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London Mathematical Society Lecture Note series: 401

A Double Hall Algebra Approach toAffine Quantum SchurWeyl Theory

BANGMING DENGBeijing Normal University

JIE DUUniversity of New South Wales, Sydney

QIANG FUTongji University, Shanghai

C A M B R I D G E U N I V E R S I T Y P R E S SCambridge, New York, Melbourne, Madrid, Cape Town,

Singapore, So Paulo, Delhi, Mexico City

Cambridge University PressThe Edinburgh Building, Cambridge CB2 8RU, UK

Published in the United States of America by Cambridge University Press, New York

www.cambridge.orgInformation on this title: www.cambridge.org/9781107608603

c B. Deng, J. Du and Q. Fu 2012

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in this publication, and does not guarantee that any content on suchwebsites is, or will remain, accurate or appropriate.

We dedicate the book to our teachers:Peter GabrielShaoxue Liu

Leonard ScottJianpan Wang

2010 Mathematics Subject Classification. Primary 17B37, 20G43, 20C08;Secondary 16G20, 20G42, 16T20

Key words and phrases. affine Hecke algebra, affine quantum Schur algebra,cyclic quiver, Drinfeld double, loop algebra, quantum group, SchurWeyl

duality, RingelHall algebra, simple representation

Abstract

Over its one-hundred year history, the theory of SchurWeyl dualityand its quantum analogue have had and continue to have profoundinfluences in several areas of mathematics such as Lie theory, rep-resentation theory, invariant theory, combinatorial theory, and so on.Recent new developments include, e.g., walled Brauer algebras andrational Schur algebras, quantum Schur superalgebras, and the inte-gral SchurWeyl duality for types other than A. This book takes analgebraic approach to the affine quantum SchurWeyl theory.

The book begins with a study of extended RingelHall algebrasassociated with the cyclic quiver of n vertices and the GreenXiaoHopf structure on their Drinfeld doublethe double RingelHallalgebra. This algebra is presented in terms of Chevalley type and cen-tral generators and is proved to be isomorphic to the quantum loopalgebra of the general linear Lie algebra. The rest of the book inves-tigates the affine quantum SchurWeyl duality on three levels. Thisincludes

the affine quantum SchurWeyl reciprocity; the bridging role played by the affine quantum Schur algebra

between the quantum loop algebra and the corresponding affineHecke algebra;

Morita equivalence of certain representation categories; the presentation of affine quantum Schur algebras; and the realization conjecture for the double RingelHall algebra which

is proved to be true in the classical case.

Connections with various existing works by Lusztig, VaragnoloVasserot, Schiffmann, Hubery, ChariPressley, FrenkelMukhin, andothers are also discussed throughout the book.

Contents

Introduction page 1

1 Preliminaries 91.1 The loop algebra gln and some notation 91.2 Representations of cyclic quivers and RingelHall algebras 121.3 The quantum loop algebra U(sln) 171.4 Three types of generators and associated monomial bases 201.5 Hopf structure on extended RingelHall algebras 25

2 Double RingelHall algebras of cyclic quivers 312.1 Drinfeld doubles and the Hopf algebra D(n) 312.2 SchiffmannHubery generators 372.3 Presentation of D(n) 412.4 Some integral forms 452.5 The quantum loop algebra U(gln) 492.6 Semisimple generators and commutator formulas 55

3 Affine quantum Schur algebras and the SchurWeylreciprocity 623.1 Cyclic flags: the geometric definition 633.2 Affine Hecke algebras of type A: the algebraic definition 683.3 The tensor space interpretation 743.4 BLM bases and multiplication formulas 773.5 The D(n)-H(r)-bimodule structure on tensor spaces 793.6 A comparison with the VaragnoloVasserot action 883.7 Triangular decompositions of affine quantum Schur algebras 953.8 Affine quantum SchurWeyl duality, I 1023.9 Polynomial identity arising from semisimple generators 1063.10 Appendix 115

vii

viii Contents

4 Representations of affine quantum Schur algebras 1214.1 Affine quantum SchurWeyl duality, II 1224.2 ChariPressley category equivalence and classification 1264.3 Classification of simple S(n, r)C-modules:

the upward approach 1324.4 Identification of simple S(n, r)C-modules: the n > r case 1364.5 Application: the setS (n)r 1414.6 Classification of simple S(n, r)C-modules:

the downward approach 1434.7 Classification of simple U(n, r)C-modules 150

5 The presentation and realization problems 1535.1 McGertys presentation for U(n, r) 1545.2 Structure of affine quantum Schur algebras 1575.3 Presentation of S(r, r) 1625.4 The realization conjecture 1695.5 Lusztigs transfer maps on semisimple generators 172

6 The classical (v = 1) case 1796.1 The universal enveloping algebra U(gln) 1796.2 More multiplication formulas in affine Schur algebras 1856.3 Proof of Conjecture 5.4.2 at v = 1 1906.4 Appendix: Proof of Proposition 6.2.3 194

Bibliography 201Index 205

Introduction

Quantum SchurWeyl theory refers to a three-level duality relation. At Level I,it investigates a certain double centralizer property, the quantum Schur Weylreciprocity, associated with some bimodules of quantum gln and the Heckealgebra (of type A)the tensor spaces of the natural representation of quan-tum gln (see [43], [21], [27]). This is the quantum version of the well-knownSchurWeyl reciprocity which was beautifully used in H. Weyls influentialbook [77]. The key ingredient of the reciprocity is a class of important finitedimensional endomorphism algebras, the quantum Schur algebras or q-Schuralgebras, whose classical version was introduced by I. Schur over a hundredyears ago (see [69], [70]). At Level II, it establishes a certain Morita equiv-alence between quantum Schur algebras and Hecke algebras. Thus, quantumSchur algebras are used to bridge representations of quantum gln and Heckealgebras. More precisely, they link polynomial representations of quantum glnwith representations of Hecke algebras via the Morita equivalence. The thirdlevel of this duality relation is motivated by two simple questions associatedwith the structure of (associative) algebras. If an algebra is defined by gen-erators and relations, the realization problem is to reconstruct the algebra asa vector space with hopefully explicit multiplication formulas on elementsof a basis; while, if an algebra is defined in terms of a vector space such asan endomorphism algebra, it is natural to seek their generators and definingrelations.

As one of the important problems in quantum group theory, the realizationproblem is to construct a quantum group in terms of a vector space and certainmultiplication rules on basis elements. This problem is crucial to understandtheir structure and representations (see [47, p. xiii] for a similar problem forKacMoody Lie algebras and [60] for a solution in the symmetrizable case).Though the RingelHall algebra realization of the -part of quantum envelop-ing algebras associated with symmetrizable Cartan matrices was an important

1

2 Introduction

breakthrough in the early 1990s, especially for the introduction of the geomet-ric approach to the theory, the same problem for the entire quantum groupsis far from completion. However, BeilinsonLusztigMacPherson (BLM) [4]solved the problem for quantum gln by exploring further properties comingfrom the quantum SchurWeyl reciprocity. On the other hand, as endomor-phism algebras and as homomorphic images of quantum gln , it is natural tolook for presentations for quantum Schur algebras via the presentation of quan-tum gln . This problem was first considered in [18] (see also [26]). Thus, as aparticular feature in the type A theory, realizing quantum gln and presentingquantum Schur algebras form Level III of this duality relation. For a completeaccount of the quantum SchurWeyl theory and further references, see Parts 3and 5 of [12] (see also [17] for more applications).

There are several developments in the establishment of an affine analogueof the quantum SchurWeyl theory. Soon after BLMs work, Ginzburg andVasserot [32, 75] used a geometric and K -theoretic approach to investigateaffine quantum Schur algebras1 as homomorphic images of quantum loopalgebra U(gln) of gln in the sense of Drinfelds new presentation [20], calledquantum affine gln (at level 0) in this book. This establishes at Level I thefirst centralizer property for the affine analogue of the quantum SchurWeylreciprocity. Six years later, investigations around affine quantum Schur alge-bras focused on their different definitions and, hence, different applications.For example, Lusztig [56] generalized the fundamental multiplication formu-las [4, 3.4] for quantum Schur algebras to the affine case and showed thatthe extended quantum affine sln , U(n), does not map onto affine quan-tum Schur algebras; VaragnoloVasserot [73] investigated RingelHall algebraactions on tensor spaces and described the geometrically defined affine quan-tum Schur algebras in terms of the endomorphism algebras of tensor spaces.Moreover, they proved that the tensor space definition coincides with Greensdefinition [35] via q-permutation modules. Some progress on the second cen-tralizer property has also been made recently by Pouchin [61]. The approachesused in these works are mainly geometric. However, like the non-affine case,there would be more favorable algebraic and combinatorial approaches.

At Level II, representations at non-roots-of-unity of quantum affine slnand gln over the complex number field C, including classifications of finitedimensional simple modules, have been thoroughly investigated by ChariPressley [6, 7, 8], and FrenkelMukhin [28] in terms of Drinfeld polynomials.Moreover, an equivalence between the module category of the Hecke algebra

1 Perhaps they should be called quantum affine Schur algebras. Since our purpose is to establishan affine analogue of the quantum SchurWeyl theory, this terminology seems moreappropriate to reflect this.

Introduction 3

H(r)C and a certain full subcategory of quantum affine sln (resp., gln) hasalso been established algebraically by ChariPressley [9] (resp., geometricallyby GinzburgReshetikhinVasserot [31]) under the condition n > r (resp.,n r ). Note that the approach in [31] uses intersection cohomology com-plexes. It would be interesting to know how affine quantum Schur algebraswould play a role in these works.

Much less progress has been made at Level III. When n > r , DotyGreen[18] and McGerty [58] have found a presentation for affine quantum Schuralgebras, while the last two authors of this book have investigated the realiza-tion problem in [24], where they first developed an approach without using thestabilization property, a key property used in the BLM approach, and presentedan ideal candidate for the realization of quantum affine gln .

This book attempts to establish the affine quantum SchurWeyl theory as awhole and is an outcome of algebraically understanding the works mentionedabove.

First, building on Schiffmann [67] and Hubery [40], our starting point isto present the double RingelHall algebra D(n) of the cyclic quiver with nvertices in terms of Chevalley type generators together with infinitely manycentral generators. Thus, we obtain a central subalgebra Z(n) such thatD(n) = U(n)Z(n) = U(n) Z(n). We then establish an isomorphismbetween D(n) and Drinfelds quantum affine gln in the sense of [20]. Inthis way, we easily obtain an action on the tensor space which upon restric-tion coincides with the RingelHall algebra action defined geometrically byVaragnoloVasserot [73] and commutes with the affine Hecke algebra action.

