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Quivers with relations for symmetrizable Cartanmatrices II : Convolution algebras

C. Geiss, B. Leclerc, and J. Schroer

December 13, 2018

Abstract

We realize the enveloping algebra of the positive part of a symmetrizable Kac-Moody al-gebra as a convolution algebra of constructible functions on module varieties of some Iwanaga-Gorenstein algebras of dimension 1.

1 Introduction

Let Q be a finite quiver without oriented cycles. LetC be the symmetric generalized Cartan ma-trix corresponding to the undirected graph underlyingQ, and letg = g(C) be the Kac-Moody Liealgebra attached toC. Kac [K1] has shown that the dimension vectors of the indecomposablerepresentations ofQ form the roots of the positive partn of g. For quiversQ of finite type, Ringel[Rin1, Rin2] found a direct construction of the Lie algebran itself, and of its enveloping algebraU(n), in terms of the representation theory ofQ. He used Hall polynomials counting extensionsof representations overFq, and recoveredU(n) as the Hall algebra of the path algebraFqQ spe-cialized atq= 1. Later, Schofield [S] replaced counting points of varieties overFq by taking Eulercharacteristic of complex varieties, and extended Ringel’s result to an arbitrary quiver (see also[Rie] in the finite-type case). Finally, Lusztig [L1] reformulated Schofield’s construction and ob-tainedU(n) as a convolution algebra of constructible functions over the affine spaces rep(Q,d) ofrepresentations ofCQ with dimension vectord.

In this paper we prove a broad generalization of Schofield’s theorem. We take forC an arbitrarysymmetrizable generalized Cartan matrix [K2, §1.1, §2.1]. This means that there exists a diagonalmatrixD with positive integer diagonal entries such thatDC is symmetric. The corresponding Kac-Moody algebrag = g(C) is called symmetrizable. With this datum together with an orientationΩof the graph naturally attached toC, we have associated in [GLS] a finite-dimensional algebraH = H(C,D,Ω) defined by a quiver with relations. This algebra makes sense over an arbitraryfield K, but here we fixK = C so that varieties ofH-modules are complex varieties. WhenC issymmetric andD is the unit matrix,H is just the path algebra of the quiverQ corresponding toCandΩ. In general, it is shown in [GLS] that H is Iwanaga-Gorenstein of dimension 1, and thatits category of locally free modules (i.e. modules of homological dimension⩽ 1) carries an Eulerform whose symmetrization is given byDC. The affine varieties repl.f.(H,r) of locally freeH-modules with rank vectorr are smooth and irreducible (Proposition 3.4). By analogy with [S, L1],we then introduce a convolution bialgebraM=M(H) of constructible functions on the varietiesrepl.f.(H,r).We show thatM is a Hopf algebra isomorphic to the enveloping algebra of theLiealgebra of its primitive elements (Proposition 3.8). Let again n denote the positive part of thesymmetrizable Kac-Moody algebrag. Our main result is then:

1

Theorem 1.1. The Hopf algebraM(H) is isomorphic to U(n).

This generalizes Schofield’s theorem in two directions. Firstly, to the best of our knowledge,Theorem 1.1 gives the first complex geometric construction of U(n) for a symmetrizable general-ized Cartan matrix. Secondly, note that ifD is a symmetrizer forC, thenkD is also a symmetrizerfor everyk ∈ Z>0. As k increases, the categories of locally free modules overH(C,kD,Ω) be-come more and more rich and complicated, the dimension of thevarieties repl.f.(H(C,kD,Ω),r)increase and their orbit structure gets finer, but the convolution algebrasM(H(C,kD,Ω)) remainthe same. Thus for every symmetrizable Kac-Moody algebrag we get an infinite series of exactcategories repl.f.(H(C,kD,Ω)) (k≥ 1) whose convolution algebrasM(H(C,kD,Ω)) are isomor-phic toU(n). This appears to be new, even whenC is symmetric.

Theorem 1.1 implies that every positive root ofg is the rank vector of an indecomposableobject of repl.f.(H(C,kD,Ω)). However, the converse is not true, and already for a matrixC offinite typeB3 with minimal symmetrizerD, one can find indecomposable locally freeH-moduleswhose rank vector is not a root, see [GLS, §13.7]. But by Theorem 1.1, any primitive element ofM vanishes on such an indecomposable module.

The proof of Theorem 1.1 is largely inspired from [S], but with some non-trivial modifications.In fact whenC is symmetric andD is the unit matrix the algebraM coincides with the algebradenoted byR+(Q) in [S]. However in all other cases,M is defined using filtrations which are notcomposition series, and so we have to adapt all the argumentsof [S] to our setting (compare forinstance [S, Lemma 3.6.2] with Proposition 4.13 below). We also note that, following [L1] andin contrast to [S], we systematically use the language of constructible functions and convolutionproducts. This allows us to simplify some key steps (comparefor instance the proof of [S, Theorem4.3] with that of Proposition 5.2 below).

We hope that our complete and detailed exposition of the proof of Theorem 1.1 will also beuseful for readers interested in Schofield’s original theorem. Indeed, although this important resulthas been cited by many authors (e.g. [L1], [J], [BT]), the manuscript [S] remains unpublished,and we do not know of any other proof available in the literature. The reader only interested in thecase of path algebras may read our paper assuming everywherethatC is symmetric andD is theunit matrix.

Let us outline the structure of the paper and the main steps ofthe proof of Theorem 1.1. InSection 2 we review the definition ofH(C,D,Ω) and the results of [GLS] which will be neededin the sequel. In Section 3 we introduce the algebra of constructible functionsM(H) and showthat it is a homomorphic image ofU(n) (Corollary 3.11). The rest of the paper is devoted to theproof that this homomorphism is an isomorphism. To do that wefollow Schofield’s strategy andintroduce in Section 4 a new algebraD constructed as a “limit” of certain algebras of constructiblefunctionsDP indexed by the projectiveH-modulesP. This algebraD contains two copies ofM(Proposition 4.4 and Corollary 4.6). Various relations satisfied by the generators ofD are obtainedin §4.4. These relations are very close to the relations satisfied by the Chevalley generators ofU(g), but a few relations do not match. To overcome this problem, in Section 5 one considers acertain quotientE of a “Borel” subalgebraD≥0 ofD, and one constructs a Lie algebraL of partiallydefined derivations onE . One then shows that the generators ofL satisfy all the defining relationsof the Kac-Moody algebrag (Theorem 5.8). Applying the Gabber-Kac theorem, one deduces thatL is in fact isomorphic tog, and finally thatM is isomorphic toU(n) (Theorem 5.10). The papercloses with the description of two basic examples of algebrasM (Section 6).

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2 The algebraH

2.1 Definition of H(C,D,Ω)

We retain the notation of [GLS]. Let C = (ci j ) ∈Mn(Z) be a symmetrizable generalized Cartanmatrix, and letD = diag(c1, . . . ,cn) be a symmetrizer ofC. This means thatci ∈Z>0, and

cii = 2, ci j ⩽ 0 for i /= j, cici j = c jc ji .

Whenci j < 0 definegi j ∶= ∣gcd(ci j ,c ji)∣, fi j ∶= ∣ci j ∣/gi j .

An orientation ofC is a subsetΩ ⊂ 1,2, . . . ,n×1,2, . . . ,n such that the following hold:

(i) (i, j),( j, i)∩Ω /= ∅ if and only if ci j < 0;

(ii) For each sequence of the form((i1, i2),(i2, i3), . . . ,(it , it+1)) with t ≥ 1 and(is, is+1) ∈Ω forall 1⩽ s⩽ t we havei1 /= it+1.

For an orientationΩ of C let Q ∶=Q(C,Ω) ∶= (Q0,Q1) be the quiver with vertex setQ0 ∶= 1, . . . ,nand with arrow set

Q1 ∶= α(g)i j ∶ j → i ∣ (i, j) ∈Ω,1⩽ g⩽ gi j ⋃ εi ∶ i → i ∣ 1⩽ i ⩽ n.

LetH ∶=H(C,D,Ω) ∶=CQ/I

whereCQ is the path algebra ofQ, andI is the ideal ofCQ defined by the following relations:

(H1) (nilpotency) for eachi we haveεci

i = 0;

(H2) (commutativity) for each(i, j) ∈Ω and each 1⩽ g⩽ gi j we have

ε f ji

i α(g)i j = α(g)i j ε fi jj .

