Invariant ideals of Abelian group algebras under the torus action of a field. I.
type: Item
instance of: scholarly article
title: Invariant ideals of Abelian group algebras under the torus action of a field. I.
zbMATH Open document ID: 1217.16020
published in: Journal of Algebra
publication date: 2011-02-18T00:00:00Z
review text: Let \(K\) be a field, \(V\) an Abelian group, \(G\) a group of automorphisms of \(V\). Then \(G\) acts on the group algebra \(K[V]\) and it is an interesting problem to determine all \(G\)-stable ideals of this algebra. In the papers [\textit{C. J. B. Brookes, D. M. Evans}, Math. Proc. Camb. Philos. Soc. 130, No. 2, 287-294 (2001; Zbl 1005.20005), \textit{D. S. Passman, A. E. Zalesskiĭ}, Proc. Am. Math. Soc. 130, No. 4, 939-949 (2002; Zbl 0992.16021) and \textit{J. M. Osterburg, D. S. Passman, A. E. Zalesskiĭ}, Proc. Am. Math. Soc. 130, No. 4, 951-957 (2002; Zbl 0992.16022)] the special case, when \(V\) is a vector space over an infinite field and \(G\) is the multiplicative group of scalars, was settled. Let \(V=V_1\oplus V_2\) be a finite-dimensional vector space over an infinite locally-finite field \(F\). Then \(V\) admits the torus action of \(G=F^\bullet\) by defining \((v_1\oplus v_2)^g=v_1g^{-1}\oplus v_2g\). If \(K\) is a field of characteristic different from that of \(F\), then \(G\) acts on the group algebra \(K[V]\). In this paper, the case when \(V_1\) and \(V_2\) are both 1-dimensional is examined and the author shows that there are just four \(G\)-stable proper ideals of \(K[V]\), namely, with \(A\) a subgroup of \(V\), the augmentation ideal \(\omega(A,V)\) of \(A\), the kernel of the natural epimorphism \(K[V]\to K[V/A]\), these are \(\omega(V,V)\), \(\omega(V_1,V)\), \(\omega(V_2,V)\) and \(\omega(V_1,V)\cap\omega(V_2,V)\). Furthermore, it is announced that, applying the same techniques, the analogous problem for the torus action of the special linear group on the coordinate space of dimension \(n\) can be handled analogously, which will be the subject of a forthcoming paper.
reviewed by: János Kurdics
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 16D25
Mathematics Subject Classification ID: 20C07
zbMATH DE Number: 5853359
zbMATH Keywords: group algebras
zbMATH Keywords: group actions
zbMATH Keywords: torus actions
zbMATH Keywords: invariant ideals
author: D. S. Passman
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2010.01.023
OpenAlex ID: W2085088843
cites work: Augmentation modules for affine groups
cites work: Q3996998
cites work: Inversion invariant additive subgroups of division rings
cites work: Invariant ideals of Abelian group algebras under the multiplicative action of a field. II
cites work: Observations on group rings
cites work: Q4448997
cites work: Invariant ideals of Abelian group algebras under the multiplicative action of a field. I
DOI: 10.1016/J.JALGEBRA.2010.01.023
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description: scientific article; zbMATH DE number 5853359