Let
kbe a finite field and assume that Λ is a finite dimensional associative
k-al
gebra 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
ges/glyphs/CD4.GIF>
MNLfor an
L,
M,
Nges/glyphs/BOA.GIF>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-finite
k-al
gebra has Hall polynomials. In this investigation, we shall show that if Λ is a representation-finite trivial extension al
gebra, then, for any
L,
M,
Nges/glyphs/BOA.GIF>modΛ with
Nindecomposable, Λ has the Hall polynomials
ges/glyphs/CD4.GIF>
MLNand
ges/glyphs/CD4.GIF>
MNL. Using these Hall polynomials, we can naturally structure the free abelian group with a basis indΛ, denoted by
K(modΛ), into a Lie al
gebra and the universal enveloping al
gebra of
K(modΛ)
ges/glyphs/BSS.GIF>
ZQis just H(Λ)
1ges/glyphs/BSS.GIF>
ZQ, where H(Λ)
1is the de
generated Hall al
gebra of Λ (see Section 5 below).