This definition is illustrated by many examples in [GLS, Section 13].Clearly, H is a finite-dimensionalC-algebra. It is known [GLS, Theorem 1.1] thatH is

Iwanaga-Gorenstein of dimension 1. This means that for anH-moduleM we have

(proj.dim(M) <∞)⇐⇒ (inj.dim(M) <∞)⇐⇒ (proj.dim(M) ⩽ 1)⇐⇒ (inj.dim(M) ⩽ 1).

Moreover, if M is a submodule of a projectiveH-module and if proj.dim(M) ⩽ 1, thenM isprojective. Dually, ifM is a quotient module of an injectiveH-module and inj.dim(M) ⩽ 1 thenM is injective.

Note that ifC is symmetric and ifD = diag(1, . . . ,1), thenH is isomorphic to the path algebraCQ, whereQ is the acyclic quiver obtained fromQ by deleting all loopsεi . More generally, itis easy to see that ifC is symmetric andD = diag(k, . . . ,k) for somek> 0, thenH is isomorphic toRQ ∶=R⊗CCQ, whereR is the truncated polynomial ringC[x]/(xk). In that case,H-modulesare nothing else than representations ofQ over the ringR. WhenC is only symmetrizable, onehas a similar picture by replacing the path algebraRQ by a modulated graph over a family oftruncated polynomial rings, as we shall now explain.

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2.2 Modulated graphs

It was shown in [GLS, §5] thatH gives rise to a modulated graph, and that the category ofH-modules is isomorphic to the category of representations ofthis modulated graph. This viewpoint,which is very close to Dlab and Ringel’s theory of species [DR], will be useful in several placesbelow.

For i = 1, . . . ,n, let Hi be the subalgebra ofH generated byεi. Thus Hi is isomorphic toC[x]/(xci ). For(i, j) ∈Ω we define

iH j ∶=Hi SpanC(α(g)i j ∣ 1⩽ g⩽ gi j )H j .

It is shown in [GLS] that iH j is anHi-H j -bimodule, which is free as a leftHi-module and free asa rightH j -module. AnHi-basis ofiH j is given by

α(g)i j ,α(g)i j ε j , . . . ,α(g)i j ε fi j−1

j ∣ 1⩽ g⩽ gi j .In particular, we have an isomorphism of leftHi-modules:iH j ≅H

∣ci j ∣i .

The tuple(Hi , iH j (i, j) ∈Ω) is called theoriented modulationassociated withH(C,D,Ω). Arepresentation(Mi,Mi j ) of this modulation consists of finite-dimensionalHi-modulesMi for eachi = 1, . . . ,n, and ofHi-linear maps

Mi j ∶ iH j ⊗H j M j →Mi

for each(i, j) ∈ Ω. Representations of this modulation form an abelian category rep(C,D,Ω)isomorphic to the category ofH-modules [GLS, Proposition 5.1]. (Here we identify the categoryof H-modules with the category of representations of the quiverQ(C,Ω) satisfying the relations(H1) and (H2).) Given a representation(Mi,Mi j ) in rep(C,D,Ω) the correspondingH-module

(Mi ,M(α(g)i j ),M(εi)) is obtained as follows: theC-linear mapM(εi)∶Mi →Mi is given by

M(εi)(m) ∶= εim.

(here we use thatMi is anHi-module), and for(i, j) ∈Ω, theC-linear mapM(α(g)i j )∶M j →Mi isdefined by

M(α(g)i j )(m) ∶=Mi j (α(g)i j ⊗m).The mapsM(α(g)i j ) andM(εi) satisfy the defining relations (H1) and (H2) ofH because the mapsMi j areHi-linear.

2.3 Locally freeH-modules

We say that anH-moduleM = (Mi,M(α(g)i j ),M(εi)) is locally free if for every i the Hi-moduleMi is free. By [GLS, Theorem 1.1],M is locally free if and only if proj.dim(M) ⩽ 1. The fullsubcategory repl.f.(H) whose objects are the finite-dimensional locally free modules is closedunder extensions, kernels of epimorphisms and cokernel of monomorphisms, and it has Auslander-Reiten sequences [GLS, Lemma 3.8, Theorem 3.9].

Therank vectorof M ∈ repl.f.(H) is then-tuple of integers rank(M)= (rk(Mi)). For j = 1, . . . ,nwe denote byE j the unique locally freeH-module with rank vector(δi j ∣ i = 1, . . . ,n). In otherwords,E j is nothing else thanH j regarded as anH-module in the obvious way.

ForM,N ∈ repl.f.(H) the integer

⟨M,N⟩H ∶= dimHomH(M,N)−dimExt1H(M,N)4

depends only on the rank vectors rank(M) and rank(N), see [GLS, Proposition 4.1]. The map(M,N)↦ ⟨M,N⟩H thus descends to a bilinear form on the Grothendieck groupZn of repl.f.(H),

given on the basisαi = rank(Ei) by

⟨αi ,α j⟩H =⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

cici j if ( j, i) ∈Ω,

ci if i = j,

0 otherwise.

Let (−,−)H be the symmetrization of⟨−,−⟩H defined by(a,b)H ∶= ⟨a,b⟩H + ⟨b,a⟩H , and letqH bethe quadratic form defined byqH(a) ∶= ⟨a,a⟩H . The formsqH and⟨−,−⟩H are called thehomolog-ical bilinear formsof H.

Note that(−,−)H is nothing else than the symmetric bilinear form

(αi ,α j) = cici j , (1⩽ i, j ⩽ n)associated with the symmetric matrixDC.

3 The convolution algebraM

3.1 Definition of the algebraM

Let d ∈Nn be a dimension vector. Let rep(H,d) be the affine complex variety ofH-modules withdimension vectord = (d1, . . . ,dn). By definition the closed points in rep(H,d) are tuples

M = (M(a))a∈Q1 ∈ ∏a∈Q1

HomC(Cds(a),C

dt(a))of C-linear maps such that

M(εi)ci = 0

and for each(i, j) ∈Ω and 1⩽ g⩽ gi j we have

M(εi) f ji M(α(g)i j ) =M(α(g)i j )M(ε j) fi j.

The groupGd ∶=GLd1×⋯×GLdn acts on rep(H,d) by conjugation. TheGd-orbit of M ∈ rep(H,d)is denoted byOM . TheGd-orbits of rep(H,d) are in one-to-one correspondence with isomorphismclasses ofH-modules with dimension vectord.

Recall that aconstructible functionon a complex algebraic varietyV is a mapϕ ∶V →C suchthat the image ofϕ is finite, and for eacha ∈ C the preimageϕ−1(a) is a constructible subsetof V. LetFd be the complex vector space of constructible functionsf ∶ rep(H,d)→ C which areconstant onGd-orbits, and let

F =F(H) = ⊕d∈NnFd.

We endowF with a convolution product∗ defined as in [L1, §10.12] or [L2, §2.1], using Eulercharacteristics of constructible subsets. Namely, we put

( f ∗g)(X) =∫Y⊆X

f (Y)g(X/Y)dχ , ( f ,g ∈F , X ∈ rep(H)).Here, the integral is taken on the variety of allH-submodulesY of X, and for a constructiblefunctionϕ on a varietyV, we set

∫Y∈V

ϕ(Y)dχ =∑a∈C

a⋅χ(ϕ−1(a)).

5

It is well-known that(F ,∗) has the structure of anNn-graded associativeC-algebra, seee.g.[BT,§4.2].

Let ei = (0, . . . ,ci , . . . ,0) ∈Nn be the dimension vector ofEi. Letθi ∈F denote the characteristicfunction of theGei -orbit of rep(H,ei) corresponding toEi.

Definition 3.1. We denote byM =M(H) the subalgebra of(F ,∗) generated byθi (1 ⩽ i ⩽ n),and we setMd =M∩Fd (d ∈Nn).

Note thatMd is a finite-dimensional vector space. The unit element1M ofM is the charac-teristic function of the zeroH-module.

Lemma 3.2. Let f ∈Md and X∈ rep(H,d). If X is not locally free then f(X) = 0.

