Cyclic Complexes, Hall Polynomials and Simple Lie Algebras

CYCLIC COMPLEXES, HALL POLYNOMIALS AND SIMPLE LIE ALGEBRAS

QINGHUA CHEN AND BANGMING DENG

Abstract. In this paper we study the category Cm(P) of m-cyclic complexes over P, where P is the category of projective modules over a finite dimensional hereditary algebra A, and describe almost split sequences in Cm(P). This is applied to prove the existence of Hall polynomials in Cm(P) when A is representation finite and m 6= 1. We further introduce the Hall algebra of Cm(P) and its localization in the sense of Bridgeland. In the case when A is representation finite, we use Hall polynomials to define the generic Bridgeland–Hall algebra of A and show that it contains a subalgebra isomorphic to the integral form of the corresponding quantum enveloping algebra. This provides a construction of the simple Lie algebra associated with A.

1. Introduction

The Ringel–Hall algebra Hv(A) of a finite dimensional algebra A over a finite field Fq was introduced by Ringel [28] in 1990. By definition, the algebra Hv(A) has a basis the isoclasses (isomorphism classes) of A-modules, and the structure constants are given by counting certain submodules. Ringel then showed that when A is hereditary and representation finite, Hv(A) is isomorphic to the positive part of the corresponding quantum enveloping algebra. Later on, Green [13] obtained a comultiplication formula for Ringel–Hall algebras of hereditary algebras and extended Ringel’s alge- braic realization to arbitrary types. When A is hereditary of finite representation type, Ringel [30] proved that the structure constants of the Ringel–Hall algebra Hv(A) are actually polynomials in q, called Hall polynomials. By evaluating Hall polynomials at q = 1, it was shown in [29] that the degenerate Ringel–Hall algebra H1(A) is isomorphic to the positive part of the associated universal enveloping algebra. In particular, this gives a realization of nilpotent parts of the semisimple Lie algebra associated with A. Based on Ringel’s idea, Peng and Xiao [25] obtained a realization of the whole semisimple Lie algebra in terms of the root category of A. It turns out that Ringel–Hall algebra approach provides a nice framework for the realization of quantum enveloping algebras and Kac–Moody Lie algebras, see, e.g., [28, 30, 13, 29, 21, 22, 26, 34]. After Ringel’s discovery, some efforts have been made in order to obtain the whole quantum group. For example, many authors have studied Hall algebras associated with triangulated categories; see [37, 20, 35, 39]. Recently, Bridgeland [5] introduced the Hall algebra of 2-cyclic complexes of projective modules over a finite dimensional hereditary algebra A and proved that by taking localization and reduction, the result- ing algebra admits a subalgebra isomorphic to the whole quantum enveloping algebra

2000 Mathematics Subject Classification. 16G20, 17B37, 17B20. Key words and phrases. cyclic complex; Hall polynomial; quantum group; simple Lie algebra. Supported partially by the Natural Science Foundation of China and the Specialized Research Fund for the Doctoral Program of Higher Education. 1 2 QINGHUA CHEN AND BANGMING DENG associated with A. If, moreover, A is representation finite, then the two algebras coincide. The present paper mainly deals with the category Cm(P) of m-cyclic complexes of projective modules over a finite dimensional hereditary algebra A (setting b C0(P) = C (P) by convention). We first describe almost split sequences in Cm(P) in a way similar to that given in [33] for C1(P). This allows us to prove the existence of Hall polynomials in Cm(P) when A is representation finite and m 6= 1. Second, we introduce the Hall algebra of m-cyclic complexes and, based on [5], prove that its localization is isomorphic to the tensor product of m-copies of the extended Ringel–Hall algebra of A. Finally, under the assumption that A is connected and representation finite, i.e., up to Morita equivalence, A is given by a (connected) valued Dynkin quiver ∆,~ we use Hall polynomials to define the generic Bridgeland–Hall algebra of ∆~ and show that its degenerate form is isomorphic to the universal enveloping algebra of the simple Lie algebra g∆ associated with the underlying diagram ∆. This provides a realization of the entire g∆ in terms of 2-cyclic complexes. We refer to [3, 12, 2] for basic notions concerning representations of finite dimensional algebras. Nevertheless, we want to fix some notation and terminology used throughout the paper. Given a finite dimensional algebra A over a field F, we denote by A-mod the category of finite dimensional (left) A-modules and by P = PA the full subcategory of A-mod consisting of projective A-modules. Let G0(A) be the Grothendieck group of A-mod which is the free abelian group ZI with basis the set I of isoclasses of simple A-modules. For each i ∈ I, let Si be a simple A-module belonging to the class i. Given a module M in A-mod, we denote by dim M the image of M in P G0(A), called the dimension vector of M. Hence, if dim M = i∈I xii, then xi is the number of composition factors isomorphic to Si in a composition series of M. The P dimension of M over F will be denoted by dim M. For a = aii ∈ ZI, we write P i∈I |a| = i∈I ai. Thus, |dim M| is the number of composition factors of M. 2 Now suppose A is hereditary, i.e, Ext A(M,N) = 0 for all M,N ∈ A-mod. The Euler form h−, −i : ZI × ZI → Z associated with A is defined by 1 hdim M, dim Ni = dim FHomA(M,N) − dim FExt A(M,N), where M,N ∈ A-mod. Its symmetrization (−, −): ZI × ZI → Z, called the symmet- ric Euler form, is given by (dim M, dim N) = hdim M, dim Ni + hdim N, dim Mi. We will mostly work with a finite dimensional hereditary algebra over a finite field Fq of q elements. In this case, for each simple A-module Si, ∼ Di := End A(Si) = Fqdi for some di > 1. 1 For i 6= j in I, we consider the Di-Dj-bimodule Ext A(Sj,Si) and Dj-Di-bimodule 1 Ext A(Si,Sj) and define 1 1 ci,j = −dim Di Ext A(Sj,Si) − dim Ext A(Si,Sj)Di .

We obtain a matrix CA = (ci,j)i,j∈I by setting ci,i = 2 for all i ∈ I. It is easy to see that CA is a symmetrizable generalized Cartan matrix with symmetrization matrix D = diag(di : i ∈ I). Thus, we have the Kac–Moody Lie algebra gA = g(CA) associated with CA; see [19]. Further, we have the associated universal enveloping algebra U(gA) (deﬁned over Q or C) and the associated quantum enveloping algebra CYCLIC COMPLEXES, HALL POLYNOMIALS AND SIMPLE LIE ALGEBRAS 3

U(gA) (defined over C or the field of rational functions Q(v) in indeterminate v); see [18, 23]. Following [8, 9], each finite dimensional hereditary Fq-algebra A, up to Morita ~ equivalence, can be obtained from an Fq-species associated with a valued quiver ∆ = (Γ, d; Ω), where ∆ = (Γ, d) is a valued graph and Ω is an orientation of ∆. We will call ∆ the type of A. If, moreover, A is representation finite, then ∆ is a disjoint union of Dynkin diagrams. We remark that finite dimensional hereditary Fq-algebras can be also constructed in terms of quivers with automorphisms; see [16, 6]. For a finite dimensional hereditary Fq-algebra A, CA only depends on its type ∆. Thus, we will write C∆, g∆ instead of CA and gA. Also, we denote by Φ(∆) the + root system of g∆ and by Φ (∆) the set of positive roots. The set of simple roots is denoted by {αi | i ∈ I}.

2. Category of cyclic complexes and almost split sequences In this section we recall from [25, 5] the notion of m-cyclic complexes over the module category of a finite dimensional algebra A. When A is hereditary, we describe almost split sequences in the category of m-cyclic complexes over projective A-modules. This description is analogous to that given in [33] for the case m = 1. Given an additive category A , let Cb(A ) be the category of bounded complexes of b • i i i objects in A . Each object in C (A ) will be written as M = (M , d )i∈Z, where M are objects in A (all but finitely many are zero), and di : M i → M i+1 are morphisms in A satisfying di+1di = 0. There is a shift functor [1] : Cb(A ) −→ Cb(A ),M • 7−→ M •[1], • i i i i+1 i i+1 where M [1] = (X , f ) is defined by X = M and f = −d for all i ∈ Z. We also define M •[−1] to be the complex N • with M • = N •[1]. Inductively, we can define • b b • • M [s] for each s ∈ Z and, thus, a functor [s]: C (A ) → C (A ) taking M 7→ M [s]. For each m > 1, write Zm = Z/mZ = {0, 1, . . . , m − 1}. By definition, an m-cyclic • i i i complex M = (M , d )i∈Zm over A consists of objects M in A and morphisms i i i+1 i+1 i d : M → M for i ∈ Zm satisfying d d = 0. A morphism f between two m- cyclic complexes M • = (M i, di) and N • = (N i, ci) is given by a family of morphisms i i i i fi : M → N satisfying fi+1d = c fi for all i ∈ Zm. The category of m-cyclic complexes over A is denoted by Cm(A ). b For notational simplicity, we write C0(A ) for C (A ) and set Z0 = Z. For each m > 0, Cm(A ) admits an exact structure such that a sequence 0 −→ M • −→ E• −→ N • −→ 0 i i i is exact if and only if for each i ∈ Zm, the sequence 0 → M → E → N → 0 is split exact in A . By applying [15, Sect. 3.2] and [25, Sect. 7], we obtain that Cm(A ) becomes a Frobenius category in which projective objects are finite direct sums of i i KX [s] for X ∈ A and s ∈ Zm, where KX = (M , d ) ∈ Cm(A ) is defined by setting ( ( X, if i = 0, −1; id , if i = −1; M i = di = X if m 6= 1; and 0, otherwise, 0, otherwise

0 0 0 idX M = X ⊕ X, d = 0 0 if m = 1. For each m > 0, let Km(A ) be the corresponding homotopy category of Cm(A ) and let πm = πm,A : Cm(A ) → Km(A ) denote the natural functor. Note that 4 QINGHUA CHEN AND BANGMING DENG

b K0(A ) = K (A ). If m > 1, there exists a functor b Fm : C (A ) = C0(A ) −→ Cm(A ) • s s • i i taking M = (M , d )s∈Z to Fm(M ) = (X , d )i∈Zm with M Xi = M s and di = (−1)mdiag{ds | s ∈ i}. s∈i

b This induces a functor K (A ) = K0(A ) → Km(A ), still denoted by Fm. It is direct to check the following commutative diagram

Fm C0(A ) / Cm(A )

π0 πm

Fm K0(A ) / Km(A )

Remark 2.1. The m-cyclic complexes have been first introduced in [25] for positive even integer m, but they are called m-cycle complexes there; see also [40]. In case m = 2, they are called Z2-graded complexes in [5]. Let now A be a finite dimensional algebra over a field F. Applying the above construction to A-mod and P = PA, we obtain categories Cm(A-mod), Cm(P), Km(A-mod), and Km(P) for all m ∈ N. We also have functors

Fm : C0(A-mod) −→ Cm(A-mod), Fm : C0(P) −→ Cm(P), Fm : K0(A-mod) −→ Km(A-mod), and Fm : K0(P) −→ Km(P).

b It is well known that K0(A-mod) = K (A-mod) is a triangulated category. If, moreover, A has ﬁnite global dimension, then there is a triangulated equivalence b b from K0(P) = K (P) to the bounded derived category D (A-mod); see [15].