Second, by a thorough investigation of a BLM type basis for affine quantumSchur algebras, we introduce certain triangular relations for the correspondingstructure constants and, hence, a triangular decomposition for affine quantumSchur algebras. With this decomposition, we establish explicit algebra epi-morphisms r = r,Q(v) from the double RingelHall algebra D(n) to affinequantum Schur algebras S(n, r) := S(n, r)Q(v) for all r 0. This alge-braic construction has several nice applications, especially at Levels II and III.For example, the homomorphic image of commutator formulas for semisimplegenerators gives rise to a beautiful polynomial identity whose combinatorialproof remains mysterious.

Like the quantum Schur algebra case, we will establish for n r aMorita equivalence between affine quantum Schur algebras S(n, r)F andaffine Hecke algebrasH(r)F of type A over a field F with a non-root-of-unityparameter. As a by-product, we prove that every simple S(n, r)F-moduleis finite dimensional. Thus, applying the classification of simple H(r)C-modules by Zelevinsky [81] and Rogawski [66] yields a classification of simple

4 Introduction

S(n, r)C-modules. Hence, inflation via the epimorphisms r,C gives manyfinite dimensional simple UC(gln)-modules. We will also use r,C togetherwith the action on tensor spaces and a result of ChariPressley to prove thatfinite dimensional simple polynomial representations of UC(gln) are all infla-tions of simple S(n, r)C-modules. In this way, we can see the bridging roleplayed by affine quantum Schur algebras between representations of quantumaffine gln and those of affine Hecke algebras. Moreover, we obtain a classifica-tion of simple S(n, r)C-modules in terms of Drinfeld polynomials and, whenn > r , we identify them with those arising from simple H(r)C-modules.

Our findings also show that, if we regard the category S(n, r)C-Mod ofS(n, r)C-modules as a full subcategory of UC(gln)-modules, this category isquite different from the category Chi C considered in [54, 6.2]. For exam-ple, the latter is completely reducible and simple objects are usually infinitedimensional, while S(n, r)C-Mod is not completely reducible and all simpleobjects are finite dimensional. As observed in [23, Rem. 9.4(2)] for quantumgl and infinite quantum Schur algebras, this is another kind of phenomenonof infinite type in contrast to the finite type case.

The discussion of the realization and presentation problems is also based onthe algebra epimorphisms r and relies on the use of semisimple generators andindecomposable generators forD(n) which are crucial to understand the inte-gral structure and multiplication formulas. We first use the new presentation forD(n) to give a decomposition for S(n, r) = U(n, r)Z(n, r) into a productof two subalgebras, where Z(n, r) is a central subalgebra and U(n, r) is thehomomorphic image of U(n), the extended quantum affine sln . By taking aclose look at this structure, we manage to get a presentation for S(r, r) forall r 1 and acknowledge that the presentation problem is very complicatedin the n < r case. On the other hand, we formulate a realization conjecturesuggested by the work [24] and prove the conjecture in the classical (v = 1)case.

We remark that, unlike the geometric approach in which the ground ringmust be a field or mostly the complex number field C, the algebraic, or rather,the representation-theoretic approach we use in this book works largely over aring or mostly the integral Laurent polynomial ring Z[v, v1].

We have organized the book as follows.In the first preliminary chapter, we introduce in 1.4 three different types of

generators and their associated monomial bases for the RingelHall algebrasof cyclic quivers, and display in 1.5 the GreenXiao Hopf structure on theextended version of these algebras.

Chapter 2 introduces a new presentation using Chevalley generatorsfor Drinfelds quantum loop algebra U(gln) of gln . This is achieved by

Introduction 5

constructing the presentation for the double RingelHall algebra D(n) asso-ciated with cyclic quivers (Theorem 2.3.1), based on the work of Schiffmannand Hubery, and by lifting Becks algebra monomorphism from the quantumsln with a DrinfeldJimbo presentation into U(gln) to obtain an isomorphismbetween D(n) and U(gln) (Theorem 2.5.3).

Chapter 3 investigates the structure of affine quantum Schur algebras. Wefirst recall the geometric definition by GinzburgVasserot and Lusztig, theHecke algebra definition by R. Green, and the tensor space definition byVaragnoloVasserot. Using the Chevalley generators of D(n), we easilyobtain an action on the Q(v)-space with a basis indexed by Z and, hence,an action of D(n) on r (3.5). We prove that this action commutes withthe affine Hecke algebra action defined in [73]. Moreover, we show that therestriction of the action to the negative part ofD(n) (i.e., to the correspondingRingelHall algebra) coincides with the RingelHall algebra action geometri-cally defined by VaragnoloVasserot (Theorem 3.6.3). As an application ofthis coincidence, the commutator formula associated with semisimple gener-ators, arising from the skew-Hopf pairing, gives rise to a certain polynomialidentity associated with a pair of elements , Nn (Corollary 3.9.6). Themain result of the chapter is an elementary proof of the surjective homomor-phism r from the double RingelHall algebra D(n), i.e., the quantum loopalgebra U(gln), onto the affine quantum Schur algebra S(n, r) (Theorem3.8.1). The approach we used is the establishment of a triangular decom-position of S(n, r) (Theorem 3.7.7) through an analysis of the BLM typebases.

In Chapter 4, we discuss the representation theory of affine quantum Schuralgebras over C and its connection to polynomial representations of quan-tum affine gln and representations of affine Hecke algebras. We first establisha category equivalence between the module categories S(n, r)C-Mod andH(r)C-Mod for n r (Theorem 4.1.3). As an application, we will reinterpretChariPressleys category equivalence ([9, Th. 4.2]) between (level r ) repre-sentations of UC(sln) and those of affine Hecke algebrasH(r)C, where n > r ,in terms of representations of S(n, r)C (Proposition 4.2.1). We then developtwo approaches to the classification of simple S(n, r)C-modules. In the so-called upward approach, we use the classification of simple H(r)C-modulesof Zelevinsky and Rogawski to classify simple S(n, r)C-modules (Theorems4.3.4 and 4.5.3), while in the downward approach, we determine the classifi-cation of simple S(n, r)C-modules (Theorem 4.6.8) in terms of simple poly-nomial representations of UC(gln). When n > r , we prove an identificationtheorem (Theorem 4.4.2) for the two classifications. Finally, in 4.7, a clas-sification of finite dimensional simple U(n, r)C-modules is also completed

6 Introduction

and its connections to finite dimensional simple UC(sln)-modules and finitedimensional simple (polynomial) UC(gln)-modules are also discussed.

We move on to look at the presentation and realization problems in Chap-ter 5. We first observe S(n, r) = U(n, r)Z(n, r), where U(n, r) andZ(n, r) are homomorphic images of U(n) and Z(n), respectively, and thatZ(n, r) U(n, r) if and only if n > r . A presentation for U(n, r) is givenin [58] (see also [19] for the n > r case). Building on McGertys presentation,we first give a DrinfeldJimbo type presentation for the subalgebra U(n, r)(Theorem 5.1.3). We then describe a presentation for the central subalgebraZ(n, r) as a Laurent polynomial ring in one indeterminate over a polynomialring in r 1 indeterminates over Q(v). We manage to describe a presentationfor S(r, r) for all r 1 (Theorem 5.3.5) by adding an extra generator (and itsinverse) together with an additional set of relations on top of the relations givenin Theorem 5.1.3. What we will see from this case is that the presentation forS(n, r) with r > n can be very complicated.

We discuss the realization problem from 5.4 onwards. We first describe themodified BLM approach developed in [24]. With some supporting evidence,we then formulate the realization conjecture (Conjecture 5.4.2) as suggestedin [24, 5.5(2)], and state its classical (v = 1) version. We end the chapterwith a closer look at Lusztigs transfer maps [57] by displaying some explicitformulas for their action on the semisimple generators for S(n, r) (Corol-lary 5.5.2). These formulas also show that the homomorphism from U(sln) tolimS(n, n +m) induced by the transfer maps cannot either be extended to thedouble RingelHall algebra D(n). (Lusztig already pointed out that it cannotbe extended to U(n).) This somewhat justifies why a direct product is used inthe realization conjecture.

In the final Chapter 6, we prove the realization conjecture for the classi-cal (v = 1) case. The key step in the proof is the establishment of moremultiplication formulas (Proposition 6.2.3) between homogeneous indecom-posable generators and an arbitrary BLM type basis element. As a by-product,we display a basis for the universal enveloping algebra of the loop algebraof gln (Theorem 6.3.4) together with explicit multiplication formulas betweengenerators and arbitrary basis elements (Corollary 6.3.5).

There are two appendices in 3.10 and 6.4 which collect a number oflengthy calculations used in some proofs.

Conjectures and problems. There are quite a few conjectures and prob-lems throughout the book. The conjectures are mostly natural generalizationsto the affine case, for example, the realization conjecture 5.4.2 and the con-jectures in 3.8 on an integral form for double RingelHall algebras and

Introduction 7

the second centralizer property in the affine quantum SchurWeyl reciprocity.Some problems are designed to seek further solutions to certain questions suchas quantum Serre relations for semisimple generators (Problem 2.6.4), theAffine Branching Rule (Problem 4.3.6), and further identification of simplemodules from different classifications (Problem 4.6.11). There are also prob-lems for seeking different proofs. Problems 3.4.3 and 6.4.2 form a key steptowards the proof of the realization conjecture.

Notational scheme. For most of the notation used throughout the book, ifit involves a subscript or a superscript , it indicates that the same notationwithouthas been used in the non-affine case, say, in [4], [12], [33], etc. Herethe triangledepicts the cyclic Dynkin diagram of affine type A.

For a ground ring Z and a Z-module (or a Z-algebra) A, we often use thenotation AF := A F to represent the object obtained by base change to afield F, which itself is a Z-module. In particular, if Z = Z[v, v1], then wewriteA for AQ(v).

Acknowledgements. The main results of the book have been presented bythe authors at the following conferences and workshops. We would like tothank the organizers for the opportunities of presenting our work.

Conference on Perspectives in Representation Theory, Cologne, September2009;

International Workshop on Combinatorial and Geometric Approach toRepresentation Theory, Seoul National University, September 2009;

2010 ICM Satellite Conference, Bangalore, August 2010; 12th National Algebra Conference, Lanzhou, June 2010; Southeastern Lie Theory Workshop: Finite and Algebraic Groups and

Leonard Scott Day, Charlottesville, June 2011; 55th Annual Meeting of the Australian Mathematical Society, Wollongong,

September 2011.

