On Z-graded associative algebras and their N-graded modules

Abstract

Let $A$ be a $Z$-graded associative algebra and let $\rho$ be an irreducible $N$-graded representation of $A$ on $W$ with finite-dimensional homogeneous subspaces. Then it is proved that $\rho (\tilde{A})=g{l}_J(W)$, where $\tilde{A}$ is the completion of $A$ with respect to a certain topology and $g{l}_J(W)$ is the subalgebra of $\End W$, generated by homogeneous endomorphisms. It is also proved that an $N$-graded vector space $W$ with finite-dimensional homogeneous spaces is the only continuous irreducible $N$-graded $g{l}_J(W)$-module up to equivalence, where $g{l}_J(W)$ is considered as a topological algebra in a certain natural way, and that any continuous $N$-graded $g{l}_J(W)$-module is a direct sum of some copies of $W$. A duality for certain subalgebras of $g{l}_J(W)$ is also obtained.

AMS-LaTex 1.2, 17 pp, to appear in the Proceedings of the Conference at NCSU, Raleigh, May 1998