Remarks 2.2. (1) For m > 1, let Γ = Γm be the cyclic quiver with vertex set Zm and ∞ arrows ρi : i −→ i+1 (i ∈ Zm). We let by convention Γ0 be the quiver of type A∞ with vertex set Z0 = Z and arrows ρi : i −→ i+1 (i ∈ Z). Put Λm = FΓ/I, where FΓ is the path algebra of Γ and I is the ideal of FΓ generated by ρi+1ρi for i ∈ Zm. Note that Λ0 does not have an identity element, but enough idempotents. Then Cm(A-mod) can be identified with the category (A ⊗F Λm)-mod of finite dimensional A ⊗F Λm- modules. Note that they have different exact structures; nevertheless, Cm(P) can be viewed as an exact full subcategory of (A ⊗F Λm)-mod. 2 (2) In the case of m = 1, Λ1 is isomorphic to the algebra F[x]/(x ) of dual numbers. The categories C1(A-mod) and C1(P), where A is the path algebra of a quiver, have been studied in [33] in connection with Gorenstein-projective A ⊗F Λ1-modules.

As indicated above, Cm(P) is a Frobenius category. In the following we describe almost split sequences in Cm(P) in the case when A is hereditary. The description is similar to that in [33] for the case m = 1. From now onwards, we assume that A is a ﬁnite dimensional hereditary algebra over a ﬁeld F. Then Cm(P) is closed under subobjects, but, in general, is not abelian. We now describe all the indecomposable objects Cm(P). First of all, for each indecomposable projective P ∈ P and each i ∈ Zm, we have the indecomposable projective and injective object KP [i] in Cm(P). Furthermore, CYCLIC COMPLEXES, HALL POLYNOMIALS AND SIMPLE LIE ALGEBRAS 5

i i given a morphism f : Q → P in P, we deﬁne Cf = (X , d ) ∈ Cm(P) by setting  P, if i = 0; (  f, if i = −1; Xi = Q, if i = −1; di = if m 6= 1; and 0, otherwise 0, otherwise,

0 0 h 0 f i X = P ⊕ Q, d = 0 0 if m = 1. For each A-module M, we ﬁx a minimal projective resolution1 of M

δM εM 0 −→ ΩM −→ PM −→ M −→ 0(2.1) and set CM = CδM ∈ Cm(P). The following result follows easily from the property that A is hereditary; see, for example, [15, 33, 5] for a proof.

Lemma 2.3. The objects CM [i] and KP [i], where i ∈ Zm, M is an indecomposable A-module, and P is an indecomposable projective A-module, provide a complete set of indecomposable objects in Cm(P). Moreover, KP [i] are all indecomposable projective objects in Cm(P). In particular, Fm : C0(P) → Cm(P) is dense. • i i Remark 2.4. By the lemma above, a complex M = (M , d ) in Cm(P) is acyclic, i • i i−1 • i.e., H (M ) = Ker (d )/Im(d ) = 0, ∀ i ∈ Zm, if and only if M is isomorphic to a ﬁnite direct sum of complexes of the form KP [i]. The next lemma can be proved by a direct calculation and by applying an argument similar to that in the proof of [5, Lem. 3.3]. • • Lemma 2.5. (1) If X ,Y ∈ C0(P) are of the form CM or KP for some M ∈ A-mod and P ∈ P, then for i, j ∈ Z, • • HomC0(P)(X [i],Y [j]) 6= 0 =⇒ j = i or i + 1. • • (2) For X ,Y ∈ Cm(P), there is an isomorphism Ext 1 (X•,Y •) ∼ Hom (X•,Y •[1]). Cm(P) = Km(P) Moreover, if m 6= 1, then for M,N ∈ A-mod and i ∈ Zm, ( Ext 1 (M,N), if i = 0; Ext 1 (C ,C [i]) ∼ A Cm(P) M N = HomA(M,N), if i = −1.

• i i Let σ : C0(P) → C0(P) denote the functor such that for M = (M , d ) ∈ C0(P), • i i i m+i i m+i σ(M ) = (X , f ) is deﬁned by X = M and f = d for all i ∈ Z. Then the two lemmas above imply that for each m > 1, the functor

Fm : C0(P) −→ Cm(P) is a G-precovering as deﬁned in [1, (1.7)], where G denotes the inﬁnite cyclic group generated by σ. Obviously, Fm is G-invariant, i.e., Fm ◦ g = Fm for all g ∈ G. Moreover, it can be checked that Fm is a Galois G-covering in the sense of [4, Def. 2.8], and the induced functor b Fm : K0(P) = K (P) −→ Km(P). is a Galois G-covering, too.

1 The notation PM and ΩM will be used throughout the paper. 6 QINGHUA CHEN AND BANGMING DENG

By applying [4, Th. 3.7] to Fm, we obtain that a short exact sequence 0 −→ X• −→ Y • −→ Z• −→ 0 in C0(P) is almost split if and only if • • • 0 −→ Fm(X ) −→ Fm(Y ) −→ Fm(Z ) −→ 0 is an almost split sequence in Cm(P). Thus, the construction of almost split sequences in C1(P) given in [33] can be used to obtain those in Cm(P) for m 6= 1. In the following we describe almost split sequences in Cm(P) based on A-mod. Let M be an indecomposable non-projective A-module. By [3, Ch. 5], there is an almost split sequence ϕ ψ ηM : 0 −→ τM −→ E −→ M −→ 0 in A-mod, where τ = τA denotes the Auslander–Reiten translate in A-mod. By the Horseshoe Lemma (see [36, Sect. 2.2]), the minimal projective resolutions of M and τM as in (2.1) give the following commutative diagram with exact row and columns: (2.2) 0 0 0

ι p 0 / ΩτM / ΩτM ⊕ ΩM / ΩM / 0

δτM δe δM ι p 0 / PτM / PτM ⊕ PM / PM / 0 ε ε τM εe M ϕ ψ 0 / τM / E / M / 0

0 0 0 where ι and p denote the canonical inclusions and projections, respectively. Set E• = C ∈ C ( ). We obtain an exact sequence in C ( ): e δe m P m P

ϕe • ψe (2.3) ηeM : 0 −→ CτM −→ Ee −→ CM −→ 0. We have the following result. Proposition 2.6. Let M be an indecomposable non-projective A-module. Then the above sequence ηeM is an almost split sequence in Cm(P). Moreover, if τM is not • ∼ • ∼ simple, then Ee = CE; if τM is simple, then Ee = CE ⊕ KPτM .

Proof. It can be checked by the deﬁnition that ϕe is a left almost split morphism. Since CM is indecomposable, it follows that ηeM is an almost split sequence. Using a dual statement of [3, Ch. V, Lem. 3.2(4)], we have top (E) = top (M) ⊕ top (τM) if τM is not simple and top (E) = top (M) if τM is simple. This gives the second statement.

Now let {Si | i ∈ I} be a complete set of simple A-modules, and let Pi and Ii be the projective cover and injective hull of Si, respectively. Then

top Pi = Si = soc Ii. CYCLIC COMPLEXES, HALL POLYNOMIALS AND SIMPLE LIE ALGEBRAS 7

Thus, for each i ∈ I, we have the canonical projection pi : Pi → Si and the canonical inclusion κi : Si → Ii. Consider the projective resolution of Ii as in (2.1). We then can lift the composition κipi : Pi → Ii to a morphism si : Pi → PIi :

Pi si κipi

δIi ~ εIi 0 / ΩIi / PIi / Ii / 0 We get a commutative diagram

1 κ= 0 π=[0 1] (2.4) 0 / ΩIi / ΩIi ⊕ Pi / Pi / 0

δI i δe=[δIi si] 1 0 / PIi / PIi / 0 / 0 which gives the exact sequence

ϕ • ψ ηi : 0 −→ CIi −→ H −→ CPi [1] −→ 0(2.5) in C ( ), where H• = C . By Lemma 2.5, under the isomorphism m P δe Ext 1 (C [1],C ) ∼ Hom (C [1],C [1]), Cm(P) Pi Ii = Km(P) Pi Ii the sequence ηi corresponds to the non-zero morphism deﬁned by si. Hence, ηi is non-split. The next result can be obtained by combining [33, Lem. 5.2 & 5.3] with [4, Th. 3.7].

Proposition 2.7. For each i ∈ I, the above sequence ηi is an almost split sequence • ∼ in Cm( ). Moreover, if Ii = Si is simple, then H = K ⊕ C [1]. Otherwise, P Pi rad Pi H• =∼ C ⊕ C [1]. Ii/soc Ii rad Pi The following corollary is an easy implication of Propositions 2.6 and 2.7. Corollary 2.8. Let A be a representation-ﬁnite hereditary algebra. Then the Auslander– b Reiten quiver of C0(PA) = C (PA) is directed. We refer the reader to some examples given in [33, Sect. 8] which illustrate the Auslander–Reiten quivers of C1(PA).