The research was partially supported by the Australian Research Coun-cil, the Natural Science Foundation of China, the 111 Program of China, theProgram NCET, the Fok Ying Tung Education Foundation, the FundamentalResearch Funds for the Central Universities of China, and the UNSW Gold-star Award. The first three and last two chapters were written while Deng andFu were visiting the University of New South Wales at various times. Thehospitality and support of UNSW are gratefully acknowledged.

The second author would like to thank Alexander Kleshchev and Arun Ramfor helpful comments on the Affine Branching Rule (4.3.5.1), and Vyjayanthi

8 Introduction

Chari for several discussions and explanations on the paper [9] and somerelated topics. He would also like to thank East China Normal University, andthe Universities of Mainz, Virginia, and Auckland for their hospitality duringhis sabbatical leave in the second half of 2009.

Finally, for all their help, encouragement, and infinite patience, we thankour wives and children: Wenlian Guo and Zhuoran Deng; Chunli Yu, AndyDu, and Jason Du; Shanshan Xia.

Bangming DengJie DuQiang Fu

Sydney5 December 2011

1Preliminaries

We start with the loop algebra of gln(C) and its interpretation in terms of matrixLie algebras. We use the subalgebra of integer matrices of the latter to intro-duce several important index sets which will be used throughout the book.RingelHall algebrasH(n) associated with cyclic quivers (n) and their geo-metric construction are introduced in 1.2. In 1.3, we discuss the compositionsubalgebra C(n) of H(n) and relate it to the quantum loop algebra U(sln).We then describe in 1.4 three types of generators for H(n), which con-sist of all simple modules together with, respectively, the SchiffmannHuberycentral elements, homogeneous semisimple modules, and homogeneous inde-composable modules, and their associated monomial bases (Corollaries 1.4.2and 1.4.6). These generating sets will play different roles in what follows.Finally, extended RingelHall algebras and their Hopf structure are discussedin 1.5.

1.1. The loop algebra gln and some notation

For a positive integer n, let gln(C) be the complex general linear Lie algebra,and let

gln(C) := gln(C) C[t, t1]be the loop algebra of gln(C); see [47]. Thus, gln(C) is spanned by Ei, j tmfor all 1 i, j n, and m Z, where Ei, j is the matrix (k,i j,l)1k,ln . The(Lie) multiplication is the bracket product associated with the multiplication

(Ei, j tm)(Ek,l tm ) = j,k Ei,l tm+m .We may interpret the Lie algebra gln(C) as a matrix Lie algebra. Let

M, n(C) be the set of all ZZ complex matrices A = (ai, j )i, jZ with ai, j C

9

10 1. Preliminaries

such that

(a) ai, j = ai+n, j+n for i, j Z, and(b) for every i Z, the set { j Z | ai, j = 0} is finite.Clearly, conditions (a) and (b) imply that there are only finitely many non-zeroentries in each column of A. For A, B M, n(C), let [A, B] = AB B A.Then (M, n(C), [ , ]) becomes a Lie algebra over C.

Denote by Mn,(C) the set of n Z matrices A = (ai, j ) over C satisfying(b) with i [1, n] := {1, 2, . . . , n}. Then there is a bijection

1 : M, n(C) Mn,(C), (ai, j )i, jZ (ai, j )1in, jZ. (1.1.0.1)For i, j Z, let Ei, j M, n(C) be the matrix (ei, jk,l )k,lZ defined by

ei, jk,l =

{1, if k = i + sn, l = j + sn for some s Z;0, otherwise.

The set {Ei, j |1 i n, j Z} is a C-basis of M, n(C). SinceEi, j+ln Ep,q+kn = j,p Ei,q+(l+k)n,

for all i, j, p, q, l, k Z with 1 j, p n, it follows that the mapM, n(C) gln(C), Ei, j+ln Ei, j t l , 1 i, j n, l Z

is a Lie algebra isomorphism. We will identify the loop algebra gln(C) withM, n(C) in the sequel.

In Chapter 6, we will consider the loop algebra gln := gln(Q) = M,n(Q)defined over Q and its universal enveloping algebra U(gln) and triangular partsU(gln)+, U(gln), and U(gln)0. Here U(gln)+ (resp., U(gln), U(gln)0) is thesubalgebra of U(gln) generated by Ei, j for all i < j (resp., Ei, j for all i > j ,Ei,i for all i). We will also relate these algebras in 6.1 with the specializationsat v = 1 of the RingelHall algebra H(n) and the double RingelHall algebraD(n).

We now introduce some notation which will be used throughout the book.Consider the subset M,n(Z) of M, n(C) consisting of matrices with integer

entries. For each A M,n(Z), letro(A) = (

jZai, j)

iZ and co(A) =(

iZai, j)

jZ.

We obtain functionsro, co : M,n(Z) Zn,

1.1. The loop algebra gln and some notation 11

whereZn := {(i )iZ | i Z, i = in for i Z}.

For = (i )iZ Zn, A M,n(Z), and i0 Z, let() =

i0+1ii0+n

i and (A) =

i0+1ii0+njZ

ai, j =

i0+1 ji0+niZ

ai, j .

Clearly, both () and (A) are defined and independent of i0. We sometimesidentify Zn with Zn via the following bijection

2 : Zn Zn, 2() = (1, . . . , n). (1.1.0.2)For example, we define a dot product on Zn by := 2() 2() =n

i=1 ii , and define the order relation on Zn by setting

2() 2() i i for all 1 i n. (1.1.0.3)Also, let ei Zn be defined by 2(ei ) = ei = (0, . . . , 0, 1

(i), 0, . . . , 0).

Let(n) :={A = (ai, j ) M,n(Z) | ai, j N} = M,n(N),

Nn :={(i )iZ Zn | i 0},and, for r 0, let

(n, r) :={A (n) | (A) = r} and(n, r) :={ Nn | () = r}.

The set Mn(Z) can be naturally regarded as a subset of Mn,(Z) by sending(ai, j )1i, jn to (ai, j )1in, jZ, where ai, j = 0 if j Z\[1, n]. Thus, (theinverse of) 1 induces an embedding

1 : Mn(Z) M,n(Z). (1.1.0.4)By removing the subscripts, we define similarly the subsets (n), (n, r) ofMn(Z) and subset (n, r) of Nn , etc. Note that 2((n, r)) = (n, r).

LetZ = Z[v, v1], where v is an indeterminate, and let Q(v) be the fractionfield of Z . For integers N , t with t 0, let[

Nt

]=

1it

vNi+1 v(Ni+1)vi vi Z and

[N0

]= 1. (1.1.0.5)

If we put [m] = vmvmvv1 =

[m1]

and [N ]! := [1][2] [N ], then[

Nt

]=

[N ]![t]![Nt]! for all 1 t N . Given a polynomial f Z and z C := C\{0},we sometimes write fz for f (z), e.g., [m]z , [m]!z , etc.

12 1. Preliminaries

When counting occurs, we often use[[Nt

]]:= vt (Nt)

[Nt

]to denote the Gaussian polynomials in v2.

Also, for any Q(v)-algebra A and an invertible element X A , let[X; at

]=

ts=1

Xvas+1 X1va+s1vs vs and

[X; a0

]= 1, (1.1.0.6)

for all a, t Z with t 1.

1.2. Representations of cyclic quivers and RingelHallalgebras

Let (n) (n 2) be the cyclic quivern

1 2 3 n2 n1

with vertex set I = Z/nZ = {1, 2, . . . , n} and arrow set {i i + 1 | i I }.Let F be a eld. By Rep0(n) = Rep0F(n) we denote the category of nitedimensional nilpotent representations of (n) over F, i.e., representationsV = (Vi , fi )iI of (n) such that all Vi are nite dimensional and the com-position fn f2 f1 : V1 V1 is nilpotent. The vector dim V = (dimF Vi ) NI = Nn is called the dimension vector of V . (We shall sometimes identify NIwith Nn under (1.1.0.2).) For each vertex i I , there is a one-dimensional rep-resentation Si in Rep0(n) satisfying (Si )i = F and (Si ) j = 0 for j = i . It isknown that the Si form a complete set of simple objects in Rep0(n). Hence,each semisimple representation Sa in Rep0(n) is given by Sa = iI ai Si ,where a = (a1, . . . , an) NI . A semisimple representation Sa is calledsincere if a is sincere, namely, all ai are positive. In particular, the vector

:= (1, . . . , 1) Nn

will often be used.Moreover, up to isomorphism, all indecomposable representations in

Rep0(n) are given by Si [l] (i I and l 1) of length l with top Si . Thus,the isoclasses of representations in Rep0(n) are indexed by multisegments

1.2. Representations of cyclic quivers and RingelHall algebras 13

=iI, l1 i,l[i; l), where the representation M() corresponding to isdefined by

M() = MF() =

1in,l1i,l Si [l].

Since the set of all multisegments can be identified with the set

+ (n) = {A = (ai, j ) (n) | ai, j = 0 for i j}of all strictly upper triangular matrices via

3 : + (n) , A = (ai, j )i, jZ

i< j,1inai, j [i; j i),

we will use + (n) to index the finite dimensional nilpotent representations. Inparticular, for any i, j Z with i < j , we have

Mi, j := M(Ei, j ) = Si [ j i ], and Mi+n, j+n = Mi, j .Thus, for any A = (ai, j ) + (n) and i0 Z,

M(A) = MF(A) =

1in,i< jai, j Mi, j =

i0+1ii0+n,i< j

ai, j Mi, j .

For A = (ai, j ) + (n), setd(A) =

i< j,1in

ai, j ( j i).

Then dimF M(A) = d(A). Moreover, for each = (i ) NI , set A = (ai, j )with ai, j = j,i+1i , i.e., A =ni=1 i Ei,i+1. Then

M(A) =

1ini Si =: S (1.2.0.1)

is semisimple. Also, for A + (n), we write d(A) = dim M(A) ZI ,the dimension vector of M(A). Hence, ZI is identified with the Grothendieckgroup of Rep0(n).

A matrix A = (ai, j ) + (n) is called aperiodic if, for each l 1, thereexists i Z such that ai,i+l = 0. Otherwise, A is called periodic. A nilpo-tent representation M(A) is called aperiodic (resp., periodic) if A is aperiodic(resp., periodic).

It is well known that there exist AuslanderReiten sequences in Rep0(n);see [1]. More precisely, for each i I and each l 1, there is an AuslanderReiten sequence

0 Si+1[l] Si [l + 1] Si+1[l 1] Si [l] 0,

14 1. Preliminaries

where we set Si+1[0] = 0 by convention. Si+1[l] is called the AuslanderReiten translate of Si [l], denoted by Si [l]. In this case, indeed defines anequivalence from Rep0(n) to itself, called the AuslanderReiten translation.For each A = (ai, j ) + (n), we define (A) + (n) by M( (A)) = M(A). Thus, if we write (A) = (bi, j ) + (n), then bi, j = ai1, j1 forall i, j .

