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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Weighted representations of a primitive algebra

Author: E. G. Goodaire
Journal: Proc. Amer. Math. Soc. 42 (1974), 61-66
MSC: Primary 16A64
MathSciNet review: 0325696
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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.

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Keywords: Diagonable subspace, weighted representation
Article copyright: © Copyright 1974 American Mathematical Society

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