Invariant ideals and polynomial forms
Author:
D. S. Passman
Journal:
Trans. Amer. Math. Soc. 354 (2002), 3379-3408
MSC (2000):
Primary 16S34; Secondary 20F50, 20G05
DOI:
https://doi.org/10.1090/S0002-9947-02-03006-4
Published electronically:
April 1, 2002
MathSciNet review:
1897404
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: Let denote the group algebra of an infinite locally finite group
. In recent years, the lattice of ideals of
has been extensively studied under the assumption that
is simple. From these many results, it appears that such group algebras tend to have very few ideals. While some work still remains to be done in the simple group case, we nevertheless move on to the next stage of this program by considering certain abelian-by-(quasi-simple) groups. Standard arguments reduce this problem to that of characterizing the ideals of an abelian group algebra
stable under the action of an appropriate automorphism group of
. Specifically, in this paper, we let
be a quasi-simple group of Lie type defined over an infinite locally finite field
, and we let
be a finite-dimensional vector space over a field
of the same characteristic
. If
acts nontrivially on
by way of the homomorphism
, and if
has no proper
-stable subgroups, then we show that the augmentation ideal
is the unique proper
-stable ideal of
when
. The proof of this result requires, among other things, that we study characteristic
division rings
, certain multiplicative subgroups
of
, and the action of
on the group algebra
, where
is the additive group
. In particular, properties of the quasi-simple group
come into play only in the final section of this paper.
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Additional Information
D. S. Passman
Affiliation:
Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
Email:
passman@math.wisc.edu
DOI:
https://doi.org/10.1090/S0002-9947-02-03006-4
Received by editor(s):
November 16, 2001
Published electronically:
April 1, 2002
Additional Notes:
Research supported in part by NSF Grant DMS-9820271.
Dedicated:
Dedicated to Idun Reiten on the occasion of her $60$th birthday
Article copyright:
© Copyright 2002
American Mathematical Society