We now introduce the degeneration order on + (n) and generic extensionsof nilpotent representations. These notions play an important role in the studyof bases for both the RingelHall algebra H(n) of (n) and its compositionsubalgebra C(n); see, for example, [11, 13]. For two nilpotent representationsM, N in Rep0(n) with dim M = dim N , defineN dg M dimF Hom(X, N )dimF Hom(X, M), for all X Rep0(n);

(1.2.0.2)see [82]. This gives rise to a partial order on the set of isoclasses of representa-tions in Rep0(n), called the degeneration order. Thus, it also induces a partialorder on + (n) by letting

A dg B M(A) dg M(B).By [62] and [11, 3], for any two nilpotent representations M and N , thereexists a unique extension G (up to isomorphism) of M by N with mini-mal dim End(G). This representation G is called the generic extension of Mby N and will be denoted by M N in the sequel. Moreover, for nilpotentrepresentations M1, M2, M3,

(M1 M2) M3 = M1 (M2 M3).Also, taking generic extensions preserves the degeneration order. More pre-cisely, if N1 dg M1 and N2 dg M2 , then N1 N2 dg M1 M2. ForA, B + (n), let A B + (n) be defined by M(A B) = M(A) M(B).

As above, let Z = Z[v, v1] be the Laurent polynomial ring in indetermi-nate v. By [65] and [36], for A, B1, . . . , Bm + (n), there is a polynomialAB1,...,Bm Z[v2] in v2, called the Hall polynomial, such that for any finitefield F of q elements, AB1,...,Bm |v2=q (the evaluation of AB1,...,Bm at v2 = q)equals the number F MF(A)MF(B1),...,MF(Bm ) of the filtrations

0 = Mm Mm1 M1 M0 = MF(A)such that Mt1/Mt = MF(Bt ) for all 1 t m.

Moreover, for each A = (ai, j ) + (n), there is a polynomial aA =aA(v

2) Z in v2 such that, for each finite field F with q elements, aA|v2=q =| Aut(MF(A))|; see, for example, [59, Cor. 2.1.1]. For later use, we give an

1.2. Representations of cyclic quivers and RingelHall algebras 15

explicit formula for aA. Let m A denote the dimension of rad End(MF(A)),which is known to be independent of the field F. We also have

End(MF(A))/rad End(MF(A)) =

1in, ai, j >0Mai, j (F),

where Mai, j (F) denotes the full matrix algebra of ai, j ai, j matrices over F.Hence,

| Aut(MF(A))| = |F|m A

1in, ai, j >0|GLai, j (F)|.

Consequently,

aA = v2m A

1in, ai, j>0(v2ai, j 1)(v2ai, j v2) (v2ai, j v2ai, j2). (1.2.0.3)

In particular, if z C is not a root of unity, then aA|v2=z = 0.Let H(n) be the (generic) RingelHall algebra of the cyclic quiver (n),

which is by definition the free Z-module with basis {u A = u[M(A)] | A + (n)}. The multiplication is given by

u AuB = vd(A),d(B)

C+ (n)CA,BuC

for A, B + (n), whered(A), d(B) = dim Hom(M(A), M(B)) dim Ext1(M(A), M(B))

(1.2.0.4)is the Euler form associated with the cyclic quiver (n). If we write d(A) =(ai ) and d(B) = (bi ), then

d(A), d(B) =iI

ai bi iI

ai bi+1. (1.2.0.5)

Since both dimF End(MF(A)) and dimF MF(A) = d(A) are independent ofthe ground field, we put for each A + (n),

d A = dimF End(MF(A)) dimF MF(A) and u A = vdA u A; (1.2.0.6)

cf. [13, (8.1)].1 As seen in [13], it is sometimes convenient to work with thePBW type basis {u A | A + (n)} of H(n).

The degeneration order gives rise to the following triangular relation inH(n): for A1, . . . , At + (n),

u A1 u At = v

1r

16 1. Preliminaries

There is a natural NI -grading on H(n):

H(n) =

dNIH(n)d, (1.2.0.8)

where H(n)d is spanned by all u A with d(A) = d. Moreover, we will fre-quently consider in the sequel the algebra H(n) = H(n) Z Q(v) obtainedby base change to the fraction field Q(v).

In order to relate RingelHall algebras H(n) of cyclic quivers with affinequantum Schur algebras later on, we recall Lusztigs geometric construction ofH(n) (specializing v to a square root of a prime power) developed in [53, 54](cf. also [55]).

Let F = Fq be the finite field of q elements and d NI . Fix an I -gradedF-vector space V = iI Vi of dimension vector d, i.e., dimF Vi = di for alli I . Then each element in

EV ={( fi )

iI

HomF(Vi , Vi+1) | fn f1 is nilpotent}

can be viewed as a nilpotent representation of (n) over F of dimension vectord. The group GV = iI GL(Vi ) acts on EV by conjugation. Then there is abijection between the GV -orbits in EV and the isoclasses of nilpotent represen-tations of (n) of dimension vector d. For each A + (n) with d(A) = d,we will denote by OA the orbit in EV corresponding to the isoclass of M(A).

Define Hd = CGV (EV ) to be the vector space of GV -invariant functionsfrom EV to C. Now let a,b Nn with d = a + b and fix I -graded F-vectorspaces U = iI Ui and W = iI Wi with dimension vectors a and b,respectively. Let E be the set of triples (x, , ) such that (1) x EV , (2)the sequence

0 W V U 0of I -graded spaces is exact, and (3) (W ) is stable by x . Let F be the set ofpairs (x, W ), where x EV and W V is an x-stable I -graded subspace ofdimension vector b. Consider the diagram

EU EW p1 E p2 F p3 EV ,where p1, p2, p3 are projections defined in an obvious way. Given f CGU (EU ) = Ha and g CGW (EW ) = Hb, define the convolution productof f and g by

f g = q 12 m(a,b)(p3)!h CGV (EV ) = Hd,

1.3. The quantum loop algebra U(sln) 17

where h C(F) is the function such that p2h = p1( f g) andm(a,b) =

1in

ai bi +

1inai bi+1.

Consequently, H = dNn Hd becomes an associative algebra over C.2 Foreach A + (n), let O A be the characteristic function of OA and put

OA = q 12 dimOAO A .By [13, 8.1]), there is an algebra isomorphismH(n) Z C H, u A vd(A)d(A)d(A)O A , for all A + (n),(1.2.0.9)

where C is viewed as a Z-module by specializing v to q 12 . In particular, thisisomorphism takes u A to OA.

1.3. The quantum loop algebra U(sln)As mentioned in the introduction, an important breakthrough for the struc-ture of quantum groups associated with semisimple complex Lie algebras isRingels Hall algebra realization of the -part of the quantum enveloping alge-bra associated with the same quiver; see [63, 64]. For the RingelHall algebraH(n) associated with a cyclic quiver, it is known from [65] that a subalgebra,the composition algebra, is isomorphic to the -part of a quantum affine sln .We now describe this algebra and use it to display a certain monomial basis.

The Z-subalgebra C(n) of H(n) generated by u[mSi ] (i I and m 1) is called the composition algebra of (n). By (1.2.0.8), C(n) inherits anNI -grading by dimension vectors:

C(n) =

dNIC(n)d,

where C(n)d = C(n) H(n)d. Let ui := u Ei,i+1 = u[Si ] and define thedivided power

u(m)i =

1[m]! u

mi C(n),

for i I and m 1. In fact,u(m)i = vm(m1)u[mSi ] C(n) H(n).

2 The algebra H defined also geometrically in [73, 3.2] has a multiplication opposite to the onefor H here.

18 1. Preliminaries

We now use the strong monomial basis property developed in [11, 13] toconstruct an explicit monomial basis of C(n). For each A = (ai, j ) + (n),define

= A = max{ j i | ai, j = 0}.In other words, is the Loewy length of the representation M(A).

Suppose now A is aperiodic. Then there is i1 [1, n] such that ai1,i1+ = 0,but ai1+1,i1+1+ = 0. If there are some ai1+1, j = 0, we let p 1 satisfyai1+1,i1+1+p = 0 and ai1+1, j = 0 for all j > i1 + 1 + p; if ai1+1, j = 0 for allj > i1 + 1, let p = 0. Thus, > p. Now set

t1 = ai1,i1+1+p + + ai1,i1+and define A1 = (bi, j ) + (n) by letting

bi, j =

0, if i = i1, j i1 + 1 + p;ai1+1, j + ai1, j , if i = i1 + 1 < j, j i1 + 1 + p;ai, j , otherwise.

Then, A1 is again aperiodic. Applying the above process to A1, we get i2 andt2. Repeating the above process (ending with the zero matrix), we finally gettwo sequences i1, . . . , im and t1, . . . , tm . This gives a word

wA = i t11 i t22 i tmm ,where i1, . . . , im are viewed as elements in I = Z/nZ, and define themonomial

u(A) = u(t1)i1 u(t2)i2 u

(tm)im C(n).

The algorithm above can be easily modified to get a similar algorithm forquantum gln . We illustrate the algorithm with an example in this case.

Example 1.3.1. If A =1 2 3 4

5 0 06 0

7, then = 4, i1 = 1, p = 1 and t1 = 2+3+4 = 9.

Here we ignore all zero entries on and below the diagonal for simplicity. Thus,

A1 =1 0 0 0

7 3 46 0

7, = 3, i2 = 2, p = 1, and t2 = 3 + 4 = 7,

A2 =1 0 0 0

7 0 09 4

7, = 2, i3 = 3, p = 1, and t3 = 4, and

A3 =1 0 0 0

7 0 09 0

11, = 1, i4 = 4, p = 0, and t4 = 11.

1.3. The quantum loop algebra U(sln) 19

Now, for a matrix defining a semisimple representation, we have all = 1and p = 0. So the remaining cases are i5 = 3, t5 = 9; i6 = 2, t6 = 7; andi7 = 1, t7 = 1. Hence,

u(A) = u(9)1 u(7)2 u(4)3 u(11)4 u(9)3 u(7)2 u1.Proposition 1.3.2. The set

{u(A) | A + (n) aperiodic}is a Z-basis of C(n).Proof. Let A + (n) be aperiodic and let wA = i t11 i t22 i tmm be the corre-sponding word constructed as above. By [11, Th. 5.5], wA is distinguished, thatis, AA1,...,Am = 1, where As = ts Eis ,is+1 for 1 s m. By [13, Th. 7.5(i)],the u(A) with A + (n) aperiodic form a Z-basis of C(n).

