⊗ µD) = (µD ⊗M) (D ⊗ rM ) (rM ⊗D)(e4-2) r′M (µD ⊗M) = (M ⊗ µD) (r′M ⊗D) (D ⊗ r′M )(e4-3) (D ⊗ rM ) (rM ⊗D) (M ⊗ δD) = (δD ⊗M) rM(e4-4) (r′M ⊗D) (D ⊗ r′M ) (δD ⊗M) = (M ⊗ δD) r′M(e4-5) sM (µD ⊗M) = (M ⊗ µD) (sM ⊗D) (D ⊗ sM )(e4-6) s′M (M ⊗ µD) = (µD ⊗M) (D ⊗ s′M ) (s′M ⊗D)(e4-7) (sM ⊗D) (D ⊗ sM ) (δD ⊗M) = (M ⊗ δD) sM(e4-8) (D ⊗ s′M ) (s′M ⊗D) (M ⊗ δD) = (δD ⊗M) s′M .

Denición 1.2. Sea D un álgebra de Hopf trenzada débil en C (Ver [1]) y M un objeto enC. Un operador débil entreM y D se dene como una cuádrupla ordenada (rM , r

′M , sM , s

′M )

satisfaciendo las condiciones (e1), (e2), (e3), (e4) de la Denición 1.1 y además:

(e5) Se cumple:(e5-1) (M ⊗ λD) ∇rM = ∇rM (M ⊗ λD)(e5-2) (λD ⊗M) ∇r′M

= ∇r′M (λD ⊗M)

(e5-3) (λD ⊗M) ∇sM = ∇sM (λD ⊗M)(e5-4) (M ⊗ λD) ∇s′M

= ∇s′M (M ⊗ λD).

Denición 1.3. Sea D una biálgebra trenzada débil, M un objeto en C, (rM , r′M , sM , s

′M )

un (M,D)-OD y (M,ϕM ) un D-módulo por la izquierda. Se dirá que el (M,D)-OD escompatible con la estructura de D-módulo si se cumplen las siguientes condiciones:

(i) rM (ϕM ⊗D) = (D ⊗ ϕM ) (tD,D ⊗M) (D ⊗ rM ),(ii) r′M (D ⊗ ϕM ) = (ϕM ⊗D) (D ⊗ r′M ) (t′D,D ⊗M),

(iii) sM (D ⊗ ϕM ) = (ϕM ⊗D) (D ⊗ sM ) (tD,D ⊗M),

MÓDULOS YETTER-DRINFELD SOBRE ÁLGEBRAS DE HOPF TRENZADAS DÉBILES 3

(iv) s′M (ϕM ⊗D) = (D ⊗ ϕM ) (t′D,D ⊗M) (D ⊗ s′M ).

Si (M,%M ) es un D-comódulo por la izquierda, se dirá que el (M,D)-OD es compatiblecon la estructura de D-comódulo si se cumplen las condiciones análogas correspondientes.

Del mismo modo se dene la noción de compatibilidad con (co)estructuras de (co)módulopor la derecha.

2. La categoría de módulos Yetter-Drinfeld y su estructura monoidal

Denición 2.1. Sea D un álgebra de Hopf trenzada débil en C. Se dice que (M,ϕM , %M )es un módulo Yetter-Drinfeld izquierda-izquierda sobre D si (M,ϕM ) es un D-módulo porla izquierda, (M,%M ) es un D-comódulo por la izquierda y se cumplen las siguientes condi-ciones:

(yd1-ii) %M = (µD ⊗ ϕM ) (D ⊗ tD,D ⊗M) (δD ⊗ %M ) (ηD ⊗M)(yd2-ii) Existe un (M,D)-OD (rM , r

′M , sM , s

′M ) compatible con las estructuras de

(co)módulo de M tal que:(µD ⊗ ϕM ) (D ⊗ tD,D ⊗M) (δD ⊗ %M )

= (µD ⊗M) (D ⊗ rM ) ((%M ϕM )⊗D) (D ⊗ sM ) (δD ⊗M)

Si (M,ϕM , %M ) y (N,ϕN , %N ) dos módulos Yetter-Drinfeld izquierda-izquierda sobre Dcon operadores débiles asociados (rM , r

′M , sM , s

′M ) y (rN , r

′N , sN , s

′N ) respectivamente, el

morsmo f : M → N se dice de módulos Yetter-Drinfeld si:

(i) f es un morsmo de (co)módulos por la izquierda.(ii) rN (f ⊗D) = (D ⊗ f) rM , sN (D ⊗ f) = (f ⊗D) sMLa clase de todos los módulos Yetter-Drinfeld izquierda-izquierda sobre D junto con los

morsmos de módulos Yetter-Drinfeld forman una categoría que denotaremos por DDYD.

Nota 2.2. Cuando la categoría C es simétrica y se denen tanto el operador Yang-Baxterdébil como el operador débil a partir de la trenza de la categoría, se recuperan las condicionesintroducidas en [7].

Teorema 2.3. Sea D un AHTD en C con antípodo inversible. Entonces DDYD es una

categoría monoidal no estricta.

3. Ejemplos

Sea D un AHTD en C. La acción ϕD : D ⊗ D → D y la coacción %D : D → D ⊗ Dadjuntas se denen como

ϕD = µD (µD ⊗ λD) (D ⊗ tD,D) (δD ⊗D),

%D = (µD ⊗D) (D ⊗ tD,D) (δD ⊗ λD) δD.Entonces los morsmos ωa

D = ϕD (ηD ⊗D) : D → D y ωcD = (εD ⊗D) %D : D → D son

idempotentes en C.Si para x ∈ a, c, se designan por Ωx(D), pxD : D → Ωx(D) y ixD : Ωx(D)→ D el objeto

imagen de ωxD y los mormos tales que ixD pxD = ωx

D y pxD ixD = idΩx(D), entonces se cumpleque:

(i) El triple (Ωa(D), ϕΩa(D), ρΩa(D)) es un objeto enDDYD con ϕΩa(D) = paDϕD(D⊗iaD)

y %Ωa(D) = (D ⊗ paD) δD iaD : Ωa(D)→ D ⊗ Ωa(D).

4 C. SONEIRA CALVO

(ii) El triple (Ωc(D), ψΩc(D), %Ωc(D)) es un objeto en DDYD con ψΩc(D) = pcD µD (D⊗

icD) : D ⊗ Ωc(D)→ Ωc(D) y %Ωc(D) = (D ⊗ pcD) %D icD : Ωc(D)→ D ⊗ Ωc(D).

