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