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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Enveloping algebras of Lie color algebras: Primeness versus graded-primeness
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by Jeffrey Bergen and D. S. Passman PDF
Proc. Amer. Math. Soc. 126 (1998), 1627-1635 Request permission

Abstract:

Let $G$ be a finite abelian group and let $L$ be a, possibly restricted, $G$-graded Lie color algebra. Then the enveloping algebra $U(L)$ is also $G$-graded, and we consider the question of whether $U(L)$ being graded-prime implies that it is prime. The first section of this paper is devoted to the special case of Lie superalgebras over a field $K$ of characteristic $\neq 2$. Specifically, we show that if $i=\sqrt {-1}\in K$ and if $U(L)$ has a unique minimal graded-prime ideal, then this ideal is necessarily prime. As will be apparent, the latter result follows quickly from the existence of an anti-automorphism of $U(L)$ whose square is the automorphism of the enveloping algebra associated with its ${\mathbb {Z}}_{2}$-grading. The second section, which is independent of the first, studies more general Lie color algebras and shows that if $U(L)$ is graded-prime and if most homogeneous components $L_{g}$ of $L$ are infinite dimensional over $K$, then $U(L)$ is prime. Here we use $\Delta$-methods to study the grading on the extended centroid $C$ of $U(L)$. In particular, if $G$ is generated by the infinite support of $L$, then we prove that $C=C_{1}$ is homogeneous.
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Additional Information
  • Jeffrey Bergen
  • Affiliation: Department of Mathematics, DePaul University, Chicago, Illinois 60614
  • MR Author ID: 191461
  • Email: jbergen@condor.depaul.edu
  • D. S. Passman
  • Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
  • MR Author ID: 136635
  • Email: passman@math.wisc.edu
  • Received by editor(s): November 22, 1996
  • Additional Notes: The first author’s research was supported by the Faculty Research and Development Fund of the College of Liberal Arts & Sciences at DePaul University. The second author’s research was supported in part by NSF Grant DMS-9622566.
  • Communicated by: Ken Goodearl
  • © Copyright 1998 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 126 (1998), 1627-1635
  • MSC (1991): Primary 16S30, 16W55, 17B35
  • DOI: https://doi.org/10.1090/S0002-9939-98-04372-X
  • MathSciNet review: 1452792