Quotient structures in \(C\)-algebras
type: Item
instance of: scholarly article
title: Quotient structures in \(C\)-algebras
zbMATH Open document ID: 0895.20005
author: Harvey I. Blau
published in: Journal of Algebra
publication date: 1995-08-29T00:00:00Z
review text: A \(C\)-algebra is a commutative algebra \(A\) over the complex numbers with a distinguished basis \({\mathbf B}\) which has certain specified properties. In particular, the structure constants are real. The definition of a \(C\)-algebra was first stated explicitly by Kawada, following Hoheisel, although the notion is implicit in previous work of Schur. Kawada's goal was to abstract the relationship between the center of the group algebra and the ring of class functions for a finite group. Thus, for any \(C\)-algebra \((A,{\mathbf B})\), he constructed a dual \(C\)-algebra \((A,\widehat{\mathbf B})\), so that \(\widehat{\widehat{\mathbf B}}={\mathbf B}\), and so that when \(A\) is the center of a group algebra with basis \(\mathbf B\) consisting of the sums over conjugacy classes, then \((A,\widehat{\mathbf B})\) may be identified with the algebra of class functions with basis the set of irreducible characters of the group. The central aim of this paper is to clarify the relationship between substructures and quotient structures in \(C\)-algebras. Theorem 1 gives an explicit bijection between the quotient structures (resp. substructures) of an arbitrary \(C\)-algebra and the substructures (resp. quotient structures) of its dual. This extends a result of \textit{Z. Arad} and \textit{E. Fisman} [Commun. Algebra 19, No. 11, 2955-3009 (1991; Zbl 0790.20016), Theorem 2.9]. Theorem 2 then shows that one need assume only that \((A,{\mathbf B})\) (and not necessarily also its dual) is a table algebra in order to guarantee that any substructure of \((A,{\mathbf B})\) yields a quotient structure of \((A,{\mathbf B})\), and conversely. Then the Jordan-Hölder type theorem proved by Rao, Ray-Chaudhuri, and Singhi under Kawada's assumptions [\textit{E. Bannai}, \textit{T. Ito}, Algebraic combinatorics I, Theorem II.9.11]\ holds for all table algebras (Theorem 5), as does an analog (Theorem 6) of the Krull-Schmidt type theorem proved for commutative association schemes by \textit{…
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 05E30
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 19A22
zbMATH DE Number: 783910
zbMATH Keywords: \(C\)-algebras
zbMATH Keywords: structure constants
zbMATH Keywords: center
zbMATH Keywords: group algebras
zbMATH Keywords: rings of class functions
zbMATH Keywords: finite groups
zbMATH Keywords: irreducible characters
zbMATH Keywords: substructures
zbMATH Keywords: quotient structures
zbMATH Keywords: Jordan-Hölder type theorem
zbMATH Keywords: table algebras
zbMATH Keywords: Krull-Schmidt type theorem
zbMATH Keywords: commutative association schemes
MaRDI profile type: Publication
full work available at URL: jabr.1995.1174
OpenAlex ID: W2026315950
DOI: 10.1006/JABR.1995.1174
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description: scientific article; zbMATH DE number 783910