Enveloping algebras of lie color algebras:

primeness versus graded-primeness

Authors:
Jeffrey Bergen and D. S. Passman

Journal:
Proc. Amer. Math. Soc. **126** (1998), 1627-1635

MSC (1991):
Primary 16S30, 16W55, 17B35

MathSciNet review:
1452792

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Abstract: Let be a finite abelian group and let be a, possibly restricted, -graded Lie color algebra. Then the enveloping algebra is also -graded, and we consider the question of whether 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 of characteristic . Specifically, we show that if and if 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 whose square is the automorphism of the enveloping algebra associated with its -grading. The second section, which is independent of the first, studies more general Lie color algebras and shows that if is graded-prime and if most homogeneous components of are infinite dimensional over , then is prime. Here we use -methods to study the grading on the extended centroid of . In particular, if is generated by the infinite support of , then we prove that is homogeneous.

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Additional Information

**Jeffrey Bergen**

Affiliation:
Department of Mathematics, DePaul University, Chicago, Illinois 60614

Email:
jbergen@condor.depaul.edu

**D. S. Passman**

Affiliation:
Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706

Email:
passman@math.wisc.edu

DOI:
https://doi.org/10.1090/S0002-9939-98-04372-X

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

Article copyright:
© Copyright 1998
American Mathematical Society