Proof. For anH-moduleX and a sequencei = (i1, . . . , ik) ∈ 1, . . . ,nk, we have, by definition ofthe convolution product∗, (θi1 ∗⋯∗θik)(X) = χ(FX,i),whereFX,i is the constructible set of flags ofH-modules

0= X0 ⊂ X1 ⊂⋯ ⊂Xk = X with Xj/Xj−1 ≃Ei j (1⩽ j ⩽ k).By [GLS, Lemma 3.8] the category of locally freeH-modules is stable under extensions, hence ifX is not locally free we haveFX,i =∅ for every sequencei. This shows that(θi1 ∗⋯∗θik)(X) = 0for every sequencei, and thus, by definition ofM, that f (X) = 0 for every f ∈M.

Remark 3.3. When the Cartan matrixC is symmetric andD is the unit matrix, the algebraH isthe path algebraCQ (see §2.1). In that caseM(H) coincides with the algebraR+(CQ) of [S,§2.3], and with the algebraM0(Ω) of [L1, §10.19].

3.2 Varieties of locally freeH-modules

Let d ∈Nn be a dimension vector. IfM ∈ rep(H,d) is locally free, its rank vector isr = (r1, . . . ,rn)wherer i ∶= di/ci . Hence locally free modules can only exist ifdi is divisible byci for every i. Inthis case we say thatd is c-divisible. Let repl.f.(H,r) be the union of all orbitsOM of locally freemodulesM of rank vectorr . By Lemma 3.2, the support of every constructible functionf ∈Md

is contained in repl.f.(H,r). Consider the natural projection

π ∶ repl.f.(H,r)→ rep(H1,d1)×⋯× rep(Hn,dn), (M(α))α∈Q1 ↦ (M(ε1), . . . ,M(εn)).The image ofπ is O

E⊕r11×⋯×OE⊕rn

n, whereOE

⊕rii

is the Gdi -orbit of the freeHi-moduleE⊕r ii

of rank r i . (Note that rep(Hi ,di) is just a point ifci = 1.) We identify Im(π) with the Gd-orbitOEr of the locally freeH-moduleEr

∶=⊕ni=1E⊕r i

i . In particular,OEr is smooth and irreducible ofdimension

n

∑i=1

c2i r2

i −

n

∑i=1

cir2i .

Here, the summands of the first sum are the dimensions of the groupsGdi , while the summands ofthe second sum are the dimensions of the endomorphism rings EndHi(E⊕r i

i ).Proposition 3.4. Letd bec-divisible as above. Set ri ∶= di/ci andr = (r1, . . . ,rn). Thenrepl.f.(H,r)is a non-empty open subset ofrep(H,d). Moreover we have:

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(i) The restriction π ∶ repl.f.(H,r) → OEr of π to its image defines a vector bundle of rank∑(i, j)∈Ω ci ∣ci j ∣r ir j . In particular, repl.f.(H,r) is smooth and irreducible of dimension

n

∑i=1

ci(ci −1)r2i + ∑(i, j)∈Ω

ci ∣ci j ∣r i r j = dim(Gd)−qH(r).

(ii) If qH(r) ⩽ 0 thenrepl.f.(H,r) has infinitely many Gd-orbits.

Proof. Since the locally free modules are exactly the modules with projective dimension at most 1(see [GLS, Proposition 3.5]), upper semicontinuity yields that repl.f.(H,r) is open in rep(H,d).

By §2.2, the fibreπ−1(E⊕r11 , . . . ,E⊕rn

n ) can be identified with

⊕(i, j)∈Ω

HomHi (iH j ⊗H j E⊕r j

j , E⊕r ii ) = ⊕

(i, j)∈ΩHomHi (E⊕∣ci j ∣r j

i , E⊕r ii ) .

Therefore we easily calculate that

dim(π−1(E⊕r11 , . . . ,E⊕rn

n )) = ∑(i, j)∈Ω

ci ∣ci j ∣r ir j .

Next, notice thatπ is by constructionGd-equivariant. SinceOEr is a singleGd-orbit, all fibers ofπ are isomorphic, and, in particular, are vector spaces of thesame dimension.

Consider the trivial vector bundle

X ∶=⎛⎝⊕α∈Q1 HomC(Cds(α)

,Cdt(α))⎞⎠×OEr .

overOEr . A point of X is given by a tupleM = ((M(α(g)i j ))(i, j)∈Ω;1≤g≤gi j, (M(εi))i=1,...,n) of C-

linear maps. Obviously, the mapµ ∶X→X defined by

µ(M) ∶= ((M(εi) f ji M(α(g)i j )−M(α(g)i j )M(ε j) fi j ) , (M(εi)))is an endomorphism of the vector bundleX, and by construction Ker(µ) = repl.f.(H,d). Since bythe above consideration, the fibre

Ker(µ)(M(εi )) = π−1(M(εi))is of constant dimension for all(M(εi)) ∈ OEr , we have that Ker(µ) is a vector bundle of theclaimed rank overOEr . This proves (i).

The one-dimensional torus(λ idd1, . . . ,λ iddr ) ∣ λ ∈C∗ ⊂Gd acts trivially on repl.f.(H,r). Sothe maximal dimension of aGd-orbit is dim(Gd)−1. Hence, by (i), ifqH(r) ⩽ 0, everyGd-orbithas dimension at most dim(repl.f.(H,r))−1. This proves (ii).

Remark 3.5. The vector bundle structure of Proposition 3.4 is inspired by [Bo, Sec. 2]. The samestatement remains true for any algebraically closed field, with basically the same proof. Noticethat we have a natural action of AutH(Er) =∏n

i=1GLr i(Hi) on π−1(Er) by conjugation, such thattheGd-orbits on repl.f.(H,d) are in bijection with the AutH(Er)-orbits onπ−1(Er).

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3.3 Bialgebra structure ofM

Consider the direct product of algebrasH ×H. Modules forH ×H are pairs(X1,X2) of modulesfor H. An H×H-submodule of(X1,X2) is a pair(Y1,Y2) whereY1 is anH-submodule ofX1 andY2

is anH-submodule ofX2. Note that we can regardH ×H as the algebraH(C⊕C,D⊕D,Ω⊕Ω),whereC⊕C (resp. D⊕D) means the block diagonal matrix with two diagonal blocks equal toC (resp. D), andΩ⊕Ω is the obvious orientation ofC⊕C induced by the orientationΩ of C.Therefore we can define as above a convolution algebraF(H ×H).

We have an algebra embedding ofF(H)⊗F(H) intoF(H ×H) by setting

( f ⊗g)(X,Y) = f (X)g(Y).Following Ringel [Rin2] (see also [BT, §4.3]), one introduces a mapc ∶ F(H)→F(H ×H) by

c( f )(X,Y) = f (X⊕Y).Proposition 3.6. The map c restricts to a homomorphism c∶M→M⊗M for the convolutionproduct such that

c(θi) = θi ⊗1+1⊗θi .

This makesM into a cocommutative bialgebra.

Proof. We first show thatc is a homomorphism fromF(H) to F(H ×H) for the convolutionproduct. Indeed, on the one hand we have forf ,g ∈F(H)

(c( f ∗g))(X,Y) =∫Z⊆X⊕Y

f (Z)g((X⊕Y)/Z)dχ , (1)

and on the other hand

(c( f )∗c(g))(X,Y) =∫Z1⊆X,Z2⊆Y

f (Z1⊕Z2)g((X⊕Y)/(Z1⊕Z2))dχ . (2)

To show that the two integrals are the same, we consider theC∗-action onX⊕Y given by

λ ⋅(x,y) = (λx,y), (x ∈X, y ∈Y, λ ∈C∗).This induces aC∗-action on the variety of submodulesZ of X⊕Y, whose fixed points are exactlythe submodules of the formZ = Z1⊕Z2 with Z1 ⊆ X andZ2 ⊆Y. Moreover, for a submoduleZof X⊕Y and λ ∈ C∗, the H-moduleλ ⋅Z is isomorphic toZ, so for every f ∈ F(H) we havef (λ ⋅Z) = f (Z), and therefore (1) and (2) are equal. It follows thatc restricts to a homomorphismfromM(H) toM(H ×H). SinceEi is indecomposable, we have

c(θi)(X,Y) = θi(X⊕Y) = 1 if X ≅Ei andY = 0, or X = 0 andY ≅ Ei,0 otherwise.

Thusc(θi) can be identified withθi ⊗1+1⊗θi ∈M(H)⊗M(H) ⊂M(H ×H). Finally, sinceM(H) is generated by theθi ’s andc is multiplicative, this implies that the imagec(M(H)) isindeed contained inM(H)⊗M(H).

