Hall algebras, hereditary algebras and quantum groups

Abstract

Let R be an associative, hereditary algebra over a finite field k, and let R-fin be the full subcategory of R-mod whose objects are those left R-modules X which are finite as sets, IX] < oc. Assume also that R isfinitary in C. Ringel's sense, i.e. that IExt l (s , s ' ) ] < c~ for all simple S,S' in R-fin; this condition is met, for example, if R is finitely generated as k-algebra [4, pp. 435,436]. Let ~ be the set of all isomorphism classes in R-fin. If 2 E ~ , then U~ will denote an R-module in class 2. The class of all zero left R-modules is denoted 0. Let I C_ ~ be the set of all isomorphism classes of simple modules in R-fin. Thus {U~ : i E I} is a complete set of simple, finite left R-modules. We identify the Grothendieck group K0(R-fin) with the free Abelian group E1 = {~ , v~i : v~ E 7Z} having I as free basis, so that if X E R-fin then the corresponding element of K0(R-fin) is the "dimension vector" dimX = Y~,~I vii, where for each i E /, vi is the multiplicity of the simple module U, in any composition series of X. Clearly d imX lies in the subsemigroup