3. Hall polynomials for cyclic complexes In this section we deal with the representation-finite hereditary algebra given by a valued Dynkin quiver ∆~ and show that there exist Hall polynomials for m-cyclic complexes associated with ∆~ in the case when m 6= 1. The result will be used in Section 5 to define the generic Bridgeland–Hall algebra of ∆.~ Let A be a finite dimensional hereditary algebra over the finite field Fq and P = PA be the full subcategory of A-mod consisting of projective modules. Let Cm(P) be the category of m-cyclic complexes of P. Definition 3.1. Given objects Z,X ,...,X in C ( ) (or A-mod), let F Z 1 t m P X1,... ,Xt denote the number of filtrations

Z = Z0 ⊇ Z1 ⊇ · · · ⊇ Zt−1 ⊇ Zt = 0 8 QINGHUA CHEN AND BANGMING DENG ∼ such that Zs−1/Zs = Xs for all 1 6 s 6 t, called the Hall number associated with Z,X1,...,Xt.

For each object X in Cm(P) (or A-mod), let Aut(X) denote the automorphism group of X. The following result is taken from [27, 24].

Lemma 3.2. Let Z,X,Y be objects in Cm(P)(or A-mod). Then |Ext 1(X,Y ) | · |Aut(Z)| F Z = Z , X,Y |Aut(X)| · |Aut(Y )| · |Hom(X,Y )| 1 1 where Ext (X,Y )Z denotes the subset of Ext (X,Y ) consisting of equivalence classes of exact sequences in Cm(P)(or A-mod) of the form 0 → Y → Z → X → 0. In the following we assume that ∆~ is a valued Dynkin quiver, i.e., its underlying valued graph ∆ is a Dynkin diagram. By [8], for each prime power q (6= 1 by conven- ~ tion), we can associate a finite dimensional hereditary Fq-algebra A = A(∆, q) which is representation finite of type ∆; see the discussion in the introduction. By a well-known result in [10, 11, 8], the correspondence M 7→ dim M induces a bijection between the set of isoclasses of indecomposable A-modules and the set + + of positive roots Φ = Φ (∆) of the simple Lie algebra g∆ associated with ∆. For + each α ∈ Φ , let Mq(α) denote the corresponding indecomposable A-module. Let Si be the simple A-module corresponding to the simple root αi, i.e., Mq(αi) = Si. Let + further βi be the root in Φ such that Pi := Mq(βi) is a projective cover of Si. By Lemma 2.3, the set + {CMq(α)[s],KPi [s] | α ∈ Φ , i ∈ I, s ∈ Zm} is a complete set of indecomposable objects in Cm(P). Put + Im(∆) = (Φ ∪ I) × Zm, and identify Φ+ ∪ I with the subset (Φ+ ∪ I) × {0}. Define

Pm(∆) = {λ : Im(∆) −→ N | supp λ is ﬁnite}, where supp λ denotes the set of all x ∈ Im(∆) satisfying λ(x) 6= 0. By the Krull– Schmidt theorem, the correspondence sending λ ∈ Pm(∆) to M M C(λ) = Cq(λ) = λ(α, s)CMq(α)[s] ⊕ λ(i, s)KPi [s] + (α,s)∈Φ ×Zm (i,s)∈I×Zm induces a bijection from Pm(∆) to the set of isoclasses of objects in Cm(P). For λ, µ ∈ Pm(∆), deﬁne the addition λ ⊕ µ ∈ Pm(∆) by setting

(λ ⊕ µ)(x) = λ(x) + µ(x) for all x ∈ Im(∆).

In other words, Cq(λ ⊕ µ) = Cq(λ) ⊕ Cq(µ). An element λ ∈ Pm(∆) is said to be indecomposable if Cq(λ) is indecomposable and decomposable otherwise. We say that λ is acyclic if Cq(λ) is an acyclic complex. It is direct to see that λ is acyclic if and only if supp λ ⊆ I × Zm; see Remark 2.4. + Remark 3.3. There is a bijection from the set of functions Φ → N to the set of isoclasses of A-modules by sending λ 7→ [Mq(λ)], where M Mq(λ) = λ(α)Mq(α) ∈ A-mod. α∈Φ+ CYCLIC COMPLEXES, HALL POLYNOMIALS AND SIMPLE LIE ALGEBRAS 9

Let Z[q] be the polynomial ring over Z in indeterminate q. By a result of Ringel [29], + λ for any given three functions λ, µ, ν :Φ → N, there is a polynomial ϕµ,ν(q) ∈ Z[q] such that for each prime power q,

ϕλ (q) = F Mq(λ) . µ,ν Mq(µ),Mq(ν) They are called Hall polynomials. We are going to extend this result to the category Cm(P) for m 6= 1. λ Deﬁnition 3.4. Let λ, µ, ν ∈ Pm(∆). If there exists a polynomial ψµ,ν(q) ∈ Z[q] such that for each prime power q,

ψλ (q) = F Cq(λ) , µ,ν Cq(µ),Cq(ν) λ we say that the Hall polynomial ψµ,ν(q) exists for the triple λ, µ, ν. Using arguments similar to those in [29, Sect. 2], we obtain the following result.

Lemma 3.5. (1) For λ, µ ∈ Pm(∆), the dimensions

dim Fq HomCm(P)(Cq(λ),Cq(µ)) and dim Fq HomKm(P)(Cq(λ),Cq(µ)) only depend on λ and µ, but not on q. (2) For each λ ∈ Pm(∆), there is a monic polynomial aλ(q) ∈ Z[q] such that for each prime power q,

aλ(q) = |Aut(Cq(λ))|. b We ﬁrst consider the case m = 0. By Corollary 2.8, C0(P) = C (P) is a directed category. More precisely, there is a total ordering 6 on P0(∆) such that for x, y ∈ P0(∆),

HomC0(P)(Cq(x),Cq(y)) 6= 0 =⇒ x 6 y. Then by applying the arguments similar to those in the proof of [29, Th. 1], we obtain the following result.

λ Proposition 3.6. For λ, µ, ν in P0(∆), the Hall polynomial ψµ,ν(q) exists. This proposition together with Lemmas 3.2 and 3.5 gives the next result.

λ Corollary 3.7. For λ, µ, ν in P0(∆), there is a polynomial Eµ,ν(q) ∈ Z[q] such that for each each prime power q, Eλ (q) = |Ext 1 (C (µ),C (ν)) |. µ,ν C0(P) q q Cq(λ)

In the rest of this section, we assume m > 1. By Lemma 2.3, the map γm : P0(∆) → Pm(∆), λ 7→ γm(λ) deﬁned by X γm(λ)(x, s) = λ(x, t), ∀ (x, s) ∈ Im(∆), t∈s is surjective. By the deﬁnition, ∼ Fm(Cq(λ)) = Cq(γm(λ)).

Lemma 3.8. Let λ, µ, ν ∈ Pm(∆) with µ or ν indecomposable. Then the Hall poly- λ nomial ψµ,ν(q) exists. 10 QINGHUA CHEN AND BANGMING DENG

Proof. We only treat the case where µ is indecomposable. The case for ν to be indecomposable can be treated in a dual way. + + Let ξ, η ∈ P0(∆) with their supports contained in Φ ∪ I = (Φ ∪ I) × {0}. By Lemma 2.5(1), for t ∈ Z,

HomC0(P)(Cq(ξ),Cq(η)[t]) 6= 0 =⇒ t = 0 or 1.

Now chooseµ, ˜ ν˜ ∈ P0(∆) such that

γm(˜µ) = µ and γm(˜ν) = ν.

Thenµ ˜ is indecomposable, Fm(Cq(˜µ)) = Cq(µ), and Fm(Cq(˜ν)) = Cq(ν). Since Cq(˜µ) is indecomposable, by applying the shift functor [1], we may assume suppµ ˜ ⊆ Φ+ ∪ I, + i.e., Cq(˜µ) = CMq(α) or KPi for some α ∈ Φ or i ∈ I. Further, we may assume that suppν ˜ ⊆ (Φ+ ∪ I) × {−1, 0, . . . , m − 2}.

Applying Lemma 2.5 and the covering functor Fm : K0(P) → Km(P) gives that Ext 1 (C (µ),C (ν)) Cm(P) q q ∼ =HomKm(P)(Cq(µ),Cq(ν)[1]) = HomKm(P)(Fm(Cq(˜µ)), Fm(Cq(˜ν)[1])) ∼ M t = HomK0(P)(Cq(˜µ), σ (Cq(˜ν)[1])) = HomK0(P)(Cq(˜µ),Cq(˜ν)[1]) t∈Z ∼Ext 1 (C (˜µ),C (˜ν)). = C0(P) q q This gives a bijection of sets [ Ext 1 (C (µ),C (ν)) =∼ Ext 1 (C (˜µ),C (˜ν)) . Cm(P) q q Cq(λ) C0(P) q q Cq(λ˜) λ˜∈P0(∆), γm(λ˜)=λ Therefore, X |Ext 1 (C (µ),C (ν)) | = |Ext 1 (C (˜µ),C (˜ν)) |. Cm(P) q q Cq(λ) C0(P) q q Cq(λ˜) λ˜∈P0(∆), γm(λ˜)=λ

˜ λ˜ By Corollary 3.7, for each λ, there is a polynomial Eµ,˜ ν˜(q) ∈ Z[q] such that for each ﬁnite ﬁeld Fq, ˜ Eλ (q) = |Ext 1 (C (˜µ),C (˜ν)) |. µ,˜ ν˜ C0(P) q q Cq(λ˜) Set λ X λ˜ Eµ,ν(q) = Eµ,˜ ν˜(q)

λ˜∈P0(∆), γm(λ˜)=λ which clearly satisﬁes Eλ (q) = |Ext 1 (C (µ),C (ν)) |. µ,ν Cm(P) q q Cq(λ) The together with Lemma 3.5 shows the existence of the desired Hall polynomial λ ψµ,ν(q) ∈ Z[q].