An elementf ofM is calledprimitive if c( f ) = f ⊗1+1⊗ f .

Lemma 3.7. An element f ofM is primitive if and only if f is supported only on indecomposablemodules.

8

Proof. This follows immediately from the equalityf (X⊕Y) = c( f )(X,Y).It is easy to see that iff andg are primitive then the Lie bracket

[ f ,g] ∶= f ∗g−g∗ f

is also primitive. Hence the subspacePM ⊂M of primitive elements has the natural structure ofa Lie algebra.

Proposition 3.8. (M,∗,c) is a Hopf algebra isomorphic to the universal enveloping algebraU(PM).Proof. A nonzero elementf of M is group-like if c( f ) = f ⊗ f . Arguing as in [BT, §4.5], wesee that the only group-like element is the unit element1M. Indeed, if f is group-like for anyH-moduleX andk ∈ N we havef (X⊕k) = f (X)k. If f (X) /= 0 for a moduleX, then the decom-position of f with respect to the direct sum⊕dMd has infinitely many nonzero components, acontradiction. Hencef = λ1M whereλ = λ 2 andλ /= 0, so f = 1M.

Therefore, we can repeat the last part of the proof of [Rin2, Theorem]: by [Sw, Lemma 8.0.1],M is an irreducible cocommutative coalgebra, hence a Hopf algebra [Sw, Theorem 9.2.2]. Itthen follows from [Sw, Theorem 13.0.1] thatM is isomorphic as a Hopf algebra to the universalenveloping algebraU(PM) of the Lie algebraPM.

Remark 3.9. When the Cartan matrixC is symmetric andD is the unit matrix, the Lie algebraPM coincides with the Lie algebraL+(CQ) = L+(Q) of [S, §2.6].

3.4 Relations inM

For f ∈M we denote by adf the endomorphism ofM defined by

ad f (g) ∶= [ f ,g], (g ∈M).Proposition 3.10. The generatorsθi ofM satisfy the relations:

(adθi)1−ci j (θ j) = 0, (1⩽ i /= j ⩽ n).Proof. Sinceθi ∈ PM, we have

Θi j ∶= (adθi)1−ci j (θ j) ∈ PMfor all j /= i. By Lemma 3.2 and Lemma 3.7, to check thatΘi j = 0 it is therefore sufficient to checkthat there is no indecomposable locally freeH-module with dimension vector(1−ci j )ei +ej .

Let us assume that(i, j) ∈Ω, and consider a locally free moduleM with this dimension vector.Then, by §2.2,M is given by anHi-linear map

Mi j ∶ iH j ⊗H j M j →Mi,

whereM j =H j andMi =H⊕(1−ci j )i . Now, iH j ⊗H j M j = iH j is a freeHi-module of rankfi j gi j =−ci j ,

soMi contains a direct summandNi isomorphic toHi such thatNi ∩ Im(Mi j ) = 0. It follows thatM has a direct summand isomorphic toEi, and thereforeM is not indecomposable.

The case( j, i) ∈Ω is dual, and one can argue similarly.

Let g be the symmetrizable Kac-Moody Lie algebra overC with Cartan matrixC. It is definedby the following presentation. There are 3n generatorsei , fi , hi (1⩽ i ⩽ n) subject to the relations:

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(i) [ei , f j] = δi j hi ;

(ii) [hi ,h j] = 0;

(iii) [hi ,ej] = ci j ej , [hi , f j] = −ci j f j ;

(iv) (adei)1−ci j (ej) = 0, (ad fi)1−ci j ( f j) = 0 (i /= j).Let n be the Lie subalgebra generated byei (1 ⩽ i ⩽ n). ThenU(n) is the associativeC-algebrawith generatorsei (1⩽ i ⩽ n) subject to the relations

(adei)1−ci j (ej) = 0, (i /= j).Corollary 3.11. The assignment ei ↦ θi extends to a surjective algebra homomorphism

F ∶U(n)→M.

Proof. This follows from Proposition 3.10.

4 The algebraD

4.1 Definition ofDP

Let P be a finite-dimensional projectiveH-module. Let Gr(P) denote the variety of allH-submodules ofP. Let Grl.f(P) denote the constructible subset of Gr(P) consisting of all locallyfree submodules. The group AutH(P) of automorphisms ofP acts on Gr(P), hence it acts diago-naly on Gr(P)×Gr(P). This action restricts to an action on Grl.f(P)×Grl.f(P). DefineCP to betheC-vector space of all constructible functionsf ∶ Grl.f(P)×Grl.f(P)→C which are constant onAutH(P)-orbits. We equipCP with the convolution product defined by

( f ∗g)(X,Y) = ∫Z∈Grl.f(P)

f (X,Z)g(Z,Y)dχ , ( f ,g ∈ CP, X,Y ∈Grl.f(P)).This makes(CP,∗) into an associativeC-algebra, with unit element the function1P defined by

1P(X,Y) = 1 whenX =Y,0 otherwise.

For i = 1, . . . ,n, define elements ofCP by

xi,P(X,Y) = 1 whenX ⊂Y andY/X ≅ Ei,0 otherwise,

yi,P(X,Y) = 1 whenY ⊂ X andX/Y ≅ Ei,0 otherwise.

Definition 4.1. LetDP denote the subalgebra of(CP,∗) generated byxi,P, yi,P, (1⩽ i ⩽ n).

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4.2 Definition ofD

Let A ∶=C⟨xi , yi ; 1⩽ i ⩽ n⟩ be the free associative algebra generated by non-commutingvariablesxi , yi ,(1⩽ i ⩽ n). We have a surjective algebra homomorphismπP ∶ A→DP given by

πP(xi) = xi,P, πP(yi) = yi,P, (1⩽ i ⩽ n).DefineK as the intersection of Ker(πP) whenP runs over all isomorphism classes of projectiveH-modules.

Definition 4.2. LetD =D(H) ∶= A/K. ThenD is generated by

xi ∶= xi modK, yi ∶= yi modK, (1⩽ i ⩽ n),and we have natural surjective algebra homomorphismsπP ∶ D→DP such that

πP(xi) = xi,P, πP(yi) = yi,P, πP(1) = 1P.

Remark 4.3. When the Cartan matrixC is symmetric andD is the unit matrix, the algebraD(H)is nothing else than the algebraR′(H) of [S, §3].

The following is the extension to our setting of [S, Theorem 3.5].

Proposition 4.4. We have an injective algebra homomorphismΦ ∶ M→D such that

Φ(θi) = xi , (1⩽ i ⩽ n).Proof. If X is not a submodule ofY, we have(xi1,P∗⋯∗xik,P)(X,Y) = 0. Otherwise, ifX ⊂Y arelocally free submodules ofP then(xi1,P∗⋯∗xik,P)(X,Y) is equal to the Euler characteristic of theconstructible set of flags:

(X = X0 ⊂ X1 ⊂⋯ ⊂Xk−1 ⊂Xk =Y) ∣ Xj/Xj−1 ≅Ei j , (1⩽ j ⩽ k).Quotienting each step of the flags byX, we see that this set has the same Euler characteristic as

(0= Z0 ⊂ Z1 ⊂⋯ ⊂ Zk−1 ⊂ Zk =Y/X) ∣ Z j/Z j−1 ≅ Ei j , (1⩽ j ⩽ k).So we get

(xi1,P∗⋯∗xik,P)(X,Y) = (θi1 ∗⋯∗θik)(Y/X) if X ⊂Y,0 otherwise.

(3)

For a sequencei = (i1, . . . , ik) let us write for short

xi,P ∶= xi1,P∗⋯∗xik,P, θi ∶= θi1 ∗⋯∗θik , xi ∶= xi1 ∗⋯∗xik .

By (3), every relation∑i aiθi = 0 inM implies a relation∑i aixi,P = 0 inDP. Hence we have a well-defined homomorphismΦP fromM to DP mappingθi to xi,P. Since this can be done for everyprojective moduleP, the mapΦP can be lifted to a homomorphismΦ fromM to D mappingθi

to xi .Suppose now that∑i aixi = 0 is a relation inD, and letX be a locally freeH-module. Take a

projective coverpX ∶P→ X and setY = Ker(pX). ThenY ⊂ P andP/Y ≅ X. We have∑i aixi,P = 0in DP, so by (3) again

∑i

aixi,P(Y,P) =∑i

aiθi(X) = 0.

