Some Hall Polynomials for Representation-Finite Trivial Extension Algebras Original Research Article

Letkbe a finite field and assume that Λ is a finite dimensional associativek-algebra with 1. Denote by mod Λ the category of all finitely generated (right) Λ-modules and by ind Λ 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 phiMNLfor anL,M,N set membership, variant mod Λ (for the definition, see[7 and 8]or Section 1 below). In case Λ is directed, [7]has shown that Λ has Hall polynomials, and in case Λ is cyclic serial, the same result has also been obtained by [4]. It has been conjectured in [8]that 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 set membership, variant mod Λ withNindecomposable, Λ has the Hall polynomials phiMLNand phiMNL. Using these Hall polynomials, we can naturally structure the free abelian group with a basis ind Λ, denoted byK(mod Λ), into a Lie algebra and the universal enveloping algebra ofK(mod Λ) circle times operator Z Qis just image(Λ)1 circle times operator Z Q, where image(Λ)1is the degenerated Hall algebra of Λ (see Section 5 below).