We now define the quantum enveloping algebra of the loop algebra (thequantum loop algebra for short) of sln . Let C = C(n) = (ci, j )i, jI be thegeneralized Cartan matrix of type An1, where I = Z/nZ. We always assumethat if n 3, then ci,i = 2, ci,i+1 = ci+1,i = 1 and ci, j = 0 otherwise. Ifn = 2, then c1,1 = c2,2 = 2 and c1,2 = c2,1 = 2. In other words,

C =(

2 22 2

)or C =

2 1 0 0 11 2 1 0 00 1 2 0 0...

......

. . ....

...

0 0 0 2 11 0 0 1 2

(n 3).

(1.3.2.1)The quantum group associated to C is denoted by U(sln).

Definition 1.3.3. Let n 2 and I = Z/nZ. The quantum loop algebra U(sln)is the algebra over Q() presented by generators

Ei , Fi , Ki , K1i , i I,and relations, for i, j I ,(QSL0) K1 K2 Kn = 1;(QSL1) Ki K j = K j Ki , Ki K1i = 1;(QSL2) Ki E j = ci, j E j Ki ;(QSL3) Ki Fj = ci, j Fj Ki ;(QSL4) Ei E j = E j Ei , Fi Fj = Fj Fi if i = j 1;(QSL5) Ei Fj Fj Ei = i, j KiK

1i

1 ;(QSL6) E2i E j ( + 1)Ei E j Ei + E j E2i = 0 if i = j 1 and n 3;

20 1. Preliminaries

(QSL7) F2i Fj ( + 1)Fi Fj Fi + Fj F2i = 0 if i = j 1 and n 3;(QSL6) E3i E j (v2 +1+v2)E2i E j Ei + (v2 +1+v2)Ei E j E2i E j E3i =

0 if i = j and n = 2;(QSL7) F3i Fj (v2 + 1+ v2)F2i Fj Fi + (v2 + 1+ v2)Fi Fj F2i Fj F3i =

0 if i = j and n = 2.For later use in representation theory, let UC(sln) be the quantum loop

algebra defined by the same generators and relations (QSL0)(QSL7) with vreplaced by a non-root-of-unity z C and Q(v) by C.

A new presentation for U(sln) and UC(sln), known as Drinfelds newpresentation, will be discussed in 2.5.

In this book, quantum affine sln always refers to the quantum loop (Hopf)algebra U(sln).3 We will mainly work with U(sln) or quantum groups definedover Q(v) and mention from time to time a parallel theory over C.

Let U(sln)+ (resp., U(sln), U(sln)0) be the positive (resp., negative, zero)part of the quantum enveloping algebra U(sln). In other words, U(sln)+ (resp.,U(sln), U(sln)0) is a Q(v)-subalgebra generated by Ei (resp., Fi , K1i ),i I .

LetC(n) = C(n) Z Q(v).

Thus,C(n) identifies with the Q(v)-subalgebraH(n) generated by ui = u[Si ]for i I .Theorem 1.3.4. ([65]) There are Q(v)-algebra isomorphismsC(n) U(sln)+, ui Ei and C(n)op U(sln), ui Fi .By this theorem and the triangular decomposition

U(sln) = U(sln)+ U(sln)0 U(sln),the basis displayed in Proposition 1.3.2 gives rise to a monomial basis forU(sln).

1.4. Three types of generators and associatedmonomial bases

In this section, we display three distinct minimal sets of generators for H(n),each of which contains the generators {ui }iI for C(n). We also describe theirassociated monomial bases for H(n) in the respective generators.3 If (QSL0) is dropped, it also defines a quantum affine sln with the central extension; see,

e.g., [9].

1.4. Three types of generators and associated monomial bases 21

The first minimal set of generators contains simple modules and certain cen-tral elements. These generators are convenient for a presentation for the doubleRingelHall algebras over Q(v) (or a specialization at a non-root-of-unity)associated to cyclic quivers (see Chapter 2).

In [67] Schiffmann first described the structure of H(n) as a tensor productof C(n) and a polynomial algebra in infinitely many indeterminates. LaterHubery explicitly constructed these central elements in [39]. More precisely,for each m 1, let

cm = (1)mv2nm

A(1)dim End(M(A))aAu A H(n), (1.4.0.1)

where the sum is taken over all A + (n) such that d(A) = dim M(A) = mwith = (1, . . . , 1) Nn , and soc M(A) is square-free, i.e., dim soc M(A) in the order defined in (1.1.0.3). Note that in this case, soc M(A) is square-free if and only if top M(A) := M(A)/rad M(A) is square-free. The followingresult is proved in [67, 39].Theorem 1.4.1. The elements cm are central in H(n). Moreover, there is adecomposition

H(n) = C(n) Q(v) Q(v)[c1, c2, . . .],where Q(v)[c1, c2, . . .] is the polynomial algebra in cm for m 1. Inparticular, H(n) is generated by ui and cm for i I and m 1.

We will call the central elements cm the SchiffmannHubery generators.Let A = (ai, j ) + (n). For each s 1, define

ms = ms(A) = min{ai, j | j i = s} and A = A

1in, i< jm ji Ei, j .

(1.4.1.1)Then A is aperiodic. Moreover, for A, B + (n),

A = B A = B and ms(A) = ms(B), s 1.The next corollary is a direct consequence of Theorem 1.4.1 and Proposition1.3.2.

Corollary 1.4.2. The set

{u(A)s1

cms (A)s | A + (n)}

is a Q(v)-basis of H(n).Next, we look at the minimal set of generators consisting of simple mod-

ules and homogeneous semisimple modules. It is known from [73, Prop. 3.5]

22 1. Preliminaries

(or [13, Th. 5.2(i)]) that H(n) is also generated by ua = u[Sa] for a NI ; seealso (1.4.4.1) below. If a is not sincere, say ai = 0, then

ua =

jI, j =i

va j (1a j )

[a j ]! uai1i1 ua11 uann uai+1i+1 C(n). (1.4.2.1)

Thus, H(n) is generated by ui and ua, for i I and sincere a NI . Indeed,this result can be strengthened as follows; see also [67, p. 421].Proposition 1.4.3. The RingelHall algebra H(n) is generated by ui andum , for i I and m 1.Proof. Let H be the Q(v)-subalgebra generated by ui and um for i I andm 1. To show H = H(n), it suffices to prove ua = u[Sa] H for alla NI .

Take an arbitrary a NI . We proceed by induction on (a) = iI ai toshow ua H. If (a) = 0 or 1, then clearly ua H. Now let (a) > 1. If ais not sincere, then by (1.4.2.1), ua H. So we may assume a is sincere. Thecase where a1 = = an is trivial. Suppose now there exists i I such thatai = ai+1. Define a = (aj ), a = (aj ) NI by

aj ={

ai 1, if j = i;ai , otherwise,

and aj ={

ai+1 1, if j = i + 1;ai , otherwise.

Then, in H(n),

ui ua = vaiai+11(u X + vai1[ai ]ua) anduaui+1 = vai+1ai1(u X + vai+11[ai+1]ua),

where X + (n) is given byM(X) =

j =i,i+1

a j S j (ai 1)Si (ai+1 1)Si+1 Si [2].

Therefore,

ui ua v2ai2ai+1uaui+1 = vaiai+11(vai1[ai ] vai+11[ai+1])ua.The inequality ai = ai+1 implies vai1[ai ] vai+11[ai+1] = 0. Thus, weobtain

ua = vai+1ai+1

vai1[ai ] vai+11[ai+1]ui ua vaiai+1+1

vai1[ai ] vai+11[ai+1]uaui+1.

Since (a) = (a) = (a) 1, we have by the inductive hypothesis thatboth ua and ua belong to H. Hence, ua H. This finishes the proof.

1.4. Three types of generators and associated monomial bases 23

Remark 1.4.4. We will see that semisimple modules as generators are con-venient for the description of Lusztig type integral forms. First, by [13,Th. 5.2(ii)], they generate the integral RingelHall algebra H(n) over Z .Second, there are in 2.6 explicit commutator formulas between semisimplegenerators in the double RingelHall algebra. Thus, a natural candidate for theLusztig type form of quantum affine gln is proposed in 3.8.

Finally, we introduce a set of generators for H(n) consisting of simple andhomogeneous indecomposable modules in Rep0(n). Since indecomposablemodules correspond to the simplest non-diagonal matrices, these generatorsare convenient for deriving explicit multiplication formulas; see 3.4, 5.4,and 6.2.

For each A + (n), consider the radical filtration of M(A)M(A) rad M(A) rad t1 M(A) rad t M(A) = 0,

where t is the Loewy length of M(A). For 1 s t , we write

rad s1 M(A)/rad s M(A) = Sas , for some as NI .Write mA = ua1 uat . Applying (1.2.0.7) gives that

mA =

Bdg Af (B)u B,

where f (B) Q(v) with f (A) = v

l

24 1. Preliminaries

We use induction on m to show the following

Claim: H(n)(m) is generated by ui and uEis ,is+sn for i I and 1 s m.

Let m 1 and suppose the claim is true for H(n)(m1). Applying (1.4.4.1)to E = Eim ,im+mn gives

uE =

Bdg EBEmB = EEmE +

B

1.5. Hopf structure on extended RingelHall algebras 25

By the proof of the above proposition, we see that for m 1, each of thefollowing three sets

{ui , cs | i I, 1 s m}, {ui , us | i I, 1 s m}, and{ui , uEis ,is+sn | i I, 1 s m}

generates H(n)(m). Hence, for each m 1, there are non-zero elementsxm, ym Q(v) such that

cm xmum mod H(n)(m1) and cm ymuEis ,is+sn mod H(n)(m1).

This together with Corollary 1.4.2 gives the following result.

Corollary 1.4.6. The set{u(A

) s1

(us)ms (A) | A + (n)

}is a Q(v)-basis of H(n), where A is defined in (1.4.1.1). For each s 1,choose is Z+. Then the set{

u(A)

s1(u Eis ,is+sn

)ms (A) | A + (n)}

is also a Q(v)-basis of H(n).

1.5. Hopf structure on extended RingelHall algebras

It is known that generic RingelHall algebras exist only for Dynkin or cyclicquivers. In the finite type case, these algebras give a realization for the -partsof quantum groups. It is natural to expect that this is also true for cyclic quivers.In other words, we look for a quantum group such that H(n) is isomorphicto its -part. We will see in Chapter 2 that this quantum group is the quantumloop algebra of gln in the sense of Drinfeld [20], which, in fact, is isomorphicto the so-called double RingelHall algebra defined as the Drinfeld double oftwo Hopf algebras H(n)0 and H(n)0 together with a skew-Hopf pairing.In this section, we first introduce the pair H(n)0 and H(n)0.