Since this holds for anyX, it follows that∑i aiθi = 0 is a relation inM. ThereforeΦ is injective.

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Note that the definitions ofxi,P andyi,P immediately imply:

Lemma 4.5. The assignments xi ↦ yi , yi ↦ xi (1⩽ i ⩽ n) extend to an anti-automorphismτ ofD.

From Proposition 4.4 and Lemma 4.5 it follows that

Corollary 4.6. We have an injective algebra anti-homomorphismΨ ∶ M→D such that

Ψ(θi) = yi , (1⩽ i ⩽ n).4.3 Bialgebra structure ofD

Consider the algebrasC(P,P′)(H ×H), where(P,P′) is a finite-dimensional projectiveH ×H-module, that is, a pair of finite-dimensional projectiveH-modules. We have an algebra embeddingCP(H)⊗CP′(H)→ C(P,P′)(H ×H) given by

( f ⊗g)((X,Y),(X′,Y′)) = f (X,Y)g(X′,Y′), ( f ∈ CP(H), g ∈ CP′(H), X,Y ⊆ P, X′,Y′ ⊆P′).We introduceC-linear mapsd(P,P′) ∶ CP⊕P′(H)→ C(P,P′)(H ×H) given by

(d(P,P′)( f ))((X,Y),(X′,Y′)) = f (X⊕X′,Y⊕Y′).Proposition 4.7. The map d(P,P′) restricts to an algebra homomorphismDP⊕P′ →DP⊗DP′ suchthat

d(P,P′)(xi,P⊕P′) = xi,P⊗1P′ +1P⊗xi,P′ , d(P,P′)(yi,P⊕P′) = yi,P⊗1P′ +1P⊗yi,P′ , (1⩽ i ⩽ n).Proof. The proof is similar to the proof of Proposition 3.6. We show that d(P,P′) is a homomor-phism fromCP⊕P′(H) to C(P,P′)(H ×H) for the convolution product. Indeed, on the one hand wehave for f ,g ∈ CP⊕P′(H)

(d(P,P′)( f ∗g))((X,Y),(X′,Y′)) =∫Z⊆P⊕P′

f (X⊕X′,Z)g(Z,Y⊕Y′)dχ , (4)

and on the other hand

(d(P,P′)( f )∗d(P,P′)(g))((X,Y),(X′,Y′)) = ∫Z1⊆P,Z2⊆P′

f (X⊕X′,Z1⊕Z2)g(Z1⊕Z2,Y⊕Y′)dχ .(5)

To show that the two integrals are the same, we consider theC∗-action onP⊕P′ given by

λ ⋅(p, p′) = (λ p, p′), (p ∈P, p′ ∈P′, λ ∈C∗).This induces aC∗-action on the variety of submodulesZ of P⊕P′, whose fixed points are exactlythe submodules of the formZ = Z1⊕Z2 with Z1 ⊆ P andZ2 ⊆ P′. In particular this action fixesX⊕X′ andY⊕Y′. Since by definitionf andg are constant on the orbits ofC∗ ⊂ Aut(P⊕P′) forthe diagonal action, we see that (4) and (5) are equal.

We have:

(d(P,P′)(xi,P⊕P′))((X,Y),(X′,Y′)) = xi,P⊕P′(X⊕X′,Y⊕Y′)= 1 if X⊕X′ ⊂Y⊕Y′ and(Y⊕Y′)/(X⊕X′) ≅ Ei,

0 otherwise.

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SinceX,Y ⊆ P and X′,Y′ ⊆ P′, and sinceP∩P′ = 0, we haveX⊕X′ ⊂ Y⊕Y′ if and only ifX ⊆Y andX′ ⊆Y′. Moreover in this case,(Y⊕Y′)/(X⊕X′) = (Y/X)⊕ (Y′/X′). So sinceEi isindecomposable we have that(Y⊕Y′)/(X⊕X′) ≅ Ei if and only if Y/X ≅ Ei andY′/X′ = 0,or Y/X = 0 andY′/X′ ≅ Ei. Thusd(P,P′)(xi,P⊕P′) can be identified withxi,P⊗1P′ +1P⊗ xi,P′ .Similarly, d(P,P′)(yi,P⊕P′) can be identified withyi,P⊗1P′+1P⊗yi,P′ . It follows thatd(P,P′) restrictsto an algebra homomorphismDP⊕P′ →DP⊗DP′

We can now lift the mapsd(P,P′) ∶ DP⊕P′ →DP⊗DP′ to an algebra homomorphism

d ∶ D→D⊗D,making(D,∗,d) into a cocommutative bialgebra. It follows from Proposition 4.7 that

d(xi) = xi ⊗1+1⊗xi , d(yi) = yi ⊗1+1⊗yi , (1⩽ i ⩽ n),that is,xi andyi are primitive. LetPD be the subspace of primitive elements ofD, a complex Liealgebra. In particular,PD contains the following distinguished elements:

hi ∶= [xi ,yi], (1⩽ i ⩽ n).The following is an adaptation to our setting of [S, Theorem 3.3].

Proposition 4.8. (D,∗,d) is a cocommutative Hopf algebra isomorphic to the universalenvelop-ing algebra U(PD). It is Z

n-graded via

deg(xi) = αi , deg(yi) = −αi , (1⩽ i ⩽ n).Proof. Analogous to the proof of Proposition 3.8.

Corollary 4.9. The sub-Lie algebra ofPD generated by the elements xi is isomorphic toPM.

Proof. This follows from Proposition 4.4.

4.4 Relations inD

The following is the extension to our setting of [S, §3.6.1].

Proposition 4.10. If i /= j we have[xi ,y j]=0. On the other hand hi = [xi ,yi] /=0 for every i=1, . . . ,n.

Proof. Let i /= j. It is enough to check that for everyP we havexi,P ∗y j,P− y j,P ∗xi,P = 0. LetX,Y ∈Grl.f(P), and denote byXk (resp. Yk) the subspace ofX (resp. Y) sitting on vertexk of thequiver ofH. We have

xi,P∗y j,P(X,Y) = χ (Z ∈Grl.f(P) ∣X ⊂ Z, Y ⊂ Z, Z/X ≅Ei, Z/Y ≅E j) ,y j,P∗xi,P(X,Y) = χ (Z ∈Grl.f(P) ∣ Z ⊂ X, Z ⊂Y, Y/Z ≅E j , X/Z ≅Ei) .

Since i /= j, these two products are zero unlessXk =Yk for k /= i and k /= j. Moreover for theseproducts to be nonzero we should also haveXi ⊂Yi , Yj ⊂Xj , Yi/Xi ≅ Ei andXj/Yj ≅E j . In this case,the two constructible sets above are both reduced to a point,and have Euler characteristic 1. So,for any pair(X,Y) the difference[xi,P,y j,P](X,Y) vanishes.

On the other hand, letPi be the projective cover ofEi . Then

[xi,Pi ,yi,Pi ](Pi ,Pi) = −yi,P∗xi,P(Pi,Pi) = −1,

since the constructible setZ ⊂Pi ∣Pi/Z ≅Ei is reduced to a point. Therefore[xi,Pi ,yi,Pi ] /= 0, hence[xi ,yi] /= 0.

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Given a projectiveH-moduleP we put

hi,P ∶= [xi,P,yi,P] = πP(hi) ∈DP.

Recall from [GLS, §9] the following definition.

Definition 4.11. Let X be anH-module. Fori = 1, . . . ,n we denote by subi(X) the largest submod-uleU of X which is supported on the vertexi. Dually we denote by faci(X) the largest quotientmoduleX/U which is supported on the vertexi.

Thus subi(X) and faci(X) areHi-modules. Note that, even ifX is locally free, subi(X) andfaci(X) are not necessarily freeHi-modules. We denote bysi(X) the multiplicity of Ei as anindecomposable summand of subi(X). Similarly we denote byti(X) the multiplicity of Ei as anindecomposable summand of faci(X).Lemma 4.12. Suppose that si(X) = k≥ r ≥ 0. Then

χ(Z ⊂ X ∣ Z ≅E⊕ri ) = (kr) .

Similarly, if ti(X) = l ≥ r ≥ 0, then

χ(Z ⊂ X ∣X/Z ≅ E⊕ri ) = (l

r) .