Since Cm(P) can be viewed as a full subcategory of (A ⊗k Λm)-mod as seen in Remark 2.2(1), the ﬁrst statement of the following lemma follows from [14, Lem. 2.1], while the second one is a direct consequence of Lemma 3.2. CYCLIC COMPLEXES, HALL POLYNOMIALS AND SIMPLE LIE ALGEBRAS 11

Lemma 3.9. (1) Let 0 → X → Z → Y → 0 be an exact sequence in Cm(P). Then dim Ext 1 (Z,Z) dim Ext 1 (X ⊕ Y,X ⊕ Y ). Fq Cm(P) 6 Fq Cm(P) Moreover, the equality holds if and only if Z =∼ X ⊕ Y. µ (2) Let µ = µ1 ⊕µ2. Then ψµ1,µ2 (q) exists and, moreover, it is a monic polynomial (i.e., its leading coeﬃcient equals 1).

For each µ ∈ Pm(∆), set X d = µ(x) and l = dim Ext 1 (C (µ),C (µ)). µ µ Fq Cm(P) q q x∈Im(∆)

In other words, dµ is the number of indecomposable direct summands of Cq(µ). λ Lemma 3.10. If µ ∈ Pm(∆) satisﬁes lµ = 0, then the Hall polynomials ψµ,ν(q) exist for all λ, ν ∈ Pm(∆).

Proof. The case dµ = 0 is trivial, while the case dµ = 1 follows from Lemma 3.8.

Suppose dµ > 1 and write µ = µ1 ⊕ µ2 with dµ1 , dµ2 > 1. It follows from the assumption lµ = 0 that for all λ, ν ∈ Pm(∆) and prime powers q, X C (λ) C (θ) C (λ) F q F q = F q Cq(µ1),Cq(θ) Cq(µ2),Cq(ν) Cq(µ1),Cq(µ2),Cq(ν) θ X C (λ) C (ω) C (λ) C (µ) = F q F q = F q F q . Cq(ω),Cq(ν) Cq(µ1),Cq(µ2) Cq(µ),Cq(ν) Cq(µ1),Cq(µ2) ω By induction on d , we may suppose that both ψλ (q) and ψθ (q) exist. Since µ µ1,θ µ2,ν µ ψµ1,µ2 (q) is a monic polynomial by Lemma 3.9(2), we conclude that X λ θ µ ψµ1,θ(q)ψµ2,ν(q) /ψµ1,µ2 (q) ∈ Z[q] θ λ is the desired polynomial ψµ,ν(q). Now we are ready to prove the existence of Hall polynomials in the general case.

Theorem 3.11. Suppose m > 1. Then for arbitrary λ, µ, ν ∈ Pm(∆), the Hall λ polynomial ψµ,ν(q) exists. Proof. We order the pairs of nonnegative integers (l, d) with d > 1 lexicographically, i.e., 0 0 0 0 0 (l, d) 6 (l , d ) ⇐⇒ l < l or l = l and d 6 d . We use this ordering to proceed induction on (lµ, dµ) to show the existence of the Hall λ polynomials ψµ,ν(q). λ By Lemmas 3.8 and 3.10, ψµ,ν(q) exists whenever lµ = 0 or dµ = 1. Now let µ ∈ Pm(∆) satisfy lµ > 0 and dµ > 1. Write µ = µ1 ⊕ µ2 with µ1 6= 0 6= µ2. Then

lµ1 , lµ2 6 lµ and dµ1 , dµ2 < dµ, that is, (lµ1 , dµ1 ) < (lµ, dµ) and (lµ2 , dµ2 ) < (lµ, dµ). By the deﬁnition, for all λ, ν ∈ Pm(∆) and prime powers q, X C (λ) C (ω) C (λ) F q F q = F q Cq(ω),Cq(ν) Cq(µ1),Cq(µ2) Cq(µ1),Cq(µ2),Cq(ν) ω X C (λ) C (θ) = F q F q . Cq(µ1),Cq(θ) Cq(µ2),Cq(ν) θ 12 QINGHUA CHEN AND BANGMING DENG

Hence,

F Cq(λ) F Cq(µ) Cq(µ),Cq(ν) Cq(µ1),Cq(µ2) (3.1) X C (λ) C (θ) X C (λ) C (ω) = F q F q − F q F q . Cq(µ1),Cq(θ) Cq(µ2),Cq(ν) Cq(ω),Cq(ν) Cq(µ1),Cq(µ2) θ ω6=µ

Moreover, for ω 6= µ, F Cq(ω) 6= 0 implies the existence of a non-split exact Cq(µ1),Cq(µ2) sequence

0 −→ Cq(µ2) −→ Cq(ω) −→ Cq(µ1) −→ 0. By Lemma 3.9(1), l = dim Ext 1 (C (ω),C (ω)) < dim Ext 1 (C (µ),C (µ)) = l , ω Fq Cm(P) q q Fq Cm(P) q q µ that is, (lω, dω) < (lµ, dµ). Hence, by the induction hypothesis, all the Hall polynomials λ ω θ λ ψµ1,θ(q), ψµ1,µ2 (q), ψµ2,ν(q), and ψω,ν(q) µ exist. Since ψµ1,µ2 (q) is a monic polynomial, it follows from (3.1) that X λ θ X λ ω µ ψµ1,θ(q)ψµ2,ν(q) − ψω,ν(q)ψµ1,µ2 (q) /ψµ1,µ2 (q) ∈ Z[q] θ ω6=µ

λ is the required polynomial ψµ,ν(q). The following result is a direct consequence of the above theorem.

Corollary 3.12. Let m > 1. Then for arbitrary λ, µ, ν ∈ Pm(∆), there is a polyno- λ mial Eµ,ν(q) ∈ Z[q] such that for each prime power q, Eλ (q) = |Ext 1 (C (µ),C (ν)) |. µ,ν Cm(P) q q Cq(λ) Remark 3.13. In the proof of Lemma 3.8, the condition m > 1 is required. So our method does not work for m = 1, but we still believe that Hall polynomials should exist in this special case, too.

4. Hall algebras of m-cyclic complexes In this section we follow [28, 17, 5] to define the Hall algebra of m-cyclic complexes of projective modules over a finite dimensional hereditary algebra A and show that its localization is isomorphic to the tensor product of m copies of the extended Ringel– Hall algebra of A when m > 2. This is a slight generalization of [5, Lem. 4.7] for the case m = 2. It might be interesting to study the structure of the Hall algebra of C1(P). As in the previous sections, let A be a finite dimensional hereditary algebra over the finite field Fq and P = PA be the full subcategory of A-mod consisting of projective modules. For each m ∈ N, let Cm(P) be the category of m-cyclic complexes over P. Since Cm(P) is an exact category, we can define the Hall algebra of Cm(P) in the sense of Ringel; see [17, Th. 3]. Let Iso(Cm(P)) be the set of isoclasses of objects in • • • Cm(P). For each M ∈ Cm(P), let [M ] denote the isoclass of M . √ −1 −1 Set v = q and let Z[v, v ] be the subring of R generated by v and v . Recall L• • • • the Hall numbers FM •,N • for L ,M ,N ∈ Cm(P) given in Definition 3.1. CYCLIC COMPLEXES, HALL POLYNOMIALS AND SIMPLE LIE ALGEBRAS 13

−1 Deﬁnition 4.1. The (twisted) Hall algebra Hv(Cm(P)) of Cm(P) is the free Z[v, v ]- module with basis • {u[M •] | [M ] ∈ Iso(Cm(P))} and with multiplication given by P i i • i∈ hdim M ,dim N i X L u[M •] ∗ u[N •] = v Zm FM •,N • u[L•], [L•] • i i • i i where M = (M , dM ),N = (N , dN ) ∈ Cm(P). Note that for each i ∈ Zm, i i i i hdim M , dim N i = dim HomA(M ,N ) since M i and N i are projective A-modules. • For each [M ] ∈ Iso(C2(P)), set • (4.1) b[M •] = |Aut(M )|u[M •].

Note that the elements b[M •] are the basis elements used in [5] for the case m = 2. Since KP [r] are projective and injective objects in Cm(P), a direct calculation by using Lemma 3.2 gives the following formulas in Hv(Cm(P)).

Lemma 4.2. Suppose m 6= 1. Then for each r ∈ Zm, we have

(δr,1−δr,−1)(dim P, dim Q) b[KP [r]] ∗ b[KQ] = v b[KQ] ∗ b[KP [r]] and

(δr,0−δr,−1)(dim P, dim PM )−(δr,0−δr,1)(dim P, dim ΩM ) b[KP [r]] ∗ u[CM ] = v u[CM ] ∗ b[KP [r]], where P,Q ∈ P and M ∈ A-mod. In the rest of this section, we assume m 6= 1. We now follow ideas in [5, Sect. 4] to deduce some formulas in Hv(Cm(P)) which will be needed later on. Let M,N ∈ A-mod. On the one hand, by Lemma 2.5, Ext 1 (C ,C ) ∼ Ext 1 (M,N). Cm(P) M N = A

Hence, each extension of CM by CN is induced by an extension of M by N. More precisely, an extension L of M by N gives rise to the extension CL ⊕ KXL of CM by CN , where XL ∈ P satisﬁes XL ⊕ PL = PM ⊕ PN (so XL ⊕ ΩL = ΩM ⊕ ΩN , too). Applying Lemma 3.2 and the fact that |Aut(CZ )| = |HomA(PZ , ΩZ )| · |Aut(Z)| for Z ∈ A-mod gives that C ⊕K (4.2) F L XL = qdim HomA(PN ,ΩM )−dim EndA(XL)|Aut(X )|F L . CM ,CN L M,N On the other hand, again by Lemma 2.5, Ext 1 (C [1],C ) ∼ Hom (M,N). Cm(P) M N = A

Thus, each extension of CM [1] by CN is induced by a morphism from M to N. Namely, each morphism δ : M → N gives an exact sequence of A-modules 0 −→ X = Ker (δ) −→ M −→δ N −→ Coker (δ) = Y −→ 0.

