The isomorphism of Hall algebras (Q1330076)

This article is a valuable contribution to the theory of Hall algebras, introduced and first studied by \textit{C. M. Ringel} [see: Topics in Algebra, Pt. 1, Banach Cent. Publ. 26, Part I, 433-447 (1990; Zbl 0778.16004), Adv. Math. 84, 137-178 (1990; Zbl 0799.16013) and Invent. Math. 101, 583-591 (1990; Zbl 0735.16009)]. The Hall algebra, \({\mathcal H}(R)\) of a finitary ring \(R\) has a basis formed by isoclasses of finite modules and a multiplication that, in a sense, reflects the number of modules with given two-story filtrations. The composition algebra of \(R\) is the subalgebra \({\mathcal C}(R)\) determined by finite simple \(R\)-modules. The aim of the paper is to give necessary/sufficient conditions for the isomorphism of Hall or composition algebras associated to rings \(R\), \(R'\). One main result is that, for representation directed algebras over finite fields, \(R\), \(R'\), \({\mathcal H}(R)\cong{\mathcal H}(R')\) if and only if there exists a bijection between the isoclasses of simple modules that preserves endomorphism rings and the first and second extension groups of each pair of simples.