Proof. Clearly,Z is a submodule ofX isomorphic toE⊕ri if and only ifZ is a submodule of subi(X)

isomorphic toE⊕ri . Thus we want to calculate the Euler characteristic of

S= Z ⊂ subi(X) ∣ Z ≅E⊕ri .

We can write subi(X) ≅ E⊕ki ⊕V, whereV is anHi-module contained in Ker(εci−1

i ). The mapZ↦ εci−1

i (Z) is a fibration fromS to the Grassmannian Gr(r,Ck) with fibers isomorphic toCd,where

d = r dim(Ker((εci−1i )∣

E⊕(k−r)i ⊕V

)) = r((k− r)(ci −1)+dimV).Hence

χ(S) = χ(Gr(r,Ck))χ(Cd) = (kr) .

For the second statement, letY be the (unique) submodule ofX such thatX/Y ≅ faci(X). Usingthe mapZ↦U ∶= Z/Y we see that the setZ ⊂X ∣X/Z ≅E⊕r

i has the same Euler characteristic as

T = U ⊂ faci(X) ∣ faci(X)/U ≅E⊕ri .

As above the mapU ↦ εci−1i (U) is a fibration fromT to Gr(l − r,Cl) with fibers isomorphic to

affine spaces, so

χ(T) = χ(Gr(l − r,Cl)) = (lr) .

The following is the extension to our setting of [S, Lemma 3.6.2].

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Proposition 4.13. For X,Y ∈Grl.f(P) we have

hi,P(X,Y) = si(P/X)− ti(X) if X =Y,0 otherwise.

More generally, for f∈DP we have

(hi,P∗ f )(X,Y) = (si(P/X)− ti(X)) f (X,Y),( f ∗hi,P)(X,Y) = (si(P/Y)− ti(Y)) f (X,Y).

Proof. We have(xi,P∗yi,P)(X,Y) = χ(S) and(yi,P∗xi,P)(X,Y) = χ(I) where

S = Z ∈Grl.f(P) ∣X ⊂ Z, Y ⊂ Z, Z/X ≅Ei ≅ Z/Y,I = Z ∈Grl.f(P) ∣ Z ⊂ X, Z ⊂Y, X/Z ≅ Ei ≅Y/Z.

By the first isomorphism theorem, we have(X+Y)/Y ≅ X/(X∩Y). So sinceX andY are locallyfree,X+Y andX∩Y are either both locally free or both non locally free.

(a) SupposeX+Y andX∩Y are locally free. We have two subcases.(a1) If X =Y thenS= Z ∈Grl.f(P) ∣X ⊂Z, Z/X ≅Ei. The mapZ↦U ∶=Z/X is an isomorphim

from S to S′ = U ∣U ⊂ P/X, U ≅ Ei. Hence by Lemma 4.12,χ(S) = χ(S′) = si(P/X). Onthe other hand,I = Z ∈ Grl.f(P) ∣ Z ⊂ X, X/Z ≅ Ei. Hence by Lemma 4.12,χ(I) = ti(X). Soχ(S)−χ(I) = si(P/X)− ti(X), as required.

(a2) If X /=Y then,S /= ∅ if and only if I /= ∅, and in that caseS and I are reduced to singlepoints, namelyS= X+Y andI = X∩Y. Hence we always haveχ(S)−χ(I) = 0.

(b) SupposeX +Y andX ∩Y are not locally free. ThenX /=Y. If S /= ∅ there existsZ suchthatX+Y ⊂ Z ⊂ P andZ/Y ≅ Ei. HenceM = (X+Y)/Y is isomorphic to a non trivial submoduleof Ei. The mapZ↦U ∶= Z/Y mapsS isomorphically toU ⊂ P/Y ∣M ⊂U, U ≅ Ei. This is anaffine space (of dimension(si(P/Y)−1)(dim(M)−1)), henceχ(S) = 1. SinceN ∶= X/(X∩Y) ≅(X+Y)/Y =M, we see thatI = Z ⊂X ∣X/Z↠N, X/Z≅Ei. This is an affine space (of dimension(ti(X)−1)(dim(N)−1)), henceχ(I) = 1. So we have againχ(S)−χ(I) = 0.

In conclusion we haveχ(S)− χ(I) = 0, unlessX =Y whereχ(S)− χ(I) = si(P/X)− ti(X).This proves the first part of the proposition.

For the second part we have

(hi,P∗ f )(X,Y) = ∫Z∈Grl.f(P)

hi,P(X,Z) f (Z,Y)dχ .

Sincehi,P(X,Z) = 0 unlessZ =X, we see that(hi,P∗ f )(X,Y) = hi,P(X,X) f (X,Y). Similarly, ( f ∗hi,P)(X,Y) = f (X,Y)hi,P(Y,Y). Thus the second part follows immediately from the first one.

The following is the extension to our setting of [S, Theorem 3.6.2].

Proposition 4.14. We have[hi ,h j] = 0 in D.

Proof. We have to show that for every projectiveH-moduleP there holdsπP([hi ,h j]) = 0. ByProposition 4.13 we see that, forX,Y ∈Grl.f(P),

πP([hi ,h j])(X,Y) = [hi,P,h j,P](X,Y) = (si(P/X)− ti(X)−si(P/Y)+ ti(Y))h j,P(X,Y) = 0,

becauseh j,P(X,Y) = 0 unlessX =Y. Hence[hi,P,h j,P] = 0, as required.

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The following is the extension to our setting of [S, Corollary 3.7.1] and [S, Theorem 3.9].

Proposition 4.15. Assume that j= i, or that ( j, i) ∈Ω (so that all arrows between i and j in the

quiver of H are of the formα(g)ji ∶ i → j). Then inD we have

[h j ,xi] = c ji xi , [h j ,yi] = −c ji yi .

Proof. Let P be a projectiveH-module. LetX,Y be locally free submodules ofP. SinceH isIwanaga-Gorenstein of dimension 1,X andY are also projective (see §2.1 and §2.3). By Proposi-tion 4.13 we have

[h j,P,xi,P](X,Y) = (sj(P/X)− t j(X)−sj(P/Y)+ t j(Y))xi,P(X,Y).By definition ofxi,P we havexi,P(X,Y)=0 unlessX ⊂Y andY/X =Ei. SinceX andY are projective,this implies that

Y =Pi⊕P′, X = Ker(Pi↠Ei)⊕P′, P= Pi⊕P′⊕P′′,

wherePi is the projective cover ofEi andP′,P′′ are projective. HenceP/Y ≅P′′ andP/X ≅Ei⊕P′′.By [GLS, Proposition 3.1],

Ker(Pi↠Ei) = ⊕k∈Ω(−,i)

P∣cki∣k

whereΩ(−, i) = k = 1, . . . ,n ∣ (k, i) ∈Ω. Therefore if j = i thent j(Y)− t j(X) = 1 andsj(P/X)−sj(P/Y) = sj(Ei) = 1, thus [h j,P,xi,P](X,Y) = 2xi,P(X,Y).On the other hand, if( j, i) ∈Ω thent j(Y)− t j(X) = −∣c ji ∣ = c ji , andsj(P/X)−sj(P/Y) = 0, thus

[h j,P,xi,P](X,Y) = c ji xi,P(X,Y).So in both cases we conclude that[h j ,xi] = c ji xi . The identity[h j ,yi] = −c ji yi is proved similarly.

Note that the relation of Proposition 4.15 also holds whenc ji =0, that is, when there is no arrowbetweeni and j. In summary, in the Lie algebraPD of primitive elements ofD, the followingidentities are satisfied:

(i) [xi ,y j] = 0, if i /= j;

(ii) [xi ,yi] = hi , [hi ,xi] = 2xi , [hi ,yi] = −2yi ;

(iii) (adxi)1−ci j (x j) = 0= ad(yi)1−ci j (y j), if i /= j;

(iv) If ( j, i) ∈Ω or ci j = 0 then[h j ,xi] = c ji xi and[h j ,yi] = −c ji yi .

Relation (iii) for yi ’s follows from Corollary 4.6, taking into account that the Serre relations areleft-right symmetric.

These are almost all the defining relations of the Kac-Moody algebrag. The only relations ofthe complete presentation ofg which have not been proved are relations (iv) when(i, j) ∈Ω. Tocheck that these relations arenot satisfied, one can calculate easily that if(i, j) ∈Ω,

[h j,Pi , xi,Pi ](Ker(Pi↠Ei), Pi) = 0, xi,Pi(Ker(Pi↠Ei), Pi) = 1,

hence[h j ,xi] /= c ji xi .