References

[1] J. N. Alonso Álvarez, J.M. Fernández Vilaboa, R. González Rodríguez, Weak Hopf algebras and weakYang-Baxter operators, J. of Algebra 320 (2008), 2101-2143.

[2] J. N. Alonso Álvarez, J.M. Fernández Vilaboa, R. González Rodríguez, Weak Braided Hopf Algebras,Indiana University Mathematics Journal 57 No. 5 (2008), .

[3] J. N. Alonso Álvarez, J.M. Fernández Vilaboa, R. González Rodríguez, C. Soneira Calvo, Projectionsof Weak Braided HopfAlgebras, Science China Math. 54 No. 5 (2011), 877-906.

[4] J. N. Alonso Álvarez, J.M. Fernández Vilaboa, R. González Rodríguez, C. Soneira Calvo, The monoidalcategory of Yetter-Drinfeld modules over weak braided Hopf algebras, preprint.

[5] Böhm, G, F. Nill, K. Szlachányi, Weak Hopf algebras, I. Integral theory and C∗-structure, J. of Algebra221 (1999), 385-438.

[6] A. Nenciu, The center construction for weak Hopf algebras, Tsukuba J. Math. 26 (2002), 189-204.[7] D. E. Radford, The structure of Hopf algebras with projection, J. of Algebra 92 (1985), 322-347.[8] D.E. Radford, J. Towber, Yetter-Drinfeld categories associated to an arbitrary bialgebra, J. of Pure and

Applied Algebra 87 (1993), 259-279.[9] M. Takeuchi, Survey of braided Hopf algebras, Contemp. Math. 267 (2000), 301-323.[10] D. N. Yetter, Quantum gropus and representations of monoidal categories, Math. Proc. Cambridge

Philos. Soc. 108 (2) (1990), 261-290.

Universidade da Coruña

E-mail address: carlos.soneira@udc.es

STRONGLY PRESERVER PROBLEMS

M. BURGOS

Let A be an algebra. An element b ∈ A is called the Drazin inverse ofa ∈ A, if

ab = ba, bab = b, and akba = ak for some positive integer k.

The least of such k is the index of a, denoted by ind(a), and when ind(a) = 1,b is called the group inverse of a.

The Drazin inverse, and hence the group inverse, is unique whenever itexists. Let AD and AG denote the sets of all Drazin and group invertibleelements in A, respectively. Similarly aD and aG denote the Drazin inverseand group inverse of a, respectively.

Every unital Jordan homomorphism between Banach algebras stronglypreserves invertibility, that is, T (a−1) = T (a)−1, for every invertible ele-ment a ∈ A. Moreover, Hua’s theorem (see [6]) states that every unitaladditive map between fields that strongly preserves invertibility is eitheran isomorphism or an anti-isomorphism. In [1, 11] the authors started thestudy of Hua’s type characterizations for Banach algebras: if T : A → B is aJordan homomorphism between Banach algebras, then it strongly preservesgroup invertibility ( T (aG) = T (a)G for every a ∈ AG) and Drazin invertibil-ity (T (aD) = T (a)D for every a ∈ AD); see [11, Theorem 2.1]. Conversely,if T : A → B is a linear map and T (1) = 1 (respectively, T (1) is invertibleor 1 ∈ T (A)), strongly preserving invertibility, group invertibility or Drazininvertibility, then T (respectively T (1)T ) is a unital Jordan homomorphism(and T (1) commutes with the image of T ), [1, Theorem 4.2].

Recall that an element a ∈ A is regular if there is b ∈ A such that aba = aand b = bab. When A is a C∗-algebra, we say that b is a Moore-Penroseinverse of a if a = aba, b = bab and ab and ba are selfadjoint. It is knownthat every regular element a in A has a unique Moore-Penrose inverse thatwill be denoted by a†.

There exists a wider class of complex Banach spaces containing all C∗-algebras in which the notion of regularity makes sense and extends the con-cept given for C∗-algebras. We refer to the class of JB*-triples introducedby W. Kaup in [8]. Every C*-algebra is a JB*-triple via the triple productgiven by

x, y, z =12(xy∗z + zy∗x),1

2 M. BURGOS

and every JB*-algebra is a JB*-triple under the triple product

x, y, z = (x y∗) z + (z y∗) x− (x z) y∗.

For each x in a JB*-triple E, Q(x) will stand for the conjugate linear oper-ator on E defined by Q(x)(y) = x, y, x.

An element a in a JB*-triple E is called von Neumann regular if thereexists (a unique) b ∈ E such that Q(a)(b) = a, Q(b)(a) = b and Q(a)Q(b) =Q(b)Q(a), or equivalently Q(a)(b) = a and Q(a)(b[3]) = b (see [5, 9, 10]).The element b is called the generalized inverse of a. Denote by E∧ the setof regular elements in the JB*-triple E, and for an element a ∈ E∧ let a∧

denotes its generalized inverse.For a C∗-algebra A, an element a in A has Moore-Penrose inverse b, if

and only if a has generalized inverse b∗ in the JB*-triple structure. That isa∧ = (a†)∗ = (a∗)†.

Let A and B be C∗-algebras. We say that a linear map T : A → Bstrongly preserves regularity if T (a∧) = T (a)∧, for all a ∈ A∧. Notice thatif T is selfadjoint, then T strongly preserves Moore-Penrose invertibility ifand only if T strongly preserves regularity.

We are concerned with the study of additive and linear maps stronglypreserving Drazin, group and generalized invertibility.

Let A and B be Banach algebras. Assume that A is unital. We provethat an additive map T : A → B strongly preserves Drazin (or equivalentlygroup) invertibility, if and only if T is a Jordan triple homomorphism. WhenA and B are C∗-algebras, we characterize the linear maps strongly preservinggeneralized invertibility (in the Jordan systems’ sense), and as consequencewe determine the structure of selfadjoint linear maps strongly preservingMoore-Penrose invertibility.

References

[1] N. Boudi, M. Mbekhta, Additive maps preserving strongly generalized inverses. J.Operator Theory 64 (2010) 117-130.

[2] M. Burgos, A. C. Marquez-Garcıa, A. Morales-Campoy, Linear maps strongly preserv-ing Moore-Penrose invertibility, to appear in OaM.

