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Cyclic Complexes, Hall Polynomials and Simple Lie Algebras

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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 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.
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