Weighted representations of a primitive algebra

Author:
E. G. Goodaire

Journal:
Proc. Amer. Math. Soc. **42** (1974), 61-66

MSC:
Primary 16A64

DOI:
https://doi.org/10.1090/S0002-9939-1974-0325696-9

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 for are simultaneously diagonalizable. Denote the centralizer of *L* in *A* by . A module *V* over *A* or is *L*-weighted if for some nonzero and map for each , and *x*-weighted if for some nonzero and positive integer *n*, . In this paper we give conditions under which the following statements are equivalent:

1. All irreducible modules over *A* and are *L*-weighted.

2. For each , some irreducible *A*-module is *x*-weighted and *x* is algebraic over *F*.

**[1]**E. G. Goodaire,*Irreducible representations of algebras*, Canad. J. Math. (to appear). MR**0349763 (50:2256)****[2]**F. W. Lemire,*Weight spaces and irreducible representations of simple Lie algebras*, Proc. Amer. Math. Soc.**22**(1969), 192-197. MR**39**#4326. MR**0243001 (39:4326)****[3]**-,*Existence of weight space decompositions for irreducible representations of simple Lie algebras*, Canad. Math. Bull.**14**(1971), 113-115. MR**45**#328. MR**0291234 (45:328)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1974-0325696-9

Keywords:
Diagonable subspace,
weighted representation

Article copyright:
© Copyright 1974
American Mathematical Society