[3] M. Burgos, A. C. Marquez-Garcıa, A. Morales-Campoy, Strongly preserver problemsin Banach algebras and C*-algebra, to appear in LAA.

[4] M. P. Drazin, Pseudo-inverse in associative rings and semigroups, Amer. Math.Monthly 65 (1958) 506–514.

[5] A. Fernandez Lopez, E. Garcıa Rus, E. Sanchez Campos, M. Siles Molina, Strongregularity and generalized inverses in Jordan systems, Comm. Alg. 20 (1992) 1917–1936.

[6] L. K. Hua, On the automorphisms of a sfield, Proc. Nat. Acad. Sci. U.S.A. 35 (1949)386–389.

[7] N. Jacobson, Structure and representation of Jordan algebras. Amer. Math. Coll. Publ.39. Providence, Rhode Island (1968).

STRONGLY PRESERVER PROBLEMS 3

[8] W. Kaup, A Riemann Mapping Theorem for bounded symmentric domains in complexBanach spaces, Math. Z. 183 (1983) 503–529.

[9] W. Kaup, On spectral and singular values in JB*-triples, Proc. Roy. Irish Acad. Sect.A, 96 (1996) 95-103.

[10] O. Loos, Jordan Pairs, Lecture Notes in Math., vol. 460, Springer-Verlag, Berlin,1975.

[11] M. Mbekhta, A Hua type theorem and linear preserver problems, Math. Proc. Roy.Irish Acad. 109 (2009) 109–121.

Campus de Jerez, Facultad de Ciencias Sociales y de la Comunicacion Av.de la Universidad s/n, 11405 Jerez, Cadiz, Spain

E-mail address: maria.burgos@uca.es

FAITHFUL ACTIONS ON DIFFERENTIAL GRADED ALGEBRAS

DETERMINE THE ISOMORPHISM TYPE OF FINITE GROUPS

CRISTINA COSTOYA AND ANTONIO VIRUEL

Abstract. It is well known that the isomorphism type of a nite group cannot be decidedby enumerating the set of vector spaces (abelian groups) on which that group acts faithfully.In contrast with this situation, we show that the isomorphism type of a nite group isdetermined by the set of dierential graded algebras on which that group acts faithfully.

1. Introduction

A classical problem in Mathematics is the Isomorphism Problem:

Question 1.1. Given a category C, the Isomorphism Problem (IP in what follows) in Cconsists of providing an algorithm or procedure to determine whether or not two objects inC are isomorphic. If such a procedure exists and the techniques t in some theory T , we saythat T tells objects in C apart.

Well known examples of the IP are Dehn's Finite Presentation Problem [2] and the GraphIP [8, 6]. Here we consider the Finite Group IP by means of Representation Theory:

Does T = Representation Theory" tell nite groups apart?The answer to the question above depends on what is meant by Representation Theory,

and how it is used to compare nite groups.If we consider Representation Theory just as Linear Representation of Groups, and we

compare two nite groups by looking at their set of modules, the answer to Question 1.1 isnegative: in [7] it is shown that there exist two non isomorphic nite groups G and H, bothof size 2219728, such that Z[G] ∼= Z[H] as rings, thus every G-module admits an H-modulestructure.

Our approach to Representation Theory is broader, since we consider actions on rationalDierential Graded Algebras (DGA's for short). In this setting the answer to Question 1.1is positive:

Theorem 1.2. Let G and H be nite groups, and (A, d) be a nitely generated rationalDGA. Then the following statements are equivalent:

• G and H are isomorphic.• G ≤ Aut(A, d) if and only if H ≤ Aut(A, d).

2. Differential Graded Algebras

We follow the notation in [3].

Denition 2.1. A Graded Module V over K, is a family Vii∈Z of K-modules. By abuseof language" we say that v ∈ Vi is an element of V of degree i, and we write |v| = i.

1

2 CRISTINA COSTOYA AND ANTONIO VIRUEL

Denition 2.2. A Graded Algebra over K is a K-graded module R together with an associa-tive linear map of degree zero, R⊗R→ R, x⊗y 7→ xy, that has an identity 1 ∈ R. A deriva-tion of degree k is a linear map θ : R→ R of a degree k such that θ(xy) = θ(x)y+(−l)k|x|θ(y).

Denition 2.3. A Dierential Graded Algebra over K (DGA for short) is a K-gradedalgebra R together with a dierential in R that is a derivation. A morphism of dierentialgraded algebras f : (R, d)→ (E, d) is a morphism of graded algebras satisfying fd = df .

We are interested in the case K = Q, that is, in rational DGA's, and we consider onlythe commutative case. Here commutative means commutative in the graded sense, so ab =(−1)|a|·|b|ba.

Our interest for these DGA's comes from homotopy theory: for any topological space XSullivan [9] dened a commutative DGA APL(X), called the algebra of polynomial dier-ential forms on X with rational coecients. This DGA codies the rational homotopy typeof X.

3. Groups to Graphs to DGA's

A graph is a pair G = (V,E) of sets such that E ⊆ V 2. The elements v of V are thevertices of G, the elements of E are its edges and we will write them as couples (v, w) ofvertices. For G = (V,E) and G′ = (V ′, E′) we will say that G and G′ are isomorphic if thereexists a bijection σ : V → V ′ with (v, w) in E if and only if (σ(v), σ(w)) ∈ E′ for every(v, w) in E. Such a map is called an isomorphism; if G = G′ it is called an automorphism. Inthis work we only consider simple graphs which implies they are non directed graphs, thatis if (v, w) is in E then (w, v) is also in E, and they have no loops, that is (v, v) 6∈ E.

Groups actions on nite graphs tell nite groups apart. This follows by the result ofFrucht in [4, 5]:

Theorem 3.1. Given a nite group G, there exist innitely many non-isomorphic connected(nite) graphs G whose automorphism group is isomorphic to G.

Now, given a nite graph G, one can construct a nitely generated rational DGA (A, d)whose automorphisms are closely related to those of G. This follows by previous results ofthe authors in [1, Section 2]:

Theorem 3.2. Let G = (V,E) be a nite graph without isolated vertices, and let

M =(Λ(x1, x2, y1, y2, y3, z)⊗ Λ(xv, zv|v ∈ V ), d

)be the rational DGA where dimensions and dierential are

(1)

|x1| = 8, d(x1) = 0|x2| = 10, d(x2) = 0|y1| = 33, d(y1) = x31x2|y2| = 35, d(y2) = x21x

22

|y3| = 37, d(y3) = x1x32

|xv| = 40, d(xv) = 0,|z| = 119, d(z) = y1y2x

41x

22 − y1y3x51x2 + y2y3x

61 + x151 + x122

|zv| = 119, d(zv) = x3v +∑

(v,w)∈E xvxwx42.