Then the corresponding extension of CM [1] by CN is given by • L = CX [1] ⊕ CY ⊕ KT ⊕ KW [1], ∼ ∼ where T,W ∈ P satisfy T ⊕ PY = PN and W ⊕ ΩX = ΩM . By Deﬁnition 4.1 and Lemma 3.2, we obtain the following formulas. (For notational simplicity, we write hM,Ni = hdim M, dim Ni for M,N ∈ A-mod.) 14 QINGHUA CHEN AND BANGMING DENG

Lemma 4.3. Let M,N ∈ A-mod. Then X u ∗ u = va(L)F L b ∗ u , [CM ] [CN ] M,N [KXL ] [CL] [L] where L ∈ A-mod, XL ∈ P satisﬁes XL ⊕ PL = PM ⊕ PN , and

a(L) = hPM ,PN i + hΩM , ΩN i + 2hPN , ΩM i − hXL,PL ⊕ ΩLi − 2hXL ⊕ PL,XLi; and hPM ,ΩN i+2hΩN ,PM i+δm,2hΩM ,PN i u[CM [1]] ∗ u[CN ] =v u[CM [1]⊕CN ] X • • + vb(L )F L u ∗ u , CM [1],CN [KT ⊕KW [1]] [CX [1]⊕CY ] [L•] • ∼ where the sum is taken over L = CX [1] ⊕ CY ⊕ KT ⊕ KW [1] =6 CM [1] ⊕ CN satisfying ∼ ∼ T ⊕ PY = PN , and W ⊕ ΩX = ΩM , and • b(L ) = hPM , ΩN i − 2hPX ⊕ ΩY ,W i − 2hPY ,T i

− hT,PX ⊕ PY ⊕ ΩY i − hW, PX ⊕ ΩX ⊕ ΩY i

+ δm,2(hΩM ,PN i − 2hΩX ,T i − hT, ΩX i − hW, PY i). • • • By Remark 2.4 and Lemma 4.2, for two complexes M and N in Cm(P), if M or N • is acyclic, then t b[M •] ∗ b[N •] = v b[M •⊕N •] for some t ∈ Z. • Thus, the elements b[M •] with M acyclic are regular and satisfy both the left and right Ore condition. As in [5], we deﬁne L Hv(Cm(P)) to be the localization of • Hv(Cm(P)) with respect to the elements b[M •] with M acyclic. In other words, −1 • L Hv(Cm(P)) = Hv(Cm(P))[b[M •] : M is acyclic].

In the following we relate L Hv(Cm(P)) with the ordinary Ringel–Hall algebra Hv(A) of A as in [5, Lem. 4.7]. First, recall from [28, 31] that the (twisted) Ringel– −1 Hall algebra Hv(A) of A is by deﬁnition the free Z[v, v ]-module with basis {u[M] | M ∈ A-mod}, and the multiplication is given by hdim M,dim Ni X L u[M] ∗ u[N] = v FM,N u[L]. [L],L∈A-mod et −1 Moreover, the extended Ringel–Hall algebra Hv (A) of A is the free Z[v, v ]-module with basis {u[M] ∗ Kα | M ∈ A-mod, α ∈ ZI}, and the multiplication is given as follows: (α,dim N)+hdim M,dim Ni X L (u[M] ∗ Kα) ∗ (u[N] ∗ Kβ) = v FM,N (u[L] ∗ Kα+β), [L],L∈A-mod where M,N ∈ A-mod and α, β ∈ ZI; see [38] and [7]. For simplicity, we write u[M] = u[M] ∗ K0 and Kα = u[0] ∗ Kα. et Clearly, we have an embedding Hv(A) → Hv (A), u[M] 7→ u[M] for M ∈ A-mod. The subalgebra K spanned by Kα, α ∈ ZI, is the Laurent polynomial algebra −1 ±1 −1 Z[v, v ][Ki : i ∈ I]. Moreover, there is a Z[v, v ]-module decomposition et Hv (A) = Hv(A) ⊗ K. For each M ∈ A-mod and r ∈ Zm, set

ζ(M) = hdim ΩM , dim Mi + 2hdim PM , dim ΩM i CYCLIC COMPLEXES, HALL POLYNOMIALS AND SIMPLE LIE ALGEBRAS 15 and deﬁne (r) ζ(M) −1 (4.3) E = v b ∗ u[CM [r]] ∈ L Hv(Cm(P)). [M] [KΩM [r]] (0) (r) We simply write E[M] = E[M]. Further, let K be the subalgebra of L Hv(Cm(P)) generated by b±1 for P ∈ P and H(r) be the subalgebra generated by b±1 and [KP [r]] [KP [r]] (r) −1 E[M] for P ∈ P and M ∈ A-mod. Finally, for a Z[v, v ]-module H and a ﬁeld −1 F ⊃ Z[v, v ], we write HF = H ⊗Z[v,v−1] F. Let {Si | i ∈ I} be a complete set of simple A-modules. For i ∈ I, let Pi denote a projective cover of Si. Thus, there is a minimal projective resolution of Si

0 −→ rad Pi −→ Pi −→ Si −→ 0. The next result is an analogue of [5, Lem. 4.3 & 4.7].

Proposition 4.4. Suppose m > 2. Then (1) For P,Q ∈ P,

b[KP ] ∗ b[KQ] = b[KP ⊕Q]. (2) For each r ∈ Zm, there is an algebra isomorphism et (r) (r) −1 κr : Hv (A) −→ H , u[M] 7−→ E ,Ki 7−→ b ∗ b[KP [r]]. [M] [Krad Pi [r]] i −1 (3) Let F be a ﬁeld containing Z[v, v ]. Then multiplication induces a vector space isomorphism µ : H(0) ⊗ H(1) ⊗ · · · ⊗ H(m−1) −→ H (C ( )) . F F F L v m P F Proof. (1) This follows directly from the deﬁnition.

(2) We only consider the case r = 0 and write κ = κ0. Take M,N ∈ A-mod. By the deﬁnition and Lemmas 4.2 and 4.3, ζ(M)+ζ(N) −1 −1 E[M] ∗ E[N] = v b ∗ u[CM ] ∗ b ∗ u[CN ] [KΩM ] [KΩN ] ζ(M)+ζ(N)+(dim ΩN , dim M) −1 = v b ∗ u[CM ] ∗ u[CN ] [KΩM ⊕KΩN ]

X ζ(M)+ζ(N)+(dim ΩN , dim M)+a(L) L −1 = v FM,N b ∗ b[KX ] ∗ u[CL] [KΩM ⊕KΩN ] L [L]

X ζ(M)+ζ(N)+(dim ΩN , dim M)+a(L)−ζ(L) L = v FM,N E[L], [L] where L ∈ A-mod and a(L) is as deﬁned in Lemma 4.3. A direct calculation shows that

ζ(M) + ζ(N) + (dim ΩN , dim M) + a(L) − ζ(L) = hdim M, dim Ni. We conclude that

κ(u[M] ∗ u[N]) = E[M] ∗ E[N] = κ(u[M]) ∗ κ(u[N]). The remaining relations follow easily from Lemma 4.2. Consequently, κ is an algebra homomorphism. The injectivity of κ follows from the fact that κ sends the basis

{u[M] ∗ Kα | M ∈ A-mod, α ∈ ZI} 16 QINGHUA CHEN AND BANGMING DENG

et of Hv (A) to a linearly independent set in L Hv(Cm(P)); see [5, Lem. 4.6].

For the surjectivity of κ, it suﬃces to show that b[KP ] ∈ Im (κ) for all P ∈ P. By (1), it is enough to prove that b ∈ Im (κ) for each i ∈ I. By an induction on [KPi ] Loewy length, we may suppose that b = κ(Kα) for some α ∈ I. Then [Krad Pi ] Z

b = b κ(Ki) = κ(KαKi) ∈ Im (κ). [KPi ] [Krad Pi ] Therefore, κ is an isomorphism. (3) By Lemma 2.5(2), for M,N ∈ A-mod, Ext 1 (C [i],C [j]) = 0 for 1 i < j < m. Cm(P) M N 6

Thus, for M1,...,Mm−1 ∈ A-mod, we have u ∗ · · · ∗ u = vau [CM1 [1]] [CMm−1 [m−1]] [CM1 [1]⊕···⊕CMm−1 [m−1]] for some a ∈ Z. This together with Lemma 4.2 implies that multiplication induces an injective F-linear map H := H(1) ⊗ · · · ⊗ H(m−1) −→ H (C ( )) . 1 F F L v m P F Consider the multiplication map

µ : H(0) ⊗ H = H(0) ⊗ H(1) ⊗ · · · ⊗ H(m−1) −→ H (C ( )) . F 1 F F F L v m P F

As in the proof of [5, Lem. 4.7], we can define a filtration on L Hv(Cm(P))F. By using the formulas in Lemma 4.3, we obtain that µ is an isomorphism by dealing with the associated filtered algebra of L Hv(Cm(P))F. • Remark 4.5. A complex M ∈ C2(P) is called self-dual if it is acyclic and satisfies M •[1] =∼ M •. Bridgeland [5] has introduced the reduced localized Hall algebra of A red DHv (A) := L Hv(C2(P))/I, • where I is the ideal of L Hv(C2(P)) generated by b[M •] − 1 with M self-dual.

