Some Hall Polynomials for Representation-Finite Trivial Extension Algebras
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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 
MNLfor anL,M,N
modΛ (for the definition, seeor Section 1 below). In case Λ is directed,has shown that Λ has Hall polynomials, and in case Λ is cyclic serial, the same result has also been obtained by. It has been conjectured inthat 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,NmodΛ withNindecomposable, Λ has the Hall polynomials MLNand MNL. 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)1ZQ, where H(Λ)1is the degenerated Hall algebra of Λ (see Section 5 below).
MNLfor anL,M,NmodΛ (for the definition, seeor Section 1 below). In case Λ is directed,has shown that Λ has Hall polynomials, and in case Λ is cyclic serial, the same result has also been obtained by. It has been conjectured inthat 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,NmodΛ withNindecomposable, Λ has the Hall polynomials MLNand MNL. 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)1ZQ, where H(Λ)1is the degenerated Hall algebra of Λ (see Section 5 below).