FAITHFUL ACTIONS ON DIFFERENTIAL GRADED ALGEBRAS DETERMINE THE ISOMORPHISM TYPE OF FINITE GROUPS3

Then given f ∈ Aut(M), there exists σ ∈ AutG such that

(2)

f(x1) = x1f(x2) = x2f(y1) = y1f(y2) = y2f(y3) = y3f(xv) = xσ(v)f(z) = z + d(mz)f(zv) = zσ(v) + d(mzv)

with |mz| = |mzv | = 118 elements inM.

4. Proof of Theorem 1.2

Let G and H be nite groups, and (A, d) be a nitely generated rational DGA.If G ∼= H, then G ≤ Aut(A, d) if and only if H ≤ Aut(A, d).Assume now that G 6∼= H. Without loss of generality we may assume that H is not

(isomorphic to) a subgroup of G, that is H 6≤ G.According to Theorem 3.1, there exists a connected nite graph G such that Aut(G) = G.

LetM be the rational DGA associated to G by means of Theorem 3.2.Projection over the module of indecomposable elements of M provides an split exact

sequence of groupsK −→ Aut(M) −→ Aut(G) = G

where f ∈ K if and only iff(x1) = x1f(x2) = x2f(y1) = y1f(y2) = y2f(y3) = y3f(xv) = xvf(z) = z + d(mz)f(zv) = zv + d(mzv)

where |mz| = |mzv | = 118 elements inM, that is f(mz) = mz and f(mzv) = mzv .Therefore K is torsion free, and since Aut(M) = KoG, every a maximal nite subgroup

of Aut(M) is (up to isomorphisms) a subgroup of G.Recall that H 6≤ G, then H 6≤ Aut(M).

References

[1] C. Costoya, A. Viruel, Every nite group is the group of self homotopy equivalences of an elliptic space,arXiv:1106.1087v1.

[2] M. Dehn, Über unendliche diskontinuierliche Gruppen, Math. Ann. 71 (1911), 116144.[3] Y. Felix, S. Halperin, J.C. Thomas, Rational homotopy theory. Graduate Texts in Mathematics, 205.

Springer-Verlag, New York, 2001.[4] R. Frucht, Herstellung von Graphen mit vorgegebener abstrakter Gruppe, Compositio Math., 6 (1939),

239250.[5] R. Frucht, Graphs of degree 3 with given abstract group, Canad. J. Math., 1 (1949), 365378.

4 CRISTINA COSTOYA AND ANTONIO VIRUEL

[6] G. Gati, Further annotated bibliography on the isomorphism disease, J. of Graph Theory 3 (1979),95109.

[7] M. Hertweck, A counterexample to the isomorphism problem for integral group rings, Ann. of Math. 154(2001), 115138.

[8] R. Read and D. Corneil, The graph isomorphism disease, J. of Graph Theory 1 (1977), 339363.[9] D. Sullivan, Innitesimal computations in topology, Publications Mathématiques de l'IHÉS 47 (1977),

269331.

Dep. de Computación, Álxebra, Universidade da Coruña, Campus de Elviña, s/n, 15071 ACoruña, Spain.

E-mail address: cristina.costoya@udc.es

Dep. de Álgebra, Geometría y Topología, Campus de Teatinos, Universidad de Málaga, 29071Málaga, Spain

E-mail address: viruel@agt.cie.uma.es

GROBNER FINITE PATH ALGEBRAS

MICAH J. LEAMER

Let k be a field and Γ a finite directed multi-graph. In this talk we will give acomplete description of all path algebras kΓ and admissible orders with the propertythat all of their finitely generated ideals have finite Grobner bases and of those whichcontain a finitely generated ideal whose Grobner bases are all infinite.

We will define Grobner finite path algebras to be the path algebras such thatall of their finitely generated ideals have finite Grobner bases. We go on to classifywhich path algebras are Grobner finite. Furthermore we offer a simple descriptionof the orders in which Grobner finite path algebras have finite Grobner bases. Theseresults may be found in [2].

A graph Γ may have arrows from a vertex to itself or multiple arrows betweenthe same set of vertices. Let β denote the set of paths of finite length, β0 denotethe set of vertices and β1 denote the set of arrows. Arbitrary vertices and arrowswill be denoted by vi and αi respectively. We define functions o : β → β0 andt : β → β0, such that for any path p ∈ β, o(p) is the origin or first vertex of apath p and t(p) is the terminus or last vertex of a path p. When writing paths asa product of arrows the convention will be to write a path, α1α2 · · ·αn, from leftto right, such that t(αi) = o(αi+1). For a path p, define the length function l(p)to be the number of arrows that occur in a path p, counting multiplicities. Twopaths will be said to intersect if they share a common vertex. A cycle is a paththat begins and ends at the same vertex. A trivial cycle is a path of zero lengthbeginning and ending at the same vertex.

We define multiplication of paths, such that for p, q in β, if t(p) = o(q), thentheir product pq is the path adjoining p and q by concatenation. Otherwise, ift(p) 6= o(q) then pq = 0. The path algebra kΓ is defined to be the set of all finitelinear combinations of paths in β with coefficients in k. Addition in kΓ is the usualk−vector space addition, where β is a k-basis for kΓ. Multiplication in the pathalgebra, kΓ, extends k-linearly from the definition for multiplication of paths. Notethat the identity element is always of the form

∑v∈β0

v.A Grobner basis for an ideal in a path algebra is dependent upon choosing an

ordering for the paths in β. A path order < is considered to be an admissible orderif it satisfies the following four conditions, for all p, q, r, s ∈ β:

(1) If p 6= q, then p < q or q < p;(2) Every nonempty set of paths has a least element;(3) If p < q and rps, rqs 6= 0, then rps < rqs (r and s may be trivial); and(4) If p = qr 6= 0, then p ≥ q and p ≥ r.

