On weak crossed products, Frobenius algebras, and the weak Bruhat ordering.
type: Item
instance of: scholarly article
title: On weak crossed products, Frobenius algebras, and the weak Bruhat ordering.
zbMATH Open document ID: 1089.16019
author: A. Aljouiee
published in: Journal of Algebra
publication date: 2005-06-01T00:00:00Z
review text: Let \(K/F\) be a Galois field extension with Galois group \(G\). A map \(f\colon G\times G\to K\) is called a cosikel or weak cocycle if the \(K\)-algebra \(A_f=\bigoplus_{\sigma\in G}Ku_\sigma\), with multiplication rules \[ u_\sigma u_\tau=f(\sigma,\tau)u_{\sigma\tau};\qquad u_\sigma k=\sigma(k)u_\sigma \] is associative. Note that we do not require that \(f(\sigma,\tau)\neq 0\), as is the case with usual cocycles. In this note, it is studied when \(A_f\) is Frobenius or symmetric. A graph is associated to \(f\), and \(A_f\) is Frobenius if and only if this graph has a unique maximal element. In the case where \(f\) takes only values in \(\{0,1\}\) (that is, \(f\) is idempotent), we can consider the restricted subalgebra \(\overline A_f=\bigoplus_{\sigma\in G}Fu_\sigma\). The graph of \(f\) also allows to determine whether \(\overline A_f\) is Frobenius or symmetric.
Mathematics Subject Classification ID: 16S35
Mathematics Subject Classification ID: 16L60
Mathematics Subject Classification ID: 20F55
zbMATH DE Number: 2173130
zbMATH Keywords: weak cocycles
zbMATH Keywords: Frobenius algebras
zbMATH Keywords: weak crossed products
zbMATH Keywords: Nakayama automorphisms
zbMATH Keywords: Coxeter groups
zbMATH Keywords: symmetric algebras
reviewed by: Stefaan Caenepeel
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2004.02.001
OpenAlex ID: W1971332088
cites work: Q3724546
cites work: Q3922829
cites work: Schubert polynomials and the nilCoxeter algebra
cites work: On crossed product algebras arising from weak cocycles
cites work: A New Invariant for C Over R: Almost Invertible Cohomology Theory and the Classification of Idempotent Cohomology Classes and Algebras by Partially Ordered Sets with a Galois Group Action
cites work: Q3998721
cites work: On Frobenius algebras
cites work: Q4893217
DOI: 10.1016/J.JALGEBRA.2004.02.001
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description: scientific article; zbMATH DE number 2173130