Remote Access Transactions of the American Mathematical Society
Green Open Access

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
MathSciNet review: 1076611
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [BP] J. Bergen and D. S. Passman, Delta methods in enveloping rings, J. Algebra 133 (1990), 277-312. MR 1067408 (92b:16056)
  • [B] E. J. Behr, Enveloping algebras of Lie superalgebras, Pacific J. Math 130 (1987), 9-25. MR 910651 (89b:17023)
  • [S] M. Scheunert, The theory of Lie superalgebras, Lecture Notes in Math., vol. 716, Springer-Verlag, Berlin, 1979. MR 537441 (80i:17005)
  • [P] V. M. Petrogradskij, Identities in the enveloping algebra for modular Lie superalgebras, J. Algebra 145 (1992), 1-21. MR 1144655 (92m:17013)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 17B35, 17A70, 17B70

Retrieve articles in all journals with MSC: 17B35, 17A70, 17B70

Additional Information

Article copyright: © Copyright 1992 American Mathematical Society

American Mathematical Society