Definition 1. Let x =∑ni=1 γipi ∈ KΓ with γi ∈ K − 0. Then the support of x

is the set Supp(x) = p1, . . . , pn.1

2 MICAH J. LEAMER

Definition 2. Given an admissible order <, for any nonzero x ∈ KΓ, the tip ofx, denoted Tip(x), is the largest path in Supp(x). That is, Tip(x) ∈ Supp(x) andfor all p ∈ Supp(x), p ≤ Tip(x).

Definition 3. Given X ⊂ KΓ the set p ∈ β| p = Tip(x) for some x ∈ X isdenoted as Tip(X).

Definition 4. Let I be an ideal in the path algebra KΓ with admissible order <.We say a set G ⊂ I is a Grobner basis for I when for all x ∈ I − 0, there existsg ∈ G such that Tip(g) divides Tip(x).

Let X be a subset of KΓ and y ∈ KΓ. We say y can be reduced by X if forsome p in Supp(y), there exists x ∈ X such that Tip(x) divides p. A reduction ofy by X is given by y − kpxq, where x ∈ X, p, q ∈ β, and k ∈ K − 0 such thatkp(Tip(x))q is a term in y. A total reduction of y by X is an element resulting froma sequence of reductions that cannot be further reduced by X. In general, two totalreductions of an element y need not be the same element. However, when X is aGrobner basis for an ideal all total reductions y produce the same element. We sayan element y reduces to 0 by X if there is a total reduction of y by X which is 0.A set X ⊂ KΓ is said to be reduced if for all x ∈ X, x cannot be reduced by X−x.

Definition 5. The unique reduced monic Grobner basis is called the reduced Grobnerbasis.

Given an ideal I in the path algebra KΓ with admissible order if the reducedGrobner basis is not finite then no other Grobner basis will be finite.

Definition 6. Let f, g ∈ KΓ, with admissible order < on KΓ. Suppose there arepaths p and q, of positive length, such that Tip(f)p = qT ip(g), with l(p) < l(Tip(g)).Then f and g have an overlap relation, denoted o(f, g, p, q), given by

o(f, g, p, q) = c−1Tip(f)fp− c−1Tip(g)qg .

Given elements f and g whose tips overlap, the p and q will not necessarily beunique and consequently the same two elements f and g may have multiple overlaprelations. Additionally an element may have an overlap relation with itself i.e.o(f, f, p, q) is possibly an overlap relation.

Lemma 7. [1] Let G be a set of uniform elements that form a generating set for theideal I ⊂ KΓ, such that for all g, g′ ∈ G, T ip(g) does not divide Tip(g′). Supposethat for each f, g ∈ G every overlap relation o(f, g, p, q) reduces to 0 by G. Then, Gis a Grobner basis for the ideal I.

The following iterative construction produces a Grobner basis in the limit:

• Let G0 be generating set for an ideal I;• G′i = Gi ∪ o(f, g, p, q)| f, g ∈ Gi and p, q are overlap relations ;• Gi+1 is any reduced set produced by repeatedly replacing elements x ∈ G′i

with a reduction of x by G′i \ x.Then G = g| g ∈ Gi for i 0 is a reduced Grobner basis for I. Also for all d

there exists N such that for all n ≥ N we have

g ∈ G| l(g) ≤ d = g ∈ Gn| l(g) ≤ d.

GROBNER FINITE PATH ALGEBRAS 3

However, for an arbirary ideal over arbitrary path algebra the value of N for a givend is indeterminable. In fact the ability to determine such an N in general wouldimply a solution to the word problem.

Definition 8. Define a path algebra to be Grobner finite if there is an admissibleorder < such that all of its finitely generated ideals have finite reduced Grobnerbases.

The following examples are key to determining when a path algebra contains afinitely generated ideal with an infinite Grobner basis under a specific admissibleorder.

Example 9. Let Γ be the graph given below.

p // //////////// BB <<// "" b

BB <<// ""

b′

[[ ||oobb\\[[

q ////////////// DD BB <<// ""

c

[[ ||oobb\\[[

Let < be an admissible order on the paths of Γ such that for some i ∈ N we haveqci > b′bq. Let i be fixed for this example. In this case the 2-generated ideal 〈pbq,qci−b′bq〉 of kΓ has an infinite reduced Grobner basis pb(b′b)jq, qci−b′bq| j ∈ N0.

However, for any admissible order < such that qci < b′bq the ideal 〈pbq, b′bq−qci〉has a finite Grobner basis pbq, b′bq− qci. This is readily apparent since there areno overlap relations between the elements of the set pbq, b′bq − qci.

Example 10. Let Γ be the graph given below.

p1 // //////////// BB <<// "" p2

BB <<// ""

p3

[[ ||oobb\\[[

p4 ////////////// DD

p5

BB <<// ""

p6

[[ ||oobb\\[[

p7 //////////////

Let < be an admissible order on the paths of Γ. If p4p5p6 > p3p2p4, then the ideal〈p1p2p4, p4p5p6 − p3p2p4〉 has reduced Grobner basis

p1p2(p3p2)ip4, p4p5p6 − p3p2p4| ∈ N0.

Else if p3p2p4 > p4p5p6, then the ideal 〈p4p5p7, p3p2p4 − p4p5p6〉 of kΓ has infinitereduced Grobner basis

p4(p5p6)ip5p7, p3p2p4 − p4p5p6| i ∈ N0.

It follows that kΓ has a finitely generated ideal with an infinite reduced Grobnerbasis, under any admissible order.

Example 11. Now let Γ be a graph which contains two nontrivial cycles P and Q,which intersect at a vertex v. Let p be the path from v to itself that goes around Ponce and let q be the path from v to itself that goes around Q once. If pq2 > p2q,then the ideal 〈pqpq, pq2−p2q〉 has reduced Grobner basis pqpiq, pq2−p2q| i ∈ N.Else if p2q > pq2, then the ideal 〈pqpq, p2q− pq2〉 of KΓ has reduced Grobner basispqipq, p2q − pq2| i ∈ N. Therefore, under any admissable order a path algebrawhich contains two intersecting cycles, also contains a finitely generated ideal withan infinite reduced Grobner basis.

