Cyclic complexes, Hall polynomials and simple Lie algebras

In this paper we study the category Cm(P)Cm(P) of m  -cyclic complexes over PP, where PP is the category of projective modules over a finite dimensional hereditary algebra A  , and describe almost split sequences in Cm(P)Cm(P). This is applied to prove the existence of Hall polynomials in Cm(P)Cm(P) when A   is representation finite and m≠1m≠1. We further introduce the Hall algebra of Cm(P)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.