Configurations in abelian categories. II. Ringel-Hall algebras

Abstract

This is the second in a series math.AG/0312190, math.AG/0410267, math.AG/0410268 on configurations in an abelian category A. Given a finite partially ordered set (I,<), an (I,<)-configuration (\sigma,\iota,\pi) is a finite collection of objects \sigma(J) and morphisms \iota(J,K) or \pi(J,K) : \sigma(J) --> \sigma(K) in A satisfying some axioms, where J,K are subsets of I. Configurations describe how an object X in A decomposes into subobjects. The first paper math.AG/0312190 defined configurations and studied moduli spaces of (I,<)-configurations in A, using the theory of Artin stacks. It proved well-behaved moduli stacks Obj_A, M(I,<)_A of objects and configurations in A exist when A is the abelian category coh(P) of coherent sheaves on a projective K-scheme P, or mod-KQ of representations of a quiver Q. Write CF(Obj_A) for the vector space of constructible functions on Obj_A. Motivated by Ringel-Hall algebras, we define an associative multiplication * on CF(Obj_A) using pushforwords and pullbacks along 1-morphisms between the M(I,<)_A, making CF(Obj_A) into an algebra. We also study representations of CF(Obj_A), the Lie subalgebra CF^ind(Obj_A) of functions supported on indecomposables, and other algebraic structures on CF(Obj_A). Then we generalize these ideas to stack functions SF(Obj_A), a universal generalization of constructible functions on stacks introduced in math.AG/0509722, containing more information. Under extra conditions on A we can define (Lie) algebra morphisms from SF(Obj_A) to some explicit (Lie) algebras, which will be important in the sequels on invariants counting t-(semi)stable objects in A.