4 MICAH J. LEAMER

Theorem 12. A path algebra kΓ with admissible ordering < contains a finitelygenerated ideal with infinite reduced Grobner basis if and only if the graph Γ andthe order < satisfy one of the following conditions.(1)Γ contains 2 intersecting cycles.(2)Γ contains a subgraph of the form

p ////////////// DDb

BB <<// ""

b′

[[ ||oobb\\[[

q ////////////// DD BB <<// ""

c

[[ ||oobb\\[[

where l(b) may be zero, with b′bq < qci for some i.(3)Γ contains a subgraph of the form

BB <<// "" BB <<// ""

a

[[ ||oobb\\[[

p ////////////// DDb

BB <<// ""

b′

[[ ||oobb\\[[

q //////////////

where l(b) may be zero, with pbb′ < aip for some i.

Remark 13. A path algebra kΓ is Noetherian, if and only if, Γ both enters andexits a non-trivial cycle.

Corollary 14. A path algebra kΓ is Grobner finite if and only if no path on Γenters and exits two distinct non-trivial cycles.

By the unsolvability of the word problem it follows that whenever Γ contains twodistinct non-trivial intersecting cycles then kΓ contains a finitely generated idealfor which it is impossible to determine a Grobner basis under any admissible order.

Question 15. Which path algebras have the property that every finitely generatedideal has a determinable Grobner basis under some admissible order?

References

[1] G. Bergman: The diamond lemma for ring theory. Adv. Math. 29, 178-218 (1978).

[2] M. Leamer, Grobner finite path algebras, J. Symb. Comp., 41-1 (2006), 98-111.ract

A GRADED CRITERION IN THE CLASSIFICATION OF COFINITEHOMOGENEOUS IDEALS

ÓSCAR CORTADELLAS, PASCUAL JARA, AND F. J. LOBILLO

Abstract. A study of finiteness in a kind of finitely presented quotient algebras is dis-played in this paper. The relations generating the ideals are given by monomials or bi-nomials of same length, in order to obtain homogeneous computations. These ideals areparametrized by 3-tuples (a, b, c), being a the number of variables, b the length of mono-mials and c the number of relations conforming the ideal. We focus on the analysis ofthe (2, 3, 4)-family, constituted by 58905 elements, compute all the possible ideals, obtainthe corresponding Gröbner-Shirshov basis (up to degree 30) of each one and give informa-tion about the number of cofinite ideals in this family and the largest reached dimension.Elements of this maximal sub-family will be classified up to isomorphism, for which wedevelop a new procedure based on projections on generating sets, which is lighter thanother known routines and constitutes one of the main results of this work (Theorem 2.4).More useful information about inherent computations will be appearing during the paper,as well as some technical results over isomorphism classes of this type of graded algebras.

Introducción

En el presente trabajo vamos a abordar y resolver varias cuestiones relacionadas con lanaturaleza cofinita de algunas familias de ideales homogéneos, que van desde 45 elementoshasta varios millones. Como elemento destacado desarrollaremos un procedimiento paradeducir cuándo dos álgebras relacionadas con estos ideales son isomorfas, que mejora otrosmétodos vistos anteriormente.

Dado un álgebra de polinomios k⟨X⟩ y un ideal I E k⟨X⟩, decimos que I es un idealcofinito si el álgebra cociente k⟨X⟩/I tiene dimensión finita.

Los ideales que son objeto de este estudio son ideales de polinomios no conmutativosque están parametrizados por tres variables (a, b, c). La variable a representa el número deindeterminadas del conjunto X, b es la longitud de los monomios o binomios que conformanlas relaciones que generan el ideal y c es el número de relaciones que componen cada ideal.

De esta manera los ideales que estamos estudiando son homogéneos. Esta restricción estátomada para poder hacer cálculos consistentes sobre el álgebra cociente, ya que en estos casosuna base de Gröbner-Shirshov, aunque infinita, puede calcularse de manera exhaustiva gradoa grado.

Tenemos por tanto un doble objetivo. Por un lado estudiar el número de ideales cofinitosen cada familia y por otro estudiar cuándo de dos de estos ideales se derivan álgebrascocientes isomorfas.

Todos los cálculos respectivos a reducciones, palabras normales o bases de Gröbner-Shirshov han sido realizados por el programa Bergman ([1]). Para poder controlar mejorlos cálculos, estos se han desarrollado hasta grado 30.

1

2 ÓSCAR CORTADELLAS, PASCUAL JARA, AND F. J. LOBILLO

1. Buscando ideales cofinitos maximales en una (a, b, c)-familia

En principio estamos interesados en el estudio de la familia (2, 3, 4). Esto quiere decir quelos ideales que queremos estudiar están constituidos por cuatro relaciones formadas por ladiferencia de dos monomios de longitud tres en dos variables. Una cuenta sencilla nos diceque hay 58905 elementos en esta familia. La finitud o infinitud de cada una de las álgebrascocientes que se deducen a partir de cada ideal se estudiará a partir de la serie de Hilbertde cada ideal.

Para este fin hemos construido una rutina en C++ que usa como motor de cálculo elsistema Bergman y que nos proporciona esta información. Además nos clasifica cada uno delos ideales de acuerdo a la dimensión del álgebra cociente asociada.

Básicamente lo que hace el programa es construir todos los ideales que conforman esafamilia y calcular la serie de Hilbert asociada a cada álgebra cociente, vía el cálculo desu base de Gröbner-Shirshov. Luego revisa los coeficientes de la serie para comprobar sitiene una base finita y, si la tiene, suma el número de elementos de la base para calcular ladimensión. Finalmente agrupa toda la información en un archivo y nos lo devuelve.

En particular en esta familia tenemos 10142 ideales cofinitos, que se reparten de la sigu-iente manera

288 ideales que generan álgebras de dimensión 112446 ideales que generan álgebras de dimensión 12,3578 ideales que generan álgebras de dimensión 13,1246 ideales que generan álgebras de dimensión 14,2146 ideales que generan álgebras de dimensión 15,92 ideales que generan álgebras de dimensión 16,174 ideales que generan álgebras de dimensión 17,88 ideales que generan álgebras de dimensión 18,8 ideales que generan álgebras de dimensión 19,72 ideales que generan álgebras de dimensión 21 y4 ideales que generan álgebras de dimensión 25.

Luego tenemos que la dimensión finita máxima de las álgebras cocientes es 25 y se alcanzaen 4 casos. Podemos prácticamente asegurar que si un álgebra en esta familia tiene dimensiónsuperior a 25 va a ser no finita. Pero, ¿realmente hay cuatro casos diferentes o hay menos?

