Configurations in abelian categories. II. Ringel–Hall algebras

This is the second in a series on configurations in an abelian category A. Given a finite poset (I,≼), an (I,≼)-configuration(σ,ι,π) is a finite collection of objects σ(J) and morphisms ι(J,K) or in A satisfying some axioms, where J,K⊆I. Configurations describe how an object X in A decomposes into subobjects.The first paper defined configurations and studied moduli spaces of (I,≼)-configurations in A, using the theory of Artin stacks. It showed well-behaved moduli stacks ObjA,M(I,≼)A of objects and configurations in A exist when A is the abelian category coh(P) of coherent sheaves on a projective scheme P, or mod-KQ of representations of a quiver Q.Write CF(ObjA) for the vector space of Q-valued constructible functions on the stack ObjA. Motivated by the idea of Ringel–Hall algebras, we define an associative multiplication ∗ on CF(ObjA) using pushforwards and pullbacks along 1-morphisms between configuration moduli stacks, so that CF(ObjA) is a Q-algebra. We also study representations of CF(ObjA), the Lie subalgebra CFind(ObjA) of functions supported on indecomposables, and other algebraic structures on CF(ObjA).Then we generalize all these ideas to stack functions, a universal generalization of constructible functions, containing more information. When Exti(X,Y)=0 for all X,Y∈A and i>1, or when A=coh(P) for P a Calabi–Yau 3-fold, we construct (Lie) algebra morphisms from stack algebras to explicit algebras, which will be important in the sequels on invariants counting τ-semistable objects in A.