We need some preparation. If we define the symmetrization of the Eulerform (1.2.0.4) by

(, ) = , + , ,then I together with ( , ) becomes a Cartan datum in the sense of [54, 1.1.1].To a Cartan datum, there are associated root data in the sense of [54, 2.2.1]

26 1. Preliminaries

which play an important role in the theory of quantum groups. We shall fix thefollowing root datum throughout the book.

Definition 1.5.1. Let X = Zn , Y = Hom(X,Z), and let , rd : Y X Z be the natural perfect pairing. If we denote the standard basis of X bye1, . . . , en and the dual basis by f 1, . . . , f n , then f i , e j rd = i, j . Thus,the embeddings

I Y, i i := f i f i+1 and I X, i i = eiei+1 (1.5.1.1)with en+1 = e1 and f n+1 = f 1 define a root datum (Y, X, , rd, . . .).

For notational simplicity, we shall identify both X and Y with ZI by settingei = i = f i for all i I . Under this identification, the form , rd : ZI ZI Z becomes a symmetric bilinear form, which is different from the Eulerforms , and its symmetrization ( , ). However, they are related as follows.Lemma 1.5.2. For a = ai i ZI , if we put a = ai i and a = ai i ,then

(1) (a, b) = a,brd; (2) a, b = a,brd, for all a,b ZI.Proof. Since all forms are bilinear, (1) follows from (i, j) = i , j rd for alli, j I , and (2) from i, j = i, j i+1, j = i , jrd.

We may use the following commutative diagrams to describe the tworelations:

ZI ZI

ZI ZIZ and( ) ( )

( , )

, rd

ZI ZI

ZI ZIZ.( ) 1

,

, rd

We also record the following fact which will be used below and in 2.1.Let F be a field. A Hopf algebraA over F is an F-vector space together with

multiplication A , unit A , comultiplication A , counit A , and antipodeA which satisfy certain axioms; see, e.g., [72] and [12, 5.1].Lemma 1.5.3. If A = (A , , ,, , ) is a Hopf algebra with multipli-cation , unit , comultiplication , counit , and antipode , then A op =(A , op, ,op, , ) is also a Hopf algebra. This is called the opposite Hopfalgebra of A . Moreover, if is invertible, then both (A , op, ,, , 1)and (A , , ,op, , 1) are also Hopf algebras, which are called semi-opposite Hopf algebras.

1.5. Hopf structure on extended RingelHall algebras 27

Let

H(n)0 = H(n) Q(v) Q(v)[K11 , . . . , K1n ]. (1.5.3.1)

Putting x = x1 and y = 1y for x H(n) and y Q(v)[K11 , . . . , K1n ],H(n)0 is a Q(v)-space with basis {u+A K | ZI, A + (n)}. We arenow ready to introduce the RingelGreenXiao Hopf structure on H(n)0.Let

+ (n) := + (n)\{0}.

Proposition 1.5.4. The Q(v)-spaceH(n)0 with basis {u+A K | ZI, A + (n)} becomes a Hopf algebra with the following algebra, coalgebra, andantipode structures.

(a) Multiplication and unit (Ringel [64]): for all A, B + (n) and , ZI ,

u+Au+B =

C+ (n)

vd(A),d(B)CA,Bu+C ,

Ku+A = vd(A),u+A K,KK = K+, and

1 = u+0 = K0.

(b) Comultiplication and counit (Green [34]): for all C + (n) and ZI ,

(u+C ) =

A,B+ (n)vd(A),d(B) aAaB

aCCA,Bu

+B u+A Kd(B),

(K) = K K,(u+C ) = 0 (C = 0), and (K) = 1.

Here, if = (ai ), then K denotes (K1)a1 (Kn)an with Ki = Ki K1i+1.(c) Antipode (Xiao [78]): for all C + (n) and ZI ,

(u+C ) = C,0 +m1

(1)m

D+ (n)C1,...,Cm+ (n)

aC1 aCmaC

CC1,...,Cm DCm ,...,C1 u+D Kd(C) and(K) = K.

28 1. Preliminaries

Moreover, the inverse of is given by

1(u+C ) = C,0 +m1

(1)m

D+ (n)C1, ,Cm+ (n)

v2

i< j d(Ci ),d(C j ) aC1 aCmaC

CC1,...,CmDC1,...,Cm Kd(C)u+D and1(K) = K.

Proof. The Hopf structure on H(n)0 is almost identical to the Hopf algebraH defined in the proof of [78, Prop. 4.8] except that we used K instead of Kin the comultiplication and antipode. Thus, the comultiplication of H(n)0defined here is opposite to that defined in [78, Th. 4.5], while the antipode is theinverse. Hence, by Lemma 1.5.3, H(n)0 is the Hopf algebra semi-oppositeto a variant of the Hopf algebras considered in [78, loc cit]. (Of course, onecan directly check by mimicking the proof of [78, Th. 4.5] that H(n)0 withthe operations defined above satisfies the axioms of a Hopf algebra.)

Remarks 1.5.5. (1) Because of Lemma 1.5.2, we are able to make the rootdatum used in the second relation in (a) invisible in the definition of H(n)0.

(2) Besides the modification of changing K to K in comultiplication andantipode, we also used the Euler form , (or rather the form , rd for the rootdatum given in Definition 1.5.1) instead of the symmetric Euler form ( , ) usedin Xiaos definition of H ([78, p. 129]) for the commutator formulas betweenK and u+A . This means that H is not isomorphic to H(n)

0. However, there

is a Hopf algebra homomorphism from H to H(n)0 by sending u+A K tou+A K whose image is the (Hopf) subalgebra H(n)0 generated by Ki andu+A , for all i I and A + (n). Note that it sends the central elementK1 Kn = 1 in H to K1 Kn = 1 in H(n)0.

(3) The above modifications are necessary for compatibility with Lusztigsconstruction for quantum groups in [54] and with the corresponding relationsin affine quantum Schur algebras.

It is clear that the subalgebra of H(n)0 generated by u+A (A + (n)) isisomorphic to H(n). The subalgebra generated by K , ZI , is isomorphicto the Laurent polynomial ring Q(v)[K11 , . . . , K1n ].

Corollary 1.5.6. The Q(v)-space H(n)0 with basis {KuA | ZI, A + (n)} becomes a Hopf algebra with the following algebra, coalgebra, andantipode structures.

1.5. Hopf structure on extended RingelHall algebras 29

(a) Multiplication and unit: for all A, B + (n) and , ZI ,uAu

B =

C+ (n)

vd(B),d(A)CB,AuC ,

uA K = vd(A),KuA ,K K = K+, and

1 = u0 = K0.(b) Comultiplication and counit: for all C + (n) and ZI ,

(uC ) =

A,B+ (n)vd(B),d(A) aAaB

aCCA,B Kd(A)u

B uA ,

(K) = K K,(uC ) = 0 (C = 0), and (K) = 1.

(c) Antipode: for all C + (n) and ZI ,(uC ) = C,0 +

m1

(1)m

D+ (n)C1,...,Cm+ (n)

v2

i< j d(Ci ),d(C j ) aC1 aCmaC

CC1,...,Cm DC1,...,Cm Kd(C)uD and(K) = K.

Proof. Let H be the Q(v)-space with basis {K (uA) | ZI, A + (n)}and define the following operations on H:

(a) for all A, B + (n) and , ZI ,(uA)

(uB ) =

C+ (n)

vd(B),d(A)CB,A(uC )

,

(uA)K = vd(A),K (uA),

K K = K +,and let 1 = (u0 ) = K 0;

(b) for all C + (n) and ZI ,((uC )

) =

A,B+ (n)vd(A),d(B) aAaB

aCCA,B(u

B )

K d(B)(uA),

(K ) = K K ,((uA)

) = 0 (A = 0), and (K ) = 1,where K i = K i (K i+1)1;

30 1. Preliminaries

(c) for all C + (n) and ZI ,((uC )

) = C,0 +m1

(1)m

D+ (n)C1,...,Cm+ (n)

v2

i< j d(Ci ),d(C j ) aC1 aCmaC

CC1,...,Cm DC1,...,Cm (uD) K d(C) and(K ) = K .

By Lemma 1.5.3, if we replace the multiplication of H(n)0 by its oppositeone and by 1 and keep other structure maps unchanged, then we obtainthe semi-opposite Hopf algebra Hop of H(n)0. It is clear that the Q(v)-linear isomorphism Hop H taking u+A K (uA)K preserves all theoperations. Thus, H is a Hopf algebra with the operations (a)(c).

Now, for ZI and A + (n), set in H,K := K and uA := vd(A),d(A) K d(A)(uA). (1.5.6.1)

Then {KuA | ZI, A + (n)} is a new basis of H. It is easy to checkthat applying the operations (a)(c) to the basis elements KuA gives (a)(c). Consequently, H(n)0 with the operations (a)(c) is a Hopf algebra.

The proof above shows that H(n)0 is the semi-opposite Hopf algebra ofH(n)0 in the sense that multiplication and antipode are replaced by theopposite and inverse ones, respectively. Note that the inverse 1 of inH(n)0 is defined by

1(uC ) = C,0+m1

(1)m

D+ (n)C1,...,Cm+ (n)

aC1 aCmaC

CC1,...,Cm DCm ,...,C1 u

D Kd(C).

By (a), the subalgebra of H(n)0 generated by uA (A + (n)) isisomorphic to H(n)op. Moreover, there is a Q(v)-vector space isomorphism

H(n)0 = Q(v)[K 11 , . . . , K1n ] Q(v) H(n)op. (1.5.6.2)Remark 1.5.7. If we put A = Z[(vm 1)1]m1, then (1.2.0.3) guaranteesthat extended RingelHall algebras H(n)0A and H(n)

0A over A are well-

defined. Thus, if v is specialized to z in a field (or a ring) F which is not a rootof unity, then extended RingelHall algebras H(n)0F := H(n)0A F andH(n)0F over F are defined.

2Double RingelHall algebras of cyclic quivers

A Drinfeld double refers to a construction of gluing two Hopf algebras viaa skew-Hopf pairing between them to obtain a new Hopf algebra. We applythis construction in 2.1 to the extended RingelHall algebras H(n)0 andH(n)0 discussed in 1.5 to obtain double RingelHall algebras D(n) ofcyclic quivers.

The algebrasD(n) possess a rich structure. First, by using the SchiffmannHubery generators and the connection with the quantum enveloping algebraassociated with a BorcherdsCartan matrix, we obtain a presentation forD(n)(Theorem 2.3.1). Second, by using Drinfelds new presentation for the quan-tum loop algebra U(gln), we extend Becks (and Jings) embedding of quantumaffine sln into U(gln) to obtain an isomorphism between D(n) and U(gln)(Theorem 2.5.3). Finally, applying the skew-Hopf pairing to semisimple gen-erators yields certain commutator relations (Theorem 2.6.3). Thus, we proposea possible presentation using semisimple generators; see Problem 2.6.4.