2. Clases de isomorfía. El criterio graduado

Tras las molestias tomadas en la sección anterior para la búsqueda de ideales cofinitos,nos enfrentamos ahora al problema de clasificar las álgebras finitas encontradas, salvo iso-morfismos.

Hay varias formas de abordar este tipo de problemas: a través de su serie de Hilbert,estudiando sus grafos de Ufnarovskii ([2]), usando los estudios de combinatoria de Kostrikiny Shafarevich ([3]) y otras muchas formas. En particular nos encontramos con el algoritmodiseñado por Shirayanagi ([4]), donde el problema se reduce a demostrar si un elementodeterminado pertenece a una base de Gröbner (conmutativa).

Vamos a estudiar si las dos álgebras cocientes asociadas a los ideales

I1 = ⟨xxx, xyx, xxx− xyx, xyy − yyy⟩I2 = ⟨yxy, yyy, xxx− yxx, yxy − yyy⟩

A GRADED CRITERION IN THE CLASSIFICATION OF COFINITE HOMOGENEOUS IDEALS 3

son isomorfas. Estos dos ideales pertenecen a la familia maximal que hemos calculado enel apartado anterior. Como primer paso podemos observar que tienen la misma serie deHilbert

HAj = 1 + 2x+ 4x2 + 5x3 + 5x4 + 4x5 + 3x6 + x7, j = 1, 2

y que por lo tanto las álgebras asociadas tienen dimensión 25.Para poder aplicar el criterio de Shirayanagi, ahora tenemos que comprobar si un ele-

mento (en este caso de grado 90) pertenece a la base de Gröbner de un ideal (conmutativo)generado por cientos de elementos sobre 51 variables. Aunque técnicamente es posible,computacionalmente es inviable. De hecho la respuesta es que son isomorfas.

Vamos a sentar las bases para aplicar nuestro criterio graduado. Sean A = k⟨X⟩/IA yB = k⟨X⟩/IB dos álgebras finitamente presentadas, IA and IB dos ideales homogéneos y HA

y HB las correspondientes series de Hilbert. Supongamos que HA = HB (es un invariante

para álgebras graduadas), por lo que A =m⊕i=0

Ai, B =m⊕i=0

Bi, y que dim(Aj) = dim(Bj),

∀j. Además, A0 = B0 = k y A,B están generadas como k-álgebras por los subespacioscorrespondientes de dimensión uno.

Lema 2.1. Usando la notación anterior, si φ : A → B es un isomorfismo, φ(A1)∩B1 = ∅,

de lo cual podemos deducir el siguiente

Corolario 2.2. Si φ : A → B es un isomorfismo, entonces

φ(Ai) ⊆ Bi ⊕Bi+1 ⊕ · · · ⊕Bm con φ(Ai) ∩Bi = Bi

Esto nos viene a decir que la imagen de un elemento de grado i siempre tiene una secciónde grado i y, eventualmente, otra sección de grado mayor estricto que i.

La parte de grado i de la imagen de un elemento de grado i se llamará cabeza y a la partede grado superior la llamaremos cola.

Corolario 2.3. La matriz asociada a φ es triangular por bloques, donde cada bloque tienedimensión dim(Ai) =dim(Bi).

Sea φ : A → B un isomorfismo que verifica las condiciones anteriores. Definamos unhomomorfismo como

φ : A1 → B1

f 7→ π1 φ(f)donde π1 es la proyección sobre la componente B1. Como A1 genera A, podemos extenderφ a un homomorfismo de álgebras φex : A → B.

Teorema 2.4. Sea φ : A → B un isomorfismo entre A and B. Entonces φex define unisomorfismo graduado entre A to B.

Demostración. Vamos a probar que φ(A1) ∼= B1. Si esto fuera cierto, entonces φex(Ai) =φex(Ai

1) = φ(A1)i ∼= Bi

1 = Bi.Empecemos probando que la Imφ = π1 φ(f)\f ∈ A1 = B1. Supongamos que es falso,

luego existe un elemento g ∈ B1 tal que g /∈ Imφ. Entonces no existe un h ∈ A1 tal queπ1φ(h) = g. Pero los elementos de grado 1 de B solo pueden ser obtenidos como imagen deelementos de grado 1 en A, por el colorario (2.2). Entonces, si tal g existe, φ(A1)∩B1 = B1

y φ no podrá ser un isomorfismo.

4 ÓSCAR CORTADELLAS, PASCUAL JARA, AND F. J. LOBILLO

Por el otro lado, sea M la matriz asociada a φ (M será una matriz cuadrada de dimensióndim(A1)). Si det(M) = 0, entonces el determinante del menor correspondiente a los elemen-tos de grado uno en la matriz M asociada a φ también será cero. Por el colorario (2.3) lamatriz es triangular por bloques, luego det(M) = 0 y φ no puede ser un isomorfismo.

A partir de este resultado podemos deducir una relación directa entre el conjunto deisomorfismos entre A y B y el conjunto de isomorfismos graduados entre A y B. Conse-cuentemente, de cada isomorfismo φ : A → B podemos obtener un isomorfismo graduadoeliminando la cola de dimiensión mayor que uno de la imagen de los generadores. Y porotro lado, cada isomorfismo graduado φ : A → B puede extenderse a un isomorfismo generalencontrando unos coeficientes adecuados que respeten los elementos del bloque de dimensiónuno.

Acabamos de demostrar que todo isomorfismo da lugar a un isomorfismo graduado, luegopara comprobar si dos álgebras son isomorfas nos basta con buscar este isomorfismo gradu-ado, que además podemos ir construyendo grado a grado. En cada nuevo paso añadimos, siexistieran, nuevas restricciones sobre nuestros parámetros. Si estas restricciones resultan nocompatibles, entonces no puede existir el isomorfismo graduado y no serán isomorfas. Comoel álgebra que estamos estudiando es finita, este algoritmo termina irremediablemente dandocondiciones para que podamos definir el isomorfismo o con la certeza que ambas álgebras noson isomorfas.

Incluso podemos realizar algunos ajustes más para acelerar este proceso y evitar algunoscálculos redundantes. Estas mejoras se basan sobre todo en la estructura graduada delproblema.