5. Generic Bridgeland–Hall algebras and simple Lie algebras In this section we use Hall polynomials to deﬁne the generic Bridgeland–Hall algebra associated with a valued Dynkin quiver ∆~ and show that its degenerate form is isomorphic to the universal enveloping algebra of the simple Lie algebra g∆ associated with the underlying diagram ∆. It turns out that the Lie bracket in g∆ can be obtained by evaluating a variant of Hall polynomials at v = 1. We keep all the notation in Section 3. Let ∆~ be a valued Dynkin quiver. Then the associated Lie algebra g∆ is simple. For each i ∈ I, let Si be the simple module over + A = A(∆~ , q) corresponding to the simple root αi. Let further βi be the root in Φ such that Pi := Mq(βi) is a projective cover of Si. Since we deal with the case m = 2 in this section, we sometimes drop the subscripts 2, for example, we write + I(∆) = I2(∆) = (Φ ∪ I) × Z2 and P(∆) = P2(∆). We will view each element x ∈ I(∆) as the function f ∈ P(∆) deﬁned by f(y) = + + − δx,y for y ∈ I(∆). From now on, we identify Φ × Z2 with Φ = Φ ∪ Φ via (α, 0) 7−→ α, (α, 1) 7−→ −α. CYCLIC COMPLEXES, HALL POLYNOMIALS AND SIMPLE LIE ALGEBRAS 17

In other words, I(∆) = Φ ∪ (I × Z2), and for each prime power q and λ ∈ P(∆), the corresponding object Cq(λ) ∈ C2(PA) is given by M M Cq(λ)= λ(α)CMq(α)⊕λ(−α)CMq(α)[1] ⊕ λ(i, 0)KPi ⊕λ(i, 1)KPi [1] . α∈Φ+ i∈I Furthermore, put P± = P±(∆) = {λ ∈ P(∆) | supp λ ⊆ Φ±}, ac ac P = P (∆) = {λ ∈ P(∆) | supp λ ⊆ I × Z2}, and P0 = P0(∆) = {λ ∈ P(∆) | supp λ ⊆ I × {0}}.

Then λ ∈ P is acyclic (i.e., Cq(λ) is an acyclic complex in C2(P)) if and only if ac + + λ ∈ P . If λ ∈ P , then λ is essentially a function Φ → N. Thus, we have the corresponding module Mq(λ) over A = A(∆~ , q); see Remark 3.3, and, moreover, 0 L Cq(λ) = CMq(λ). If λ ∈ P , then Cq(λ) = KP , where P = i∈I λ(i, 0)Pi. The shift functor [1] in C2(P) induces an involution ∗ : P(∆) −→ P(∆), λ 7−→ λ∗ ∗ ∼ + such that Cq(λ ) = Cq(λ)[1]. In particular, for α ∈ Φ and i ∈ I, α∗ = −α and (i, 0)∗ = (i, 1). For each λ ∈ P+ and prime power q, consider the projective resolution

0 −→ ΩMq(λ) −→ PMq(λ) −→ Mq(λ) −→ 0. 0 of A-module Mq(λ) as in (2.1). Then there exist ℘λ, ωλ ∈ P (independent of q) such that C (℘ ) ∼ K and C (ω ) ∼ K . q λ = PMq(λ) q λ = ΩMq(λ)

In particular, if λ = αi for some i ∈ I, then Cq(℘αi ) = KPi and Cq(ωαi ) = Krad Pi . In this case, we simply write

℘i := ℘αi and ωi := ωαi . 2 −1 Let v be the indeterminate satisfying v = q and let Z = Z[v, v ] be the Laurent polynomial ring in v. Recall from [32, Sect. 6] that the generic Ringel–Hall algebra ~ ~ + Hv(∆) of ∆ is by deﬁnition the free Z-module with basis {uλ | λ :Φ → N} and with multiplication given by

hdim Mq(µ),dim Mq(ν)i X λ 2 uµ ∗ uν = v ϕµ,ν(v )uλ, λ:Φ+→N λ where ϕµ,ν(q) is the Hall polynomial deﬁned in Remark 3.3. i i i i For λ, µ ∈ P(∆), write Cq(λ) = (M , d ) and Cq(µ) = (N , c ) and deﬁne δ(λ, µ) = hdim M 0, dim N 0i + hdim M 1, dim N 1i. λ Further, by Theorem 3.11, we have Hall polynomials ψµ,ν(q) ∈ Z[q] for all λ, µ, ν ∈ P(∆).

Deﬁnition 5.1. The (generic) twisted Hall algebra Hv(∆)~ of 2-cyclic complexes associated to ∆~ is the free Z-module with basis {uλ | λ ∈ P(∆)} and with multiplication given by δ(µ,ν) X λ 2 uµ ∗ uν = v ψµ,ν(v )uλ. λ∈P(∆) 18 QINGHUA CHEN AND BANGMING DENG

However, in order to make some computations in C2(P) over ﬁnite ﬁelds Fq, we also write uλ = u[C(λ)]. For each λ ∈ P(∆), put 2 bλ = aλ(v )uλ ∈ Hv(∆)~ . By applying Lemma 4.2 and [5, Lem. 4.3 & 4.7] (see also Lemma 4.3), we have the following multiplication formulas in Hv(∆).~ ac Lemma 5.2. (1) For λ, µ ∈ P , bλ ∗ bµ = bλ⊕µ. (2) For µ, ν ∈ P+,

X a(γ) γ 2 uµ ∗ uν = v ψµ,ν(v )bξγ ∗ uγ, γ∈P+ 0 where ξγ ∈ P satisﬁes ξγ ⊕ ℘γ = ℘µ ⊕ ℘ν and a(γ) = a(Mq(γ)) ∈ Z as deﬁned in Lemma 4.3. (3) For µ, ν ∈ P+,

c(µ∗,ν) X b(ξ,η) ξ∗⊕η⊕τ(ξ,η) 2 uµ∗ ∗ uν = v uµ∗⊕ν + v ψµ∗,ν (v )uτ(ξ,η) ∗ uξ∗⊕η, ξ,η where the sum is taken over ξ, η ∈ P+ with ξ∗ ⊕ η 6= µ∗ ⊕ ν, τ(ξ, η) ∈ Pac is uniquely determined by ξ and η, c(µ∗, ν) is equal to hdim PMq(µ), dim ΩMq(ν)i + 2hdim ΩMq(ν), dim PMq(µ)i + hdim ΩMq(µ), dim PMq(ν)i, ∗ and b(ξ, η) = b(Cq(ξ ⊕ η ⊕ τ(ξ, η))) as deﬁned in Lemma 4.3.

As in Section 4, let L Hv(∆)~ denote the localization of Hv(∆)~ with respect to bλ with λ acyclic, that is, ~ ~ −1 ac L Hv(∆) = Hv(∆)[bλ : λ ∈ P ]. Remark 5.3. If q is a prime power and A = A(∆~ , q), then by Deﬁnition 4.1, special- √ −1 izing v to v = q gives Z[v, v ]-algebra isomorphisms ~ −1 ∼ Hv(∆) ⊗Z Z[v, v ] = Hv(C2(PA)) and ~ −1 ∼ L Hv(∆) ⊗Z Z[v, v ] = L Hv(C2(PA)). For λ ∈ P+, we deﬁne the elements ζ(λ) −1 `(λ) d(λ)+ζ(λ) −1 ~ (5.1) Eλ = v bω ∗ uλ,Fλ = (−1) v b ∗ ∗ uλ∗ ∈ L Hv(∆), λ ωλ where `(λ) = |dim Mq(λ)|, d(λ) = dim Mq(λ), and

ζ(λ) = hdim ΩMq(λ), dim Mq(λ)i + 2hdim PMq(λ), dim ΩMq(λ)i. By applying arguments similar to those in the proof of Proposition 4.4(2) and Lemma 5.2(2), we obtain Z-algebra embeddings + ι : Hv(∆)~ −→ L Hv(∆)~ , uλ 7−→ Eλ and (5.2) − ι : Hv(∆)~ −→ L Hv(∆)~ , uλ 7−→ Fλ.

(0) (1) Let Hv (∆)~ (resp., Hv (∆))~ be the Z-subalgebra of L Hv(∆)~ generated by Eλ ±1 ±1 + ac and bµ (resp., Fλ and bµ∗ ) for λ ∈ P and µ ∈ P . By modifying the proof of CYCLIC COMPLEXES, HALL POLYNOMIALS AND SIMPLE LIE ALGEBRAS 19

[5, Lem. 4.7] or the proof of Proposition 4.4(3), the multiplication map deﬁnes an isomorphism of vector spaces (0) ~ (1) ~ ~ (5.3) Hv (∆)Q(v) ⊗ Hv (∆)Q(v) −→ L Hv(∆)Q(v). An element λ ∈ P(∆) is said to be self-dual if Cq(λ) is self-dual, or equivalently, ∗ 0 ~ λ = µ ⊕ µ for some µ ∈ P . Let J = J∆~ be the ideal of L Hv(∆)Q(v) generated by bλ − 1 with λ self-dual and deﬁne red ~ ~ DHv (∆) := L Hv(∆)Q(v)/J , called the (generic) Bridgeland–Hall algebra of ∆.~ By Lemma 5.2(1), for λ ∈ Pac, −1 red ~ ∗ we have bλ = bλ∗ in DHv (∆) since λ ⊕ λ is self-dual. The isomorphism in (5.3) induces the following result; see [5, Lem. 4.8]. Lemma 5.4. The multiplication map ~ ~ red ~ Hv(∆)Q(v) ⊗ Q(v)[ZI] ⊗ Hv(∆)Q(v) −→ DHv (∆) uλ ⊗ α ⊗ uµ 7−→ Fλ ⊗ Kα ⊗ Eµ + P is an isomorphism of vector spaces, where λ, µ ∈ P , α = aii ∈ ZI, and Kα = Q b−1 b ai i∈I ( ωi ∗ ℘i ) .

