Weighted representations of a primitive algebra

Abstract:

Let L be a diagonable subspace of an associative algebra A with identity over a field F; that is, L is spanned by a set of pairwise commuting elements, and the linear transformations ad $x:a \mapsto ax - xa$ for $x \in L$ are simultaneously diagonalizable. Denote the centralizer of L in A by $\mathcal {C}$. A module V over A or $\mathcal {C}$ is L-weighted if for some nonzero $v \in V$ and map $\lambda :L \to F,v{(x - \lambda (x)1)^{n(x)}} = 0$ for each $x \in L$, and x-weighted if for some nonzero $v \in V,\lambda \in F$ and positive integer n, $v{(x - \lambda 1)^n} = 0$. In this paper we give conditions under which the following statements are equivalent: 1. All irreducible modules over A and $\mathcal {C}$ are L-weighted. 2. For each $x \in L$, some irreducible A-module is x-weighted and x is algebraic over F.