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 ba- sis 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 correspond- ing 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 complex- es 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 lo- calization 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 dimen- sional 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 2 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 = i2I 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 2 ZI, we write P i2I jaj = i2I ai. Thus, jdim Mj 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 2 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 2 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;j2I by setting ci;i = 2 for all i 2 I. It is easy to see that CA is a symmetrizable generalized Cartan matrix with symmetrization matrix D = diag(di : i 2 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 fαi j i 2 Ig. 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 de- scribe 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 )i2Z, 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 2 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 2 Z and, thus, a functor [s]: C (A ) ! C (A ) taking M 7! M [s]. For each m > 1, write Zm = Z=mZ = f0; 1; : : : ; m − 1g. By definition, an m-cyclic • i i i complex M = (M ; d )i2Zm over A consists of objects M in A and morphisms i i i+1 i+1 i d : M ! M for i 2 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 2 Zm.