Let Uv(g∆) be the quantum enveloping algebra of g∆ (defined over Q(v)) with ±1 + − 0 generators Ei,Fi,Ki (i ∈ I) and let U (resp., U and U ) be the subalgebra of ±1 Uv(g∆) generated by Ei (resp., Fi and Ki ) for i ∈ I. The following theorem is a generic version of [5, Th. 4.9] for finite type. Theorem 5.5. There is a Q(v)-algebra isomorphism red ~ R : Uv(g∆) −→ DHv (∆) taking b−1 b Ei 7−→ Eαi ,Fi 7−→ Fαi ,Ki 7−→ Ki = ωi ∗ ℘i . ~ Proof. By [28, 31], the elements ui = uαi in Hv(∆) satisfy the quantum Serre relations, and there are Q(v)-algebra isomorphisms + ~ − ~ U −→ Hv(∆)Q(v),Ei 7−→ ui and U −→ Hv(∆)Q(v),Fi 7−→ ui. + − Thus, the images Eαi (resp., Fαi ) of the ui under the embedding ι (resp., ι ) in (5.2) satisfy the quantum Serre relations, too. By applying Lemma 4.2 and [5, Lem. 4.3], it ±1 b∓1 b±1 can be checked directly that Eαi ,Fαi , Ki = ωi ∗ ℘i satisfy the remaining relations for defining Uv(g∆). Therefore, there is a Q(v)-algebra homomorphism red ~ R : Uv(g∆) −→ DHv (∆) b−1 b taking Ei 7→ Eαi ,Fi 7→ Fαi ,Ki 7→ ωi ∗ ℘i . On the one hand, it is well known that Uv(g∆) admits a triangular decomposition − 0 + Uv(g∆) = U ⊗ U ⊗ U . On the other hand, by Lemma 5.4, there is a triangular decomposition red ~ − 0 + DHv (∆) = D ⊗ D ⊗ D , + − 0 red ~ where D (resp., D and D ) is the subalgebra of DHv (∆) generated by Eαi ±1 (resp., Fαi and Ki ) for i ∈ I, and, moreover, there are isomorphisms ~ + ~ − Hv(∆)Q(v) −→ D , ui 7−→ Eαi and Hv(∆)Q(v) −→ D , ui 7−→ Fαi . 20 QINGHUA CHEN AND BANGMING DENG

0 ∼ 0 It is clear that U = D . We conclude that R is an isomorphism. We now introduce some notation. For integers m, d with d > 1, put dm −dm v − v −1 [m] d = ∈ Z = [v, v ]. v vd − v−d Z ! ! Set [m]vd = [m]vd ... [2]vd [1]vd for m > 1 and deﬁne [0]vd = 1 by convention. Let UZ be the integral Z-form of Uv(g∆), i.e., the Z-subalgebra generated by the divided powers r r (r) Ei (r) Fi Ei = ! and Fi = ! (i ∈ I, r > 1) [r]vi [r]vi −1 di together with Ki, Ki (i ∈ I), where vi = v ∈ Z with di = dim Fq End A(Si). Let + − (r) (r) UZ and UZ be the Z-subalgebras of UZ generated by the Ei and Fi , respectively. We remark that there are Z-algebra isomorphisms + H ~ (r) (r) r ! − H ~ (r) (r) UZ −→ v(∆),Ei 7−→ ui = ui /[r]vi and UZ −→ v(∆),Fi 7−→ ui . Put Hv(∆)~ = R(UZ ). Specializing v to 1 gives rise to a Q-algebra isomorphism ∼ ~ ~ U1 := UZ ⊗Z Q = Hv(∆) ⊗Z Q =: H1(∆).

In what follows, we identify UZ with Hv(∆)~ and U1 with H1(∆),~ and for x, y ∈ Hv(∆),~ we simply write xy instead of x ∗ y. Let U(g∆) be the universal enveloping algebra of g∆ (over Q) with standard generators ei, fi, hi, i ∈ I. By [21, 6.7] and [23], we have the result below. Lemma 5.6. There is a Q-algebra isomorphism

R : U(g∆) −→ U1/(Ki − 1 : i ∈ I) = H1(∆)~ /(Ki − 1 : i ∈ I) taking ei 7−→ Ei, fi 7−→ F i (i ∈ I).

Under the isomorphism R, the Lie algebra g∆ is identiﬁed with the Lie subalgebra ac of U1/(Ki − 1 : i ∈ I) generated by Ei, F i, [EiF i](i ∈ I). Note that for each τ ∈ P + and λ ∈ P , we have in U1/(Ki − 1 : i ∈ I), `(λ) bτ = 1, Eλ = uλ, F λ = (−1) uλ∗ .

In the following we interpret some structure coefficients in g∆ in terms of Hall polynomials arising in Hv(∆).~ For each λ ∈ P(∆), let λac ∈ Pac be defined by ac ac λ (α) = 0 and λ (x) = λ(x) for all α ∈ Φ and x ∈ I × Z2. Given a triple λ, µ, ν ∈ P(∆), define λ λ ψµ,ν(q) (5.4) Ψµ,ν(q) = ∈ Q(q). aλac (q) λ −1 In general, Ψµ,ν(q) may not lie in Z[q, q ]. However, we have the following result. Proposition 5.7. Let λ ∈ P(∆) and let α, β ∈ Φ satisfy α + β 6= 0 (i.e., α 6= −β). λ −1 Then Ψα,β(q) ∈ Z[q, q ]. CYCLIC COMPLEXES, HALL POLYNOMIALS AND SIMPLE LIE ALGEBRAS 21

Proof. Thanks to the involution ∗, we are reduced to consider the following two cases. + λ ac 0 ac + Case α, β ∈ Φ . If ψα,β(q) 6= 0, then by Lemma 5.2, λ ∈ P and γ = λ − λ ∈ P . ac Write Cq(λ ) = KP for some P ∈ P. Applying (4.2) gives λ d γ −1 ac Ψα,β(q) = δλ ⊕℘γ ,℘α⊕℘β q ϕα,β(q) ∈ Z[q, q ], γ where d = dim HomA(PMq(β), ΩMq(α))−dim End A(P ) and ϕα,β(q) is the Ringel’s Hall + polynomial associated with γ, α, β :Φ → N. − + ∗ Case α ∈ Φ and β ∈ Φ . In this case, Cq(α) = CMq(α )[1] and Cq(β) = CMq(β). ∗ 1 If HomA(Mq(α ),Mq(β))= 0, then Ext (Cq(α),Cq(β))= 0 by Lemma 2.5(2). C2(PA) Hence,

λ α⊕β dim Hom(Cq(β),Cq(α)) −1 Ψα,β(q) = δλ,α⊕βΨα,β (q) = δλ,α⊕βq ∈ Z[q, q ]. ∗ Now suppose HomA(Mq(α ),Mq(β)) 6= 0. By Lemma 5.2(3), c(α,β) X b(ξ,η) ξ∗⊕η⊕τ(ξ,η) uαuβ = v uα⊕β + v ψα,β (q)uτ(ξ,η)uξ∗⊕η, ξ,η + ∗ where c(α, β), b(ξ, η) ∈ Z, the sum is taken over all pairs ξ, η ∈ P with ξ ⊕η 6= α⊕β, and τ(ξ, η) ∈ Pac is uniquely determined by ξ and η. ∗ Since the Auslander–Reiten quiver of A is directed, we have HomA(Mq(β),Mq(α )) = 1 0, i.e., Ext (Cq(β),Cq(α)) = 0. Therefore, C2(PA) x uβuα = v uα⊕β for some x ∈ Z. ξ∗⊕η⊕τ(ξ,η) Further, by the discussion right above Lemma 4.3, Ψα,β (q) 6= 0 implies that there is an exact sequence ∗ ∗ 0 −→ Mq(ξ ) −→ Mq(α ) −→ Mq(β) −→ Mq(η) −→ 0 ∗ for some prime power q. This implies that HomA(Mq(η),Mq(ξ )) = 0 and d(ξ,η) uηuξ∗ = v uξ∗⊕η for some d(ξ, η) ∈ Z. We conclude that (5.5) c(α,β)−x X b(ξ,η)−d(ξ,η) ξ∗⊕η⊕τ(ξ,η) uαuβ = v uβuα + v ψα,β (q)uτ(ξ,η)uηuξ∗ ξ,η c(α,β)−x X b(ξ,η)−d(ξ,η) ξ∗⊕η⊕τ(ξ,η) = v uβuα + v Ψα,β (q)bτ(ξ,η)uηuξ∗ ∈ UZ . ξ,η By the embeddings in (5.2), the set + + {Eλ | λ ∈ P } (resp., {Fλ | λ ∈ P }) + − is a Z-basis of UZ (resp., UZ ). By [21, Sect. 2] (see also [6, Sect. 6.9]), there is a triangular decomposition + 0 − UZ = UZ ⊗ UZ ⊗ UZ , 0 0 0 ±1 0 where UZ = UZ ∩ U with U = Q(v)[Ki : i ∈ I] ⊂ Uv(g∆). Moreover, UZ is a free Z-module with a basis n K ; 0 o Y ci i K := Ki 0 6 ci 6 1, ti ∈ N, ∀ i ∈ I , ti i∈I vi 22 QINGHUA CHEN AND BANGMING DENG where ti −r+1 −1 r−1 Ki; 0 Y Kivi − Ki vi = r −r . ti v − v vi r=1 i i Therefore, the set + {EλκFµ | λ, µ ∈ P , κ ∈ K } + 0 ∼ 0 + forms a Z-basis of UZ . The fact that UZ ⊗ UZ = UZ ⊗ UZ implies that + {κEλFµ | λ, µ ∈ P , κ ∈ K } is also a Z-basis of UZ . By (5.1), we infer that + (5.6) {κuλuµ∗ | λ, µ ∈ P , κ ∈ K } is a Z-basis of UZ , too. An easy observation shows that if we write each term bτ(ξ,η) in (5.5) as a Z-linear combination of basis elements in K , then at least one coeﬃcient is ±1. Therefore, ξ∗⊕η⊕τ(ξ,η) applying the basis given in (5.6) forces that all Ψα,β (q) appeared in (5.5) lie −1 −1 λ in Z[v, v ], and, thus, in Z[q, q ]. Since ψα,β(q) 6= 0 implies that λ = α ⊕ β or λ ∗ has the form ξ ⊕ η ⊕ τ(ξ, η), this completes the proof. Let µ, ν ∈ P± and let γ ∈ P(∆) satisfy supp (γ) ⊆ Φ. By Lemma 5.2, there is at ac most one τ = τ(γ;µ,ν) ∈ P such that γ⊕τ ψµ,ν (q) 6= 0. If such a τ does not exist, we simply set τ = 0. + − For each α ∈ Φ, put eα = Eα if α ∈ Φ and eα = F −α if α ∈ Φ . Then the set

{eα, hi | α ∈ Φ, i ∈ I} forms a Chevalley basis of g∆. By (5.1), eα = εαuα for all α ∈ Φ, where εα = 1 if + `(α) − α ∈ Φ and εα = (−1) if α ∈ Φ . Corollary 5.8. Suppose that α, β ∈ Φ satisfy α + β 6= 0. Then

