On homogeneous integral table algebras
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 \(\mu\mid\lambda\). Finally, the last result of the paper is devoted to classify nilpotent HITAs of odd degree where the table basis contains a real faithful element. The paper contains many examples of HITAs.
scientific article; zbMATH DE number 1115987
scientific article