Some Hall polynomials for representation-finite trivial extension algebras

Let k be a finite field and assume that Lambda is a finite dimensional associative k-algebra with 1. Denote by mod Lambda the category of all finitely generated (right) Lambda-modules and by ind Lambda the full subcategory in which every object is a representative of the isoclass of an indecomposable (right) Lambda-module. We are interested in the existance of the Hall polynomial phi(NL)(M) for an L, M, N is an element of mod Lambda (for the definition, see [7, 8] or Section 1 below). In case Lambda is directed, C. M. Ringel in [7] has shown that Lambda has Hall polynomials, and in case Lambda is cyclic serial, the same result has also been obtained by J. Guo [4]. It has been conjectured in [8] that any representation-finite k-algebra has Hall polynomials. In this investigation, we shall show that if Lambda is a representation-finite trivial extension algebra, then, for any L, M, N is an element of mod Lambda with N indecomposable, Lambda has the Hall polynomials phi(NL)(M) and phi(NL)(M). Using these Hall polynomials, we can naturally structure the free abelian group with a basis ind Lambda, denoted by K(mod Lambda), into a Lie algebra and the universal enveloping algebra of K(mod Lambda) x(Z) Q is just H(Lambda)(1) x(Z) Q, where H(Lambda)(1) is the degenerated Hall algebra of Lambda (see Section 5 below). (C) 1997 Academic Press.