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5 The Lie algebraL of partially defined derivations

5.1 The idealJ

Definition 5.1. Let J be the linear subspace ofD consisting of all elementsx such that for everyprojectiveH-moduleP and every locally free submodulesX ⊆Y of P such thatP/X is injective,we haveπP(x)(X,Y) = 0.

We denote byD≥0 (resp.D>0) the subalgebra ofD generated by the elementsxi , hi (1⩽ i ⩽ n)(resp. xi , (1⩽ i ⩽ n)). By Proposition 4.4,D>0 is isomorphic toM. The following is the extensionto our setting of [S, Theorem 4.3].

Proposition 5.2. J∩D≥0 is a two-sided ideal ofD≥0.Proof. Let x∈ J∩D≥0 and let f = πP(x) for some projectiveH-moduleP. Let X ⊆Y be locally freesubmodules ofP such thatP/X is injective. Then, by definition ofJ we havef (X,Y) = 0, and wewant to show that for everyi = 1, . . . ,n,

(hi,P∗ f )(X,Y) = ( f ∗hi,P)(X,Y) = (xi,P∗ f )(X,Y) = ( f ∗xi,P)(X,Y) = 0.

By Proposition 4.13 we have

(hi,P∗ f )(X,Y) = hi,P(X,X) f (X,Y) = 0, ( f ∗hi,P)(X,Y) = hi,P(Y,Y) f (X,Y) = 0.

Next, combining Proposition 4.13 and the definition ofxi,P we easily see that every(U,V) in thesupport of an elementg ∈D≥0 satisfiesU ⊆V. We therefore have

(xi,P∗ f )(X,Y) = ∫X⊂Z⊂Y, Z/X≅Ei

f (Z,Y)dχ .

Now, sinceP/X is injective andH is Iwanaga-Gorenstein of dimension 1, for every locally freeH-moduleZ such thatX ⊂ Z ⊂ P, we have thatP/Z is injective. Hence the functionZ↦ f (Z,Y)vanishes in the rangeX ⊂ Z ⊂Y, and(xi,P ∗ f )(X,Y) = 0. The proof that( f ∗xi,P)(X,Y) = 0 issimilar.

The following is the extension to our setting of [S, Lemma 4.1].

Proposition 5.3. We have J∩D>0 = 0.Proof. Let x be a nonzero element ofD>0. Then f ∶=Φ−1(x) is a non zero element ofM. LetX be anH-module in the support off . We denote byI(X) an injective envelope ofX, and byψ ∶ P(I(X))→ I(X) a projective cover ofI(X). Then we have

(πP(I(X))(x))(ψ−1(0),ψ−1(X)) = f (ψ−1(X)/ψ−1(0)) = f (X) /= 0.

On the other hand we haveψ−1(0) ⊂ψ−1(X) ⊂P(I(X)) andP(I(X))/ψ−1(0) ≅ I(X) is injective.Thusx /∈ J.

Recall from §4.4 the definition ofsi(X) andti(X).Lemma 5.4. If X is projective and Y is injective we have

citi(X) = ⟨rank(X),αi⟩H , cisi(Y) = ⟨αi ,rank(Y)⟩H .

17

Proof. SinceX is projective⟨rank(X),αi⟩H = ⟨X,Ei⟩H = dimHomH(X,Ei) = citi(X). Similarly,sinceY is injective⟨αi ,rank(Y)⟩H = ⟨Ei,Y⟩H = dimHomH(Ei ,Y) = cisi(Y).

The following is the extension to our setting of [S, Theorem 4.4].

Proposition 5.5. Let x∈D≥0 be of degreeβ . Then

[x,hi]+ 1ci(β ,αi)x

belongs to the ideal J.

Proof. Let f = πP(x) for some projectiveH-moduleP. Let X ⊆Y be locally free submodules ofPsuch thatP/X is injective. If rank(Y)− rank(X) /= β , we havef (X,Y) = [hi,P, f ](X,Y) = 0. So wecan assume rank(Y)− rank(X) = β . By Proposition 4.13 and Lemma 5.4 we then have

[ f ,hi,P](X,Y) = −((si(P/X)−si(P/Y))+(ti(Y)− ti(X))) f (X,Y)= −

1ci(⟨αi ,β ⟩H + ⟨β ,αi⟩H) f (X,Y)

= −1ci(β ,αi) f (X,Y).

Since this holds for any projectiveH-moduleP, it follows that[x,hi]+ 1ci(β ,αi)x ∈ J.

5.2 The algebraE

The next definition is based on Proposition 5.2.

Definition 5.6. Denote by(E ,∗) the associativeC-algebra

E ∶=D≥0/(J∩D≥0).By Proposition 5.3, the assignment

xi ↦ xi modJ, (1⩽ i ⩽ n)extends to an injective algebra homomorphismD>0→ E . We can therefore write for shortxi ∈ Einstead ofxi modJ. We will also allow ourselves to writehi ∈ E for the class ofhi moduloJ. ThealgebraE is N

n-graded via

deg(xi) = αi , deg(hi) = 0, (1⩽ i ⩽ n).By Proposition 5.5, the following identity holds inE :

[hi , f ] = 1ci(β ,αi) f , (1⩽ i ⩽ n), (6)

for every f ∈ E with deg( f ) = β .

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5.3 The Lie algebraL

5.3.1 Derivations

A right derivationonE is aC-linear map

x ∶ E → E , f ↦ ( f )xsuch that ( f ∗g)x = ( f )x∗g+ f ∗(g)x, ( f ,g ∈ E).It is easy to check that right derivations onE form aC-vector space, and a Lie algebra for thebracket: ( f )[x,y] ∶= (( f )x)y−(( f )y)x, ( f ∈ E). (7)

For example, for everyi = 1, . . . ,n, the mapsxi , hi defined by

( f )xi ∶= [ f ,xi], ( f )hi ∶= [ f ,hi], ( f ∈ E),are right derivations onE . Note that, by (6), we have

( f )hi = −1ci(β ,αi) f (8)

for f ∈ E of degreeβ .

5.3.2 Partially defined derivations

We say that aC-linear mapz ∶ V → E is apartially defined right derivationonE if V is a subspaceof E of finite codimension and for everyf ,g ∈V such thatf ∗g ∈V we have

( f ∗g)z= ( f )z∗g+ f ∗(g)z.Given two partially defined derivationsz′ andz′′, we writez′ ≡ z′′ if ( f )z′ = ( f )z′′ for every f ina subspace ofE of finite codimension contained in the intersection of the domains ofz′ andz′′.This is an equivalence relation (because the intersection of three subspaces of finite codimensionhas finite codimension). We denote byR the set of congruence classes of partially defined rightderivations onE for the congruence relation≡. For simplicity, we shall use the same notation fora partially defined derivationz and its congruence class. ThenR becomes aC-vector space if wedefinez′+z′′ by ( f )(z′+z′′) ∶= ( f )z′+( f )z′′for f in the intersection of the domains ofz′ andz′′. It is also easy to check that the same formulaas (7) endowsR with the structure of a Lie algebra.

For instance, we can define elementsyi ofR by

( f )yi ∶= [ f ,yi]modJ, ( f ∈D>0, 1⩽ i ⩽ n).The fact that forf ∈D>0 we have[ f ,yi] ∈D≥0 follows from the identities

[x j ,yi] = δi j hi , [ f ∗g,yi] = [ f ,yi]∗g+ f ∗[g,yi], (1⩽ i, j ⩽ n, f ,g ∈D>0).Note that the derivationsxi andhi also give rise to well-defined elements ofR, which we continueto denote byxi andhi according to our convention.

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5.3.3 Definition ofL and relations inL

Definition 5.7. LetL denote the Lie subalgebra ofR generated byxi ,hi,yi , (1⩽ i ⩽ n).The following theorem is an analogue of [S, Theorem 4.6].

Theorem 5.8. The following identities hold inL:

(i) [xi ,y j] = δi j hi;

(ii) [hi,h j] = 0;

(iii) [h j ,xi] = c ji xi , [h j ,yi] = −c ji yi ;

(iv) (adxi)1−ci j (x j) = 0, (adyi)1−ci j (y j) = 0 (i /= j).Proof. Let us denote byu the class inE of an elementu of D≥0. By definition, for f ∈D>0, wehave ( f )[xi ,y j] = [[ f ,xi],y j]− [[ f ,y j ],xi] = [ f ,[xi ,y j]] = δi j [ f ,hi] = δi j ⋅( f )hi .