References

[1] Jörgen Backelin et al, Bergman, http://servus.math.su.se/bergman/[2] Vicktor Ufnarovskii, 1982. A growth criterion for graphs and algebras defined by words, Matematicheskie

Zametki 31, no. 3, 465–472[3] Kostrikin A. I. and Shafarevich I. R (eds.), 1995. Algebra VI, Encyclopaedia of Mathematical Sciences,

vol. 5, Springer[4] Shirayanagi, k, 1993. Decision of algebra isomorphisms using Gröbner Bases, Computational algebraic

geometry, Birkhauser, 253–265

Departmento de Álgebra, Facultad de Ciencias, Universidad de Granada, Avda. Fuentenuevas/n, E-18071, Granada, España

E-mail address: ocortad@ugr.es, pjara@ugr.es

Departmento de Álgebra, Escuela Técnica Superior de Ingenierías Informática y de Tele-comunicación, Universidad de Granada, C/ Periodista Daniel Saucedo Aranda s/n, E-18071,Granada, España

E-mail address: jlobillo@ugr.es

THE CUNTZ SEMIGROUP OF CONTINUOUS FIELDS

RAMON ANTOINE, JOAN BOSA, AND FRANCESC PERERA

Abstract. In this talk we show that, for some continuous eldsA with noK1-obstructions,the Cuntz semigroup of A can be recovered as the sheaf of continuous sections on a topo-logical space. As a consequence, these continuous elds can be classied by their CuntzSemigroup.

The rst result is analogous to a result proved by M. Dadarlat and G. Elliott [2], whichdescribes the sheaf given by K0 in terms of the continuous sections, and the second onerephrases the classication obtained in [3].

Introducción

Al nal de la década de los 80, G.A. Elliott conjeturó que las C*-álgebras simples, sepa-rables y nucleares se podrían clasicar mediante un invariante basado en la Teoría K. Dichoinvariante se ha llamado, a posteriori, Ell(_). A pesar de los buenos resultados obtenidospor dicho programa, en la última década han aparecido ejemplos descritos por M. Rør-dam y A. Toms que han hecho reescribir la conjetura de Elliott([4] y [5]). En concreto, A.Toms mostró dos C*-àlgebras que tienen el mismo invariante Ell(_), pero que pueden serdiferenciadas por el invariante conocido como semigrupo de Cuntz.

El anterior hecho ha propiciado, en los últimos años, un estudio profundo de dicho semi-grupo. Uno de los mayores logros en esta línea de investigación fue la solución del problemade continuiad del semigrupo de Cuntz hallada por Coward-Elliott-Ivanescu en 2008 ([1]). Di-cho problema fue solucionado considerando el semigrupo de Cuntz del álgebra estabilizaday deniendo la categoria Cu, donde dicho semigrupo pertenece. Gracias a esta denición,construyeron un functor covariante de la categoria de C*-álgebras a la categoría Cu.

Utilizando esta propiedad de continuidad, el resultado que voy a mostrar explica que dadoun haz continuo en la categoría Cu

S(_) : VX → CuU 7→ S(U) ,

donde X es un espacio compacto Hausdor, se satisface que el conjunto de seccionescontinuas de X a FS := tx∈XSx, denotado por Γ(X,FS), es un semigrupo en la categoríaCu. Con este resultado podemos calcular el semigrupo de Cuntz de los Continuous Fieldssin obstrucciones en K1 y con X un espacio compacto Haudor de dimensión menor o igualque 1. Así, si B es un Continuous Field compliendo las anteriores hipótesis, obtenemos que

Cu(B) ∼= Γ(X,FCu(B)) .

Gracias a la equivalencia anterior, bajo las mismas hipótesis, tenemos que los haces1

2 RAMON ANTOINE, JOAN BOSA, AND FRANCESC PERERA

S(_) : VX → CuU 7→ S(A(U))

andCuA(_) : VX → Cu

U 7→ Cu(A(U)).

son isomorfos. Este hecho es una versión continua del resultado que establece el isomorsmoentre cualquier haz continuo y su correspondiente haz de secciones contínuas cuando el límitedirecto de la categoria es el límite algebraico ([6]).

Para terminar, combinando los anteriores resultados con el hecho que Cu(B) tiene unaacción inducida por la acción de C(X) en B, explicaré el siguiente Teorema.

Teorema 0.1. Sean A y B dos continuous elds en X tales que no tienen obstrucciones

de K1, todas sus bras tienen rango real zero y X es un espacio compacto Hausdor de

dimensión menor o igual que 1. Entonces las siguientes propiedades son equivalentes:

(1) Cu(A) ∼= Cu(B) preservando la acción de Cu(C(X)),(2) CuA(_) ∼= CuB(_),(3) VA(_) ∼= VB(_).

El anterior Teorema muestra que, bajo nuestras hipotesis, el semigrupo de Cuntz contienela misma información que los haces construidos mediante dicho semigrupo o el semigrupode Murray-von Neumann. Como corolario del anterior Teorema, somos capaces de reescribirla clasicación de ciertos Continuous Fields obtenida por Dadarlat-Elliott-Niu en términosdel semigrupo de Cuntz ([3]).

Corolario 0.2. Sean A,B dos Continuous Fields separables y unitales en [0, 1] tales que sus

bras son AF -álgebras. Entonces cualquier isomorsmo φ : Cu(A) → Cu(B) que preserve

la acción de Cu(C(X)) y que cumpla φ([1A]) = [1B] se eleva a un isomorsmo A ∼= B entre

los Continuous Fields.

References

[1] K.T. Coward, G.A. Elliott and C. Ivanescu, The Cuntz Semigroup as an invariant for C∗-algebras, J.Reine Angew. Math., 623, 161193, 2008.

[2] M. Dadarlat, G. A. Elliott,One-Parameter Continuous Fields of Kirchberg Algebras, Commun. Maht.Phys., 274, 795-819, 2007.

[3] M. Dadarlat, G.A. Elliott, Z. Niu, One-Parameter Continuous Fields of Kirchberg Algebras II, Canad.J. Math. 63 (3), 500532, 2011.

[4] M.Rørdam, A simple C*-algebra with a nite and an innite projection, Acta Math., 191 (1), 109142,2003.

[5] A.S. Toms, On the classication problem for nuclear C∗-algebras, Ann. of Math. (2), 167 (3), 10291044,2008.

[6] R. O. Jr. Wells,Dierential analisys on complex manifolds volume 65 of graduate texts in Mathematics.Second Edition. New-York: Springer-Verlag 1989.

Universitat Autònoma de Barcelona

E-mail address: ramon@mat.uab.cat, jbosa@mat.uab.cat, perera@mat.uab.cat