Some Hall Polynomials for Representation-Finite Trivial Extension Algebras

Abstract

Letkbe a finite field and assume that Λ is a finite dimensional associativek-algebra with 1. Denote by modΛ the category of all finitely generated (right) Λ-modules and by indΛ the full subcategory in which every object is a representative of the isoclass of an indecomposable (right) Λ-module. We are interested in the existance of the Hall polynomial ϕMNLfor anL,M,N∈modΛ (for the definition, see977113or Section 1 below). In case Λ is directed,7has shown that Λ has Hall polynomials, and in case Λ is cyclic serial, the same result has also been obtained by4. It has been conjectured in8that any representation-finitek-algebra has Hall polynomials. In this investigation, we shall show that if Λ is a representation-finite trivial extension algebra, then, for anyL,M,N∈modΛ withNindecomposable, Λ has the Hall polynomials ϕMLNand ϕMNL. Using these Hall polynomials, we can naturally structure the free abelian group with a basis indΛ, denoted byK(modΛ), into a Lie algebra and the universal enveloping algebra ofK(modΛ)⊗ZQis just H(Λ)1⊗ZQ, where H(Λ)1is the degenerated Hall algebra of Λ (see Section 5 below).