This proves (i). The proof of (ii) follows immediately from (8). Next, assume thatf ∈ E is homo-geneous of degreeβ . Then

( f )[h j ,xi] = [[ f ,h j ],xi]− [[ f ,xi],h j] = − 1c j(β ,α j)[ f ,xi]+ 1

c j(β +αi ,α j)[ f ,xi] = c ji ⋅( f )xi .

Similarly, one has( f )[h j ,yi] =−c ji ⋅( f )yi for f ∈D>0. This proves (iii). Finally, (iv) follows fromProposition 3.10, Proposition 4.4, and Corollary 4.6.

Note thatL is aZn-graded Lie algebra via

deg(xi) = αi , deg(yi) = −αi , deg(hi) = 0.

Lemma 5.9. The dimension of the subspace ofL spanned by thehi (1⩽ i ⩽ n) is equal to the rankof the Cartan matrix C.

Proof. Let (λ1, . . . ,λn) ∈Cn be such that for everyi we have∑nj=1λ jc ji = 0. Defineh ∶=∑n

j=1λ jh j .Then, for everyβ =∑n

i=1biαi ∈Nn we haven

∑j=1

λ j

c j(α j ,β) = n

∑i=1

bi

n

∑j=1

λ jc ji = 0.

It then follows from (8) thath = 0.Conversely, leth ∶=∑n

j=1λ jh j and suppose thath = 0. We know that for everyk> 0 the homo-

geneous component ofM of degreekαi is nonzero. Indeed the constructible functionθ∗ki takesthe valuek! on E⊕k

i . SinceM≅D>0 embeds inE , this shows that the homogeneous componentEkαi of E of degreekαi is nonzero. By our assumption, we have( f )h = 0 for every f in a subspaceV of E of finite codimension. The intersection

V ∩( n

⊕i=1⊕k>0Ekαi)

must be of finite codimension in⊕ni=1⊕k>0Ekαi . Therefore, for everyi = 1, . . . ,n we can findk > 0

and a nonzero elementf ∈ Ekαi such that

0= ( f )h = ⎛⎝n

∑j=1

λ j

c j(α j ,kαi)⎞⎠ f = k( n

∑=1

λ jc ji) f = 0.

Thus for everyi we have∑nj=1λ jc ji = 0.

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5.4 Main result

Following [K2, §1.2] define a Lie algebrag by the following presentation. The generators arexi ,yi ,hi (1⩽ i ⩽ n), and the relations:

[xi ,y j] = δi j hi , [hi,h j] = 0, [hi ,x j] = ci j x j , [hi ,y j] = −ci j y j .

This is aZn-graded Lie algebra via:

deg(xi) = αi , deg(yi) = −αi, deg(hi) = 0.

By [K2, Theorem 1.2, Theorem 9.11],g has a uniqueZn-graded maximal idealr not intersectingthe subspaceh spanned by thehi (1⩽ i ⩽ n). Moreoverr is generated by the elements

(adxi)1−ci j (x j), (adyi)1−ci j (y j), (i /= j).Hence, the assigmentxi ↦ ei , yi ↦ fi , hi ↦ hi (1⩽ i ⩽ n) induces a Lie algebra isomorphismg/r ≅ g(the Gabber-Kac theorem). To be more precise, we deal here with the Lie algebras denoted byg′

andg′ in [K2]. This version of the Gabber-Kac theorem forg′ andg′ is explained in [K2, Remark1.5].

The following theorem is our main result. It is an analogue of[S, Theorem 4.7] for our algebraH(C,D,Ω).Theorem 5.10. (i) The Lie algebraL is isomorphic tog.

(ii) The Lie algebraPM is isomorphic to the positive partn of g.

(iii) The homomorphism F∶ U(n)→M is an isomorphism of Hopf algebras.

Proof. By Theorem 5.8 we have a surjective Lie algebra homomorphismfrom g to L mappingxi

to xi , hi to hi, andyi to yi . Its kernel is aZn-graded ideals of g not intersectingh, because byLemma 5.9 the space spanned by thehi ’s is isomorphic to the space spanned by thehi ’s. Hence bythe Gabber-Kac theorem,s ⊆ r andg is a homomorphic image ofL. Since all the defining relationsof g are already satisfied onL, this proves (i).

Therefore the Lie subalgebra ofL generated by thexi ’s is presented by the first relations ofTheorem 5.8(iv). But, by construction, this Lie algebra is ahomomorphic image of the Lie algebraPD>0 of primitive elements ofD>0 by mappingxi to xi . This proves that thexi ’s cannot satisfymore relations than theei ’s. SincePM ≅PD>0, this shows (ii).

Finally, (ii) implies thatF is an isomorphism of algebras. Since both families of generatorsei

andθi are primitive, this is in fact an isomorphism of Hopf algebras. This proves (iii).

6 Examples

We give a description of the algebraM in two basic cases corresponding to [GLS, §13.5, §13.6].

6.1 Dynkin type A2

Let

C = ( 2 −1−1 2

)

21

with symmetrizerD = diag(2,2) andΩ = (1,2). ThusC is a Cartan matrix of Dynkin typeA2

with a non-minimal symmetrizer. We havef12 = f21 = 1. ThusH = H(C,D,Ω) is given by thequiver

1

ε1

2α12oo

ε2

with relationsε21 = ε2

2 = 0 andε1α12 = α12ε2. There are 4 isomorphism classes of indecomposablelocally freeH-modules, displayed as follows in [GLS, §13.5]:

E1 =P1 = 11 , P2 = I1 =

21 2

1, E2 = I2 = 2

2 , X = 1 21 2 .

Here, the numbers 1, 2, in the pictures of the modules correspond to composition factors. NotethatP2 andX have the same rank vector.

Denote by1M the characteristic function of theGd-orbit of a locally freeH-moduleM ofdimension vectord. Thus,θ1 = 1E1 andθ2 = 1E2. We have

θ2∗θ1 = 1E1⊕E2, θ1∗θ2 = 1P2 +1X +1E1⊕E2, [θ1,θ2] = 1P2+1X.

The enveloping algebraM≅U(n) has a Poincare-Birkhoff-Witt basis given by

θa2 ∗[θ1,θ2]b∗θc

1 = a!b!c!b

∑k=0

1Ia2⊕Pk

2⊕Xb−k⊕Pc1, (a,b,c ∈ Z≥0).

6.2 Dynkin type B2

Let

C = ( 2 −1−2 2

)with symmetrizerD = diag(2,1) andΩ = (1,2). ThusC is a Cartan matrix of Dynkin typeB2.We havef12 = 1 and f21= 2. ThenH =H(C,D,Ω) is given by the quiver

1

ε1

2α12oo

with relationε21 = 0. There are 5 isomorphism classes of indecomposable locally freeH-modules,

displayed as follows in [GLS, §13.6]:

E1 =P1 = 11 , P2 =

21

1, I1 =

21 2

1, E2 = I2 = 2 , X = 1 2

1 .

Note thatP2 andX have the same rank vector. We have

θ1 = 1E1, θ2 = 1E2, [θ1,θ2] = 1P2+1X , [[θ1,θ2],θ2] = 2⋅1I1.

The enveloping algebraM≅U(n) has a Poincare-Birkhoff-Witt basis given by

θa2 ∗[[θ1,θ2],θ2]b∗[θ1,θ2]c∗θd

1 = 2ba!b!c!d!c

∑k=0

1Ia2⊕Ib

1⊕Pk2⊕Xc−k⊕Pd

1, (a,b,c,d ∈Z≥0).

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Christof GEISS : Instituto de Matematicas, Universidad Nacional Autonoma de Mexico,Ciudad Universitaria, 04510 Mexico D.F., Mexico.email :christof@math.unam.mx

Bernard LECLERC : Normandie Univ, France;UNICAEN, LMNO F-14032 Caen, France;CNRS UMR 6139, F-14032 Caen, France;Institut Universitaire de France.email :bernard.leclerc@unicaen.fr

Jan SCHROER : Mathematisches Institut, Universitat Bonn,Endenicher Allee 60, 53115 Bonn, Germanyemail :schroer@math.uni-bonn.de

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