On homogeneous integral table algebras
type: Item
instance of: scholarly article
title: On homogeneous integral table algebras
zbMATH Open document ID: 0948.20004
author: Harvey I. Blau
author: Bangteng Xu
published in: Journal of Algebra
publication date: 2000-11-20T00:00:00Z
review text: Let \(A\) be a finite-dimensional associative and commutative algebra over the complex field \(\mathbb{C}\) with a basis \(B=\{b_1=1,b_2,\dots,b_k\}\) such that: (1) For all \(i,j,m\), \(b_ib_j=\sum^k_{m=1}\beta_{ijm}b_m\) with \(\beta_{ijm}\in\mathbb{R}^+\cup\{0\}\). (2) There is an algebra automorphism denoted by \(-\) of order 1 or 2 with the property that if \(b_i\in B\), then \(\overline b_i\in B\). In this case \(\overline i\) is defined by \(b_{\overline i}=\overline b_i\) and \(b_i\) is called real if \(\overline b_i=b_i\). (3) For all \(i,j\), \(\beta_{ij1}\neq 0\Leftrightarrow j=\overline i\). Then \((A,B)\) is called a table algebra and \(B\) a table basis. By \textit{Z. Arad} and \textit{H. I. Blau} [J. Algebra 138, No. 1, 137-185 (1991; Zbl 0790.20015)] if \((A,B)\) is a table algebra then there is an algebra homomorphism \(f\colon A\to\mathbb{C}\) such that \(f(b_i)=f(\overline b_i)\in R^+\) for all \(i\). The positive real numbers \(f(b_i)\), \(1\leq i\leq k\), are called the degrees of \((A,B)\). A table algebra \((A,B)\) is called integral (ITA) if all the structure constants \(\beta_{ijm}\) and all the degrees \(f(b_i)\) are rational integers, and an ITA is called homogeneous (HITA) if \(f(b)=\lambda\) for all \(b\in B\setminus\{1\}\), for some \(\lambda\in\mathbb{R}^+\). In the paper under review many concepts concerning table algebras such as rescaling simplicity, isomorphism and exact isomorphism, upper central series and nilpotency are defined which we do not mention here. In Theorem 1 of the paper the authors show that any ITA has a rescaling which is also integral, and which is homogeneous of some positive integer degree. The authors in general classify all the HITAs of degree more than 2 in which the table basis contains a standard faithful element of width 2. In Theorem 4 of the paper the authors prove that if \((A,B)\) is a nilpotent HITA of degree \(\lambda\) and if \(b\in B\) has length \((b,b)=\lambda\mu\), then \(\mu\mid\lambda\). Fina…
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 16P10
Mathematics Subject Classification ID: 05E30
zbMATH DE Number: 1115987
zbMATH Keywords: homogeneous integral table algebras
zbMATH Keywords: degrees
zbMATH Keywords: upper central series
zbMATH Keywords: nilpotency
zbMATH Keywords: rescaling simplicity
zbMATH Keywords: table bases
zbMATH Keywords: faithful elements
reviewed by: M. R. Darafsheh
MaRDI profile type: Publication
full work available at URL: jabr.1997.7192
OpenAlex ID: W2093384243
cites work: On table algebras and applications to finite group theory
cites work: Real table algebras and applications to finite groups of extended Camina-Frobenius type
cites work: Homogeneous integral table algebras of degree three: a trilogy
cites work: Q3218140
cites work: Quotient structures in \(C\)-algebras
cites work: Integral table algebras, affine diagrams, and the analysis of degree two
cites work: Q4374879
cites work: On a class of integral table algebras
DOI: 10.1006/JABR.1997.7192
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description: scientific article; zbMATH DE number 1115987