2.1. Drinfeld doubles and the Hopf algebra D(n)

In this section, we first recall from [46] the notion of a skew-Hopf pairing anddefine the associated Drinfeld double. We then apply this general constructionto obtain the Drinfeld double D(n) of the RingelHall algebra H(n); see[78] for a general construction.

LetA =(A , A , A ,A , A , A ) andB= (B, B, B,B, B, B)be two Hopf algebras over a field F. A skew-Hopf pairing of A and B is anF-bilinear form : A B F satisfying:

(HP1) (1, b) = B(b), (a, 1) = A (a), for all a A , b B;(HP2) (a, bb) = (A (a), b b), for all a A , b, b B;

31

32 2. Double RingelHall algebras of cyclic quivers

(HP3) (aa, b) = (a a,opB(b)), for all a, a A , b B;(HP4) (A (a), b) = (a, 1B (b)), for all a A , b B,where (a a, b b) = (a, b)(a, b), and opB is defined by opB(b) =

b2 b1 if B(b) =

b1 b2. Note that we have assumed here that Bis invertible.

Let A B be the free product of F-algebras A and B with identity. ThenA B is the coproduct of A andB in the category of F-algebras. More pre-cisely, for any fixed bases BA and BB forA ,B, respectively, where both BAand BB contain the identity element, A B is the F-vector space spanned bythe basis consisting of all words b1b2 bm (bi (BA \{1}) (BB\{1}))of any length m 0 such that bi bi+1 is not defined (in other words,bi , bi+1 are not in the same A orB) with multiplication given by contractedjuxtaposition

(b1 bm) (b1 bm) =

b1 bmb1 bm, if bmb1 is not defined;b1 bm1cb2 bm , if bmb1 is defined,

bmb1 = 0;0, otherwise.

Note that, since c = bmb1 =

k kak , where all ak BA or all ak BB , isa linear combination of basis elements, the element b1 bm1cb2 bm is alinear combination of words and is defined inductively. Thus, 1 is replaced bythe empty word.

Let I = IA ,B be the ideal of A B generated by(b2 a2)(a1, b1)

(a1 b1)(a2, b2) (a A , b B), (2.1.0.1)

where A (a) =

a1 a2 and B(b) =

b1 b2. Moreover, I is alsogenerated by

b a

(a1, B(b1))(a2 b2)(a3, b3) (a A , b B), (2.1.0.2)

where (2)A (a) =

a1 a2 a3 and (2)B (b) =

b1 b2 b3 (see, forexample, [46, p. 72]).

The Drinfeld double of the pair A and B is by definition the quotientalgebra

D(A ,B) := A B/I.By (2.1.0.2), each element in D(A ,B) can be expressed as a linear combina-tion of elements of the form a b + I for a A and b B. Note that thereis an F-vector space isomorphism

2.1. Drinfeld doubles and the Hopf algebra D(n) 33

D(A ,B) A FB, a b + I a b;see [71, Lem. 3.1]. For notational simplicity, we write a b + I as a b.Both A andB can be viewed as subalgebras of D(A ,B) via a a 1 andb 1 b, respectively. Thus, if x and y lie in A or B, we write xy insteadof x y.

The algebra D(A ,B) admits a Hopf algebra structure induced by thoseof A and B; see [46, 3.2.3]. More precisely, comultiplication, counit, andantipode in D(A ,B) are defined by

(a b) =

(a1 b1) (a2 b2),(a b) = A (a)B(b), and(a b) = B(b) A (a),

(2.1.0.3)

where a A , b B, A (a) =

a1 a2, and B(b) =

b1 b2.By modifying [71, Lem. 3.2], we obtain that the above ideal I can be gen-

erated by the elements in certain generating sets of A and B as described inthe following result which will be used in 2.6.

Lemma 2.1.1. Let A ,B be Hopf algebras over F and let : A B Fbe a skew-Hopf pairing. Assume that XA A and XB B are generatingsets of A and B, respectively. If A (XA ) spanF XA spanF XA andB(XB) spanF XB spanF XB , then the ideal I = IA ,B is generatedby the following elements

(b2 a2)(a1, b1)

(a1 b1)(a2, b2), for all a XA , b XB.(2.1.1.1)

Proof. For a A and b B, putha,b :=

(b2 a2)(a1, b1)

(a1 b1)(a2, b2).

Let I be the ideal of A B generated by ha,b for all a XA , b XB , andset H = A B/I . We need to show I = I . Since I I, it remains toshow that for all a A and b B, ha,b I , or equivalently, ha,b = 0 inH .

First, suppose a XA and b = y1 yt with y j XB . We proceed byinduction on t to show that ha,b I . Let b = y1 yt1 and b = yt . WriteA (a) =

a1 a2, A (a1) =

a1,1 a1,2 and A (a2) =

a2,1 a2,2.

The coassociativity of A implies thata1,1a1,2a2 =

A (a1)a2 =

a1A (a2) =

a1a2,1a2,2.

34 2. Double RingelHall algebras of cyclic quivers

Further, write B(b) =

b1b2 and B(b) =

b1b2 . Then B(b) =b1b1 b2b2 . Thus, we obtain inH that

(b2b2 a2)(a1, b1b1)=

b2(b2 a2)(a1,1, b1)(a1,2, b1)=

b2(b2 a2,2)(a1, b1)(a2,1, b1)=

(b2 a2,1) b1(a1, b1)(a2,2, b2)(since a2 spanF XA and b XB)

=

(b2 a1,2) b1(a1,1, b1)(a2, b2)=

(a1,1 b1)b1(a1,2, b2)(a2, b2) (by induction for a1 and b)=

(a1 b1)b1(a2,1, b2)(a2,2, b2)=

(a1 b1b1)(a2, b2b2),that is, ha,b I .

Now suppose a = x1 xs A with xi XA and b B. We proceedby induction on s. The case s = 1 has already been treated above. So assumes > 1. Let a = x1 xs1 and a = xs . Then

(b2 a2a2 )(a1a1 , b1)=

(b(2)3 a2)a2(a1, b(2)2 )(a1 , b(2)1 )=

a1 (b(2)2 a2 )(a2, b(2)3 )(a1 , b(2)1 ) (by induction)=

a1a1 b(2)1 (a2, b(2)3 )(a2 , b(2)2 ) (since a XA )=

a1a1 b1(a2a2 , b2),

where (2)B (b) =

b(2)1 b(2)2 b(2)3 . Hence, ha,b I . This completes theproof.

A skew-Hopf pairing can be passed on to opposite Hopf algebras (seeLemma 1.5.3). The following lemma can be checked directly.Lemma 2.1.2. Let A = (A , A , A ,A , A , A ), B = (B, B, B,B, B, B) be two Hopf algebras over a field F together with a skew-Hopf pairing : A B F. Assume that A and B are bothinvertible. Then is also a skew-Hopf pairing of the semi-opposite Hopf alge-bras (A , A , A ,

opA , A ,

1A ) and (B,

opB, B,B, B,

1B ) (resp.,

(A , opA , A ,A , A ,

1A ) and (B, B, B,

opB, B,

1B )).

2.1. Drinfeld doubles and the Hopf algebra D(n) 35

We end this section with the construction of the Drinfeld double associatedwith the RingelHall algebras H(n)0 and H(n)0 introduced in 1.5.

First, we need a skew-Hopf pairing. Applying Lemma 2.1.2 to [78,Prop. 5.3] yields the following result. For completeness, we sketch a proof. Weintroduce some notation which is used in the proof. For each = iI ai i ZI , write = iI ai1i . In particular, for each A + (n), we haved(A) = d( (A)). Then, for , ZI ,

, = ( ) = , and K = K = K a11 K ann .

Proposition 2.1.3. The Q(v)-bilinear form : H(n)0 H(n)0 Q(v)defined by

(u+A K, KuB ) = vd(A),d(A)++2d(A)a1A A,B, (2.1.3.1)

where , ZI and A, B + (n), is a skew-Hopf pairing.

Proof. Condition (HP1) is obvious. We now check condition (HP2). Withoutloss of generality, we take a = u+A K , b = KuB , and b = K uC for, , ZI and A, B,C + (n). Then

(u+A K, KuB K u

C ) = (u+A K, vd(B), K+ uB uC )

=(u+A K, vd(B), K+

Dvd(C),d(B)DC,Bu

D)

= vx1AC,Ba1A ,where x1 = d(B), +d(C), d(B)+ (+ )d(A), d(A)++2d(A).

On the other hand,

((u+A K), KuB K uC )

=(B,C

vd(B ),d(C ) aBaC

aAAB,C u

+C K u+B Kd(C )K, KuB K uC )

=B,C

vd(B),d(C ) aBaC

aAAB,C (u

+C K, Ku

B )(u

+B Kd(C )K, K u

C )

= vx2AC,Ba1A ,where x2 = d(C), d(B) + d(B), d(B) + + (d(B) d(B) +) d(C), d(C) + d(B) d(B) + + 2d(A). Here we have assumedd(A) = d(B) + d(C) since AC,B = 0 otherwise. A direct calculation showsx1 = x2. Hence, (HP2) holds.

36 2. Double RingelHall algebras of cyclic quivers

Condition (HP3) can be checked similarly as (HP2). It remains to check(HP4). Take a = u+A K and b = KuB . We may suppose A = 0 = B. Then

((u+A K), KuB )

=m1

(1)mvy1

C1,...,Cm+ (n)

aC1 aCmaAaB

AC1,...,CmBCm ,...,C1

and

(u+A K, 1(KuB ))

=m1

(1)mvy2

C1,...,Cm+ (n)

aC1 aCmaAaB

AC1,...,Cm BCm ,...,C1,

where

y1 = d(B), +(d(A)+d(A))d(B), d(B)d(A)+d(A)and

y2 = d(A), d(B)d(B)+ (d(B)d(B))d(A), d(A)+.Clearly, if d(A) = d(B), then AC1,...,Cm BCm ,...,C1 = 0. Hence, we may supposed(A) = d(B). Then

y1 = d(A), + d(A), d(A) = y2.Therefore, ((u+A K), Ku

B ) = (u+A K, 1(KuB )), that is, (HP4)

holds.

