On deformations of crossed products.
type: Item
instance of: scholarly article
title: On deformations of crossed products.
zbMATH Open document ID: 1124.16021
author: Eli Aljadeff
author: Yuval Ginosar
author: Andy R. Magid
published in: Journal of Pure and Applied Algebra
publication date: 2007-04-26T00:00:00Z
review text: The subject of the paper under review is the crossed product algebra \(A*\Gamma\), where \(A\) is semisimple, finitely generated over its center, and \(\Gamma\) is a finite group. The authors give a necessary and sufficient condition in terms of the outer action of \(\Gamma\) on \(A\) for the existence of a multi-parametric semisimple deformation on the form \(A(\!(t_1,\dots,t_n)\!)*\Gamma\) (with the induced outer action). Furthermore, suppose that the base ring \(A\) is semisimple Artinian. Recall that a crossed product \(A*\Gamma\) induces an action of \(\Gamma\) on the set of simple components \(\Lambda=\{A_i\}_{i\in I}\) of \(A\). Let \(\{A_{i_j}\}_{j\in J}\) be a set of representatives for the orbits in \(\Lambda\) and let \(\Gamma_{i_j}\) be the stabilizer of \(A_{i_j}\) in \(\Gamma\). Then \(A*\Gamma\cong\prod_{j\in J}M_{n_j}(A_{i_j}*\Gamma_{i_j})\) with \(n_j=[\Gamma:\Gamma_{i_j}]\). Let \(H_{i_j}=\ker(\alpha_{i_j}\colon\Gamma_{i_j}\to\text{Out}(A_{i_j}))\) and let \(P_{i_j}\) be \(p_i\)-Sylow subgroups of \(H_{i_j}\) (in case \(\text{char}(Z(A_{i_j}))=p_i>0\)). The following conditions play an important role in the further investigation. (A) \(P_{i_j}\) is Abelian with normal complement in \(H_{i_j}\); (B) \(\text{rank}(P_{i_j})\leq p_i\)-degree of \(Z(A_{i_j})\). The following results are obtained. Theorem 1.6. Let \(R=A*\Gamma\) be a crossed product, where \(\Gamma\) is finite and \(A\) is semisimple finitely generated over its center. Assume (with the above notation) that each \(P_{i_j}\) is Abelian and has a normal complement in \(H_{i_j}\). Then there is a semisimple \(n\)-parameter homogeneous deformation \(R'=A(\!(t_1,\dots,t_n))*\Gamma\) for some integer \(n\geq 0\). Theorem 1.7. Let \(p(i)\) be the \(p_i\)-degree of \(Z(A_{i_j})\), then the minimal number of parameters is given by \(n_0=\max_{i\in I}\{1,\text{rank}(P_{i_j})-p(i)\}\). The authors also consider the so called twisting problem, which can be formulated as follows: Let \(A_\alpha^{f_0}*…
Mathematics Subject Classification ID: 16S35
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 16D60
Mathematics Subject Classification ID: 16K20
zbMATH DE Number: 5146861
zbMATH Keywords: crossed products
zbMATH Keywords: finite groups
zbMATH Keywords: multi-parameter deformations
zbMATH Keywords: twisting problem
reviewed by: Nako Nachev
MaRDI profile type: Publication
full work available at URL: j.jpaa.2006.08.015
OpenAlex ID: W2164255608
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DOI: 10.1016/J.JPAA.2006.08.015
description: scientific article; zbMATH DE number 5146861