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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Delta methods in enveloping algebras of Lie superalgebras


Authors: Jeffrey Bergen and D. S. Passman
Journal: Trans. Amer. Math. Soc. 334 (1992), 259-280
MSC: Primary 17B35; Secondary 17A70, 17B70
DOI: https://doi.org/10.1090/S0002-9947-1992-1076611-X
MathSciNet review: 1076611
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Abstract: Let $ L$ be a Lie superalgebra over a field $ K$ of characteristic $ \ne 2$ . We define

$\displaystyle \Delta (L) = \{ l \in L\vert{\dim _K}[L,l] < \infty \}. $

Then $ \Delta (L)$ is a Lie ideal of $ L$ and is restricted if $ L$ is restricted. $ \Delta (L)$ is the Lie superalgebra analog of the Lie delta ideal, used by the authors in the study of enveloping rings, and also of the finite conjugate center of a group, used in the study of group algebras and crossed products.

In this paper we examine $ U(L)$, where depending upon $ \operatorname{char}K$, $ U(L)$ denotes either the enveloping algebra or the restricted enveloping algebra of $ L$. We show that $ \Delta (L)$ controls certain properties of $ U(L)$. Specifically, we consider semiprimeness, primeness, almost constants, almost centralizers, central closures, and the Artinian condition.


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DOI: https://doi.org/10.1090/S0002-9947-1992-1076611-X
Article copyright: © Copyright 1992 American Mathematical Society