Second, the Hopf algebras H(n)0 and H(n)0 together with the skew-Hopf pairing give rise to the Drinfeld double

D(n) := D(H(n)0,H(n)0).Since as Q(v)-vector spaces, we have

D(n) = D(H(n)0,H(n)0) = H(n)0 H(n)0,we sometimes write the elements in D(n) as linear combinations of a bfor a H(n)0 and b H(n)0. Moreover, it follows from (1.5.3.1) and(1.5.6.2) that there is a Q(v)-vector space isomorphism

D(n) =H(n) Q(v) Q(v)[K11 , . . . , K1n ]Q(v) Q(v)[K11 , . . . , K1n ] Q(v) H(n)op.

Finally, we define the reduced Drinfeld double

D(n) = D(n)/I , (2.1.3.2)

2.2. SchiffmannHubery generators 37

where I denotes the ideal generated by 1 K K 1, for all ZI .By the construction,I is indeed a Hopf ideal of D(n). Thus, D(n) is againa Hopf algebra. We call D(n) the double RingelHall algebra of the cyclicquiver (n).

Let D(n)+ (resp., D(n)) be the Q(v)-subalgebra of D(n) generatedby u+A (resp., uA) for all A + (n). Let D(n)0 be the Q(v)-subalgebra ofD(n) generated by K for all ZI . ThenD(n)+ = H(n), D(n) = H(n)op, and D(n)0 = Q(v)[K 11 , . . . , K1n ].

(2.1.3.3)Moreover, the multiplication map

D(n)+ D(n)0 D(n) D(n)is an isomorphism of Q(v)-vector spaces. Also, we have

D(n)0 := D(n)+ D(n)0 = H(n)0 andD(n)0 := D(n)0 D(n) = H(n)0.

We will identify D(n)0 and D(n)0 with H(n)0 and H(n)0, respec-tively, in the sequel. In particular, we may use the PBW type basis for H(n)to display a PBW type basis for D(n):

{u+A K uB | A, B + (n), ZI }.

Remark 2.1.4. By specializing v to z C which is not a root of unity,(1.2.0.3) implies that the skew-Hopf pairing given in (2.1.3.1) is well-defined over C. Hence the construction above works over C with H(n)0,etc., replaced by the corresponding specialization H(n)0C , etc. (see Remark1.5.7). Thus, we obtain the double RingelHall algebra D,C(n) over C. Infact, if we use the ring A defined in Remark 1.5.7, the same reasoning showsthat the skew-Hopf pairing in (2.1.3.1) is defined over A. Hence, we can forma double RingelHall algebra D(n)A. Then D,C(n) = D(n)A C andD(n) = D(n)A Q(v).

2.2. SchiffmannHubery generatorsIn this and the following sections, we will investigate the structure ofD(n) byrelating it with the quantum enveloping algebra of a generalized KacMoodyalgebra based on [67, 39]; see also [38, 14].

We first construct certain primitive central elements from the central ele-ments of H(n) defined in (1.4.0.1). Recall that an element of a Hopf algebrawith comultiplication is called primitive if

38 2. Double RingelHall algebras of cyclic quivers

(x) = x 1 + 1 x .For m 1, let

cm = (1)mv2nm

A(1)dim End(M(A))aAuA D(n), (2.2.0.1)

where the sum is taken over all A + (n) such that d(A) = m and soc M(A)is square-free. We also define c0 = 1 by convention. By Theorem 1.4.1, theelements c+m and cm are central in D(n)+ and D(n), respectively.

Following [39, 4], let C(u) = 1 +m1 cm um be the generating func-tions in indeterminate u associated with the sequence {cm }m1 and defineelements xm by

X(u) =m1

xm um1 =d

dulog C(u) = 1

C(u)d

duC(u).

Thus, for each m 1,

xm = mcm m1s=1

xs cms . (2.2.0.2)

Recall from (1.2.0.6) the elementsuA = vdim End(M(A))dim M(A)uA D(n), for A + (n).

In particular, for each 1 l n and m 1,

uEl,l+mn = vmnmuEl,l+mn .

The following lemma can be deduced from [40, Lem. 12]. However, weprovide here a proof for completeness.

Lemma 2.2.1. For each m 1,

xm = vnm(vm vm)n

l=1uEl,l+mn + (v v

1)2 ym , (2.2.1.1)

where ym are Z-linear combinations of certain uA such that d(A) = m andM(A) are decomposable.

Proof. By (1.2.0.3), for each A + (n), aA Z is divisible by (vv1)(A),where (A) = 1in, jZ ai, j . Note that (A) equals the number of inde-composable summands in M(A). Since aEl,l+mn = v2(m1)(v2 1), for each1 l n and m 1, it follows from (2.2.0.1) that

cm v(1n)m1(v v1)n

l=1uEl,l+mn mod (v v

1)2I,

2.2. SchiffmannHubery generators 39

where I denote the Z-submodules of D(n) spanned by all the uAsatisfying that M(A) are decomposable (i.e., (A) > 1).

Since subrepresentations and quotient representations of each indecompos-able representation of (n) are again indecomposable, it follows that, for eachA + (n),

uAI I and IuA I.

It suffices to show that

xm vnm(vm vm)n

l=1uEl,l+mn mod (v v

1)2I.

We proceed by induction on m. If m = 1, it is trivial since x1 = c1 . Let nowm > 1. The inductive hypothesis together with (2.2.0.2) implies that

xm mv(1n)m1(v v1)n

l=1uEl,l+mn

m1s=1

(vns(vs vs)

nl=1

uEl,l+sn)

(v(1n)(ms)1(v v1) nl=1

uEl,l+(ms)n)

mod (v v1)2I.

It is clear that for 1 l, l n,

uEl,l+sn uEl,l+(ms)n

l,l uEl,l+mn mod I.

We conclude that

xm v(1n)m(m(1 v2)

m1s=1

(1 v2)(1 v2s)) nl=1

uEl,l+mn

vnm(vm vm)n

l=1uEl,l+mn mod (v v

1)2I.

We further set

zm =vnm

vm vm xm D(n), for m 1.

Applying (2.2.1.1) gives that

zm =n

l=1uEl,l+mn +

vnm(v v1)[m] y

m . (2.2.1.2)

40 2. Double RingelHall algebras of cyclic quivers

Hence, the elements z+m (resp., zm ) are central in D(n)+ (resp., in D(n)).Moreover, by Theorem 1.4.1, we have

D(n) = C(n) Q(v)[c1 , c2 , . . .] = C(n) Q(v)[z1 , z2 , . . .],(2.2.1.3)

where C(n) are the composition algebras generated by ui , 1 i n.Let be the comultiplication on D(n) induced by Greens comultipli-

cation and be the antipode of D(n) induced by Xiaos antipode; seeProposition 1.5.4 and Corollary 1.5.6.

Proposition 2.2.2. For each m 1, the elements z+m and zm satisfy(zm) = zm 1 + 1 zm and (zm) = zm .

Moreover, for all i I and m,m 1,[z+m, zm ] = 0, [z+m, ui ] = 0, [u+i , zm ] = 0, and [zm, Ki ] = 0.

In other words, zm are central elements in D(n).

Proof. By [39, Prop. 9], for each m 1,

(cm ) =m

s=0cs cms .

Applying (2.2.0.2) implies that xm are primitive; see [39, Cor. 10]. Hence, zmare also primitive, that is,

(zm) = zm 1 + 1 zm .Since D(n) is a Hopf algebra, we have ( 1)(zm) = (zm) = 0 for

each m 1. Thus,

0 = ( 1)(zm 1 + 1 zm) = (zm) + zm,i.e., (zm) = zm .

Since (z+m) = z+m 1+ 1 z+m and (zm) = zm 1+ 1 zm , applying(2.1.0.1) to z+m and zm gives

(z+m, zm) + z+m(1, zm) + zm(z+m , 1) + zmz+m(1, 1)=z+mzm(1, 1) + zm(z+m, 1) + z+m(1, zm) + (z+m, zm).

It follows from (1, 1) = 1 that zmz

+m = z+mzm , i.e., [z+m, zm ] = 0.

Similarly, we have [z+m, ui ] = [u+i , zm ] = [zm, Ki ] = 0. The last assertionfollows from (2.2.1.3).

2.3. Presentation of D(n) 41

By Theorem 1.3.4, there are isomorphisms

C(n)+ U(sln)+, u+i Ei , and C(n)

U(sln), ui Fi .Here we have applied the anti-involution U(sln)+ U(sln), Ei Fi .

By (2.2.1.3), there are decompositionsD(n)+ = C(n)+ Q(v) Z(n)+ and D(n) = C(n) Q(v) Z(n),

(2.2.2.1)where Z(n) := Q(v)[z1 , z2 , . . .] are the polynomial algebras in zm form 1.

Remark 2.2.3. If z C is not a root of unity and H(n)C := H(n) C isthe C-algebra obtained by specializing v to z, then, by (1.2.0.3), we may use(2.2.0.2) to recursively define the central elements zm in H(n)C and, hence,elements zm in D,C(n); see Remark 2.1.4. Thus, a C-basis similar to theone given in Corollary 1.4.2 can be constructed for D,C(n). In particular,we obtain decompositions

D,C(n) = C(n)C C[z1 , z2 , . . .],and hence the C-algebra D,C(n) can be presented by generatorsu+i , u

i , Ki , K

1i , z

+s , z

s , i I, s Z+ and relations (QGL1)(QGL8)

as given in the last statement of Theorem 2.3.1 below.1

2.3. Presentation of D(n)Recall from (1.3.2.1) the Cartan matrix C = (ci, j ) of affine type An1 andthe quantum group U(sln) associated with C as given in Definition 1.3.3. Byidentifying C(n) with U(sln) under the isomorphism in Theorem 1.3.4, wedescribe a presentation for D(n) as follows.

Theorem 2.3.1. The double RingelHall algebra D(n) of the cyclic quiver(n) is the Q(v)-algebra generated by Ei = u+i , Fi = ui , Ki , K1i ,z+s , zs , i I, s Z+ with relations (i, j I and s, t Z+):(QGL1) Ki K j = K j Ki , Ki K1i = 1;(QGL2) Ki E j = vi, ji, j+1 E j Ki , Ki Fj = vi, j+i, j+1 Fj Ki ;(QGL3) Ei Fj Fj Ei = i, j KiK

1i

vv1 , where Ki = Ki K1i+1;1 We will see in Proposition 2.4.5 that D,C(n) can be obtained as a specialization from theZ-algebra D(n) by the base change Z C, v z. Thus, we can use the notation D(n)Cfor this algebra.

42 2. Double RingelHall algebras of cyclic quivers

(QGL4)

a+b=1ci, j(1)a

[1 ci, j

a

]Eai E j E

bi = 0, for i = j ;

(QGL5)

a+b=1ci, j(1)a

[1 ci, j

a

]Fai Fj F

bi = 0, for i = j ;

(QGL6) z+s z+t = z+t z+s ,