X γ⊕τ(γ;α,β) γ⊕τ(γ;β,α) [eα, eβ] = εαεβ εγ Ψα,β (1) − Ψβ,α (1) eγ. γ∈Φ Proof. We only consider the case α, β ∈ Φ+ and the case α ∈ Φ−, β ∈ Φ+. The remaining two cases can be treated by applying the involution ∗. + ~ ~ Case α, β ∈ Φ . Let H1(∆) = Hv(∆)⊗Z Q denote the degenerate Ringel–Hall algebra + of ∆~ by specializing v to 1 and let n be its subspace spanned by uγ (γ ∈ Φ ). Then, by [30, Sect. 10], n is a Lie subalgebra of H1(∆)~ and, moreover, there is a Lie algebra embedding n −→ g∆, uγ 7−→ eγ. Therefore, in this case, the formula follows from the proof of Proposition 5.7 and [29, Sect. 4]. Case α ∈ Φ− and β ∈ Φ+. We have α∗ = −α ∈ Φ+. Let λ ∈ P with γ = λ − λac. a Then aλ(q) = q aγ(q)aλac (q) for some a ∈ N. By Lemma 3.2, Eλ (q)a (q) λ b α,β λ ψα,β(q) = q , aα(q)aβ(q) CYCLIC COMPLEXES, HALL POLYNOMIALS AND SIMPLE LIE ALGEBRAS 23

λ where b ∈ Z and Eα,β(q) ∈ Z[q] is such that for each prime power q,

λ 1 Eα,β(q) = |Ext (Cq(α),Cq(β))Cq(λ)|; see Corollary 3.12. Therefore,

ψλ (q) Eλ (q)a (q) Eλ (q)a (q) λ α,β b α,β λ a+b α,β γ Ψα,β(q) = = q = q . aλac (q) aα(q)aβ(q)aλac (q) aα(q)aβ(q)

α⊕β Suppose γ is decomposable. If λ = α ⊕ β, then γ = λ and Eα,β (q) = 1. Since c aα⊕β(q) = q aα(q)aβ(q) for some c ∈ N,

α⊕β a+b+c α⊕β we get that Ψα,β (q) = q , which implies that Ψα,β (1) = 1. Now let λ 6= α ⊕ β. λ If Ψα,β(q) 6= 0, then γ 6= 0 since α + β 6= 0. By an argument similar to that in the 2 proof of [29, Th. 2], aγ(q) is divided by (q−1) . Further, for big enough prime powers × 1 q, Fq = Fq\{0} acts freely on Ext (Cq(α),Cq(β))Cq(λ). This implies that

λ (q − 1) | Eα,β(q).

∗ Since Cq(α) = CMq(α )[1] and Cq(β) = CMq(β) are indecomposable, it follows that

dα dβ aα(q) = q − 1 and aβ(q) = q − 1,

λ where dα = dim End (Cq(α)) and dβ = dim End (Cq(β)). This implies Ψα,β(1) = 0. In conclusion, by applying Lemma 5.2(3), we obtain that

X λ uαuβ = uα⊕β + Ψα,β(1)bλac uλ−λac λ∈P(∆), λ6=α⊕β

X γ+τ(γ;α,β) X γ+τ(γ;α,β) = uα⊕β + Ψα,β (1)uγ = uα⊕β + Ψα,β (1)εγeγ. γ∈Φ γ∈Φ Analogously, we have

X γ+τ(γ;β,α) uβuα = uα⊕β + Ψβ,α (1)εγeγ. γ∈Φ Consequently,

X γ⊕τ(γ;α,β) γ⊕τ(γ;β,α) [eα, eβ] = εαεβ[uα, uβ] = εαεβ εγ Ψα,β (1) − Ψβ,α (1) eγ. γ∈Φ

Acknowledgement

We would like to thank the referees for helpful comments which simplify several arguments in the proofs and improve the organization of Sections 4 and 5. In particular, we are indebted to the referees for pointing out to us the reference [4] and some errors in an earlier version. 24 QINGHUA CHEN AND BANGMING DENG

References [1] H. Asashiba, A generalization of Gabriel’s Galois covering functors and derived equivalences, J. Algebra 334 (2011), 109–149. [2] I. Assem, D. Simson and A. Skowroński, Elements of Representation Theory of Associative Algebras, vol. 1: Techniques of Representation Theory, Cambridge University Press, Cambridge, 2006. [3] M. Auslander, I. Reiten, and S.O. Smalφ, Representation Theory of Artin Algebras, Cambridge Studies in Advanced Mathematics: 36. Cambridge University Press, Cambridge, 1995. [4] R. Bautista and S. Liu, Covering theory for linear categories with application to derived categories, J. Algebra 406 (2014), 173–225. [5] T. Bridgeland, Quantum groups via Hall algebras of complexes, Ann. Math. 177 (2013), 1–21. [6] B. Deng, J. Du, B. Parshall and J. Wang, Finite dimensional algebras and quantum groups, Mathematical Surveys and Monographs Volume 150, Amer. Math. Soc., Providence 2008. [7] B. Deng and J. Xiao, On double Ringel–Hall algebras, J. Algebra 251 (2002), 110–149. [8] V. Dlab and C. M. Ringel, On algebras of finite representation type, J. Algebra 33(1975), 306– 394. [9] V. Dlab and C. M. Ringel, Indecomposable representations of graphs and algebras, Memoirs Amer. Math. Soc. 6, no. 173, 1976. [10] P. Gabriel, Unzerlegbare Darstellungen I, Manuscripta Math. 6 (1972), 71–103. [11] P. Gabriel, Indecomposable representations II, Symp. Math. 11 (1973), 81–104. [12] P. Gabriel and A. V. Roiter, Representations of Finite-Dimensional Algebras, with a chapter by B. Keller, Encyclopaedia Math. Sci., no. 73, Algebra, VIII, Springer-Verlag, Berlin, 1992. [13] J. A. Green, Hall algebras, hereditary algebras and quantum groups, Invent. Math. 120(1995), 361–377. [14] J. Guo and L. Peng, Universal PBW-basis of Hall–Ringel algebras and Hall polynomials, J. Algebra, 198 (1997), 339–351. [15] D. Happel, Triangulated categories in the representation theory of finite-dimensional algebras, London Math. Soc. Lecture Note Series, 119. Cambridge University Press, Cambridge, 1988. [16] A. Hubery, Quiver representations respecting a quiver automorphism: a generalisation of a theorem of Kac, J. London Math. Soc. (2) 69(2004), 79–96. [17] A. Hubery, From triangulated categories to Lie algebras: A theorem of Peng and Xiao, Contemp. Math. 406, Amer. Math. Soc. (2006) 51–66. [18] J. C. Jantzen, Lectures on Quantum Groups, Graduate Studies in Mathematics, no. 6, Amer. Math. Soc., Providence, 1996. [19] V. Kac, Infinite Dimensional Lie Algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. [20] M. Kapranov, Heisenberg doubles and derived categories, J. Algebra 202 (1998), 712–744. [21] G. Lusztig, Finite dimensional Hopf algebras arising from quantized universal enveloping algebras, J. Amer. Math. Soc. 3 (1990), 257–296. [22] G. Lusztig, Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990), 447–498. [23] G. Lusztig, Introduction to Quantum Groups, Birkhäuser,Boston, 1993. [24] L. Peng, Some Hall polynomials for representation-finite trivial extension algebras, J. Algebra 197 (1997), 1–13. [25] L. Peng and J. Xiao, Root categories and simple Lie algebras, J. Algebra 198 (1997), 19–56. [26] L. Peng and J. Xiao, Triangulated categories and Kac–Moody algebras, Invent. Math. 140 (2000), 563–603. [27] C. Riedtmann, Lie algebras generated by indecomposables, J. Algebra 170 (1994), 526–546. [28] C. M. Ringel, Hall algebras and quantum groups, Invent. math. 101 (1990), 583–592. [29] C. M. Ringel, Hall algebras, in: Topics in Algebra, Part 1, S. Balcerzyk et al. (eds.), Banach Center Publications, no. 26, 1990, pp. 433–447. [30] C. M. Ringel, Hall polynomials for the representation-finite hereditary algebras, Adv. Math. 84 (1990), 137–178. [31] C. M. Ringel, Hall algebras revisited, in Quantum Deformations of Algebras and Their Repre- sentations, A. Joseph & S. Shnider (eds.), Israel Mathematical Conference Proceedings, no. 7, Bar-Ilan University, Bar-Ilan, 1993, pp. 171–176. CYCLIC COMPLEXES, HALL POLYNOMIALS AND SIMPLE LIE ALGEBRAS 25

[32] C. M. Ringel, The Hall algebra approach to quantum groups, Aportaciones MatemáticasComu- nicaciones 15 (1995), 85–114. [33] C. M. Ringel and P. Zhang, Representations of quivers over the algebra of dual numbers, preprint, 2012. [34] B. Sevenhant and M. Van den Bergh, A relation between a conjecture of Kac and the structure of the Hall algebra, J. Pure Appl. Algebra 160(2001), 319–332. [35] B. Töen, Derived Hall algebras, Duke Math. J. 135 (2006), 587–615. [36] C. A. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathe- matics 38, Cambridge U. Press, Cambridge, 1994. [37] J. Xiao, Hall algebra in a root category, SFB preprint 95-070, UniversitätBielefeld, 1995. [38] J. Xiao, Drinfeld double and Ringel–Green theory of Hall algebras, J. Algebra, 190 (1997), 100– 144. [39] J. Xiao and F. Xu, Hall algebras associated to triangulated categories, Duke Math. J. 143 (2008), 357–373. [40] J. Xiao, G. Zhang and B. Zhu, BGP reflection functors in root categories, Sci. China Ser. A Mathematics 48 (2005), 1033–1045.

School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China. (Current address: College of Mathematics and Computer Science, Fu Zhou university, Fu Zhou, Fujian 350108, China) E-mail address: [email protected]

Yau Mathematical Sciences Center, Tsinghua University, Beijing 100084, China. E-mail address: [email protected]