Classification of a class of integral table algebras (Q2731595)

A finite-dimensional commutative and associative algebra \(A\) with identity over \(\mathbb{C}\) is called a table algebra if it has a basis \(B=\{b_1=1,b_2,\dots,b_n\}\) such that: (i) For \(b_i,b_j\in B\), there exist \(\lambda_{ijm}\in\mathbb{R}^+\cup\{0\}\), called structure constants, such that \(b_ib_j=\sum^n_{m=1}\lambda_{ijm}b_m\). (ii) There exists an algebra automorphism -- of \(A\) whose order divides 2 such that \(b_i\in B\) implies \(\overline b_i\in B\) (\(\overline b_i=b_j\Leftrightarrow j=\overline i\)). (iii) For all \(i,j\) we have \(\lambda_{ij1}\neq 0\) if and only if \(j=\overline i\).NEWLINENEWLINENEWLINEIf \((A,B)\) is a table algebra, then by \textit{Z. Arad} and \textit{H. I. Blau} [J. Algebra 138, No. 1, 137-185 (1991; Zbl 0790.20015)], there is an algebra homomorphism \(f\colon A\to\mathbb{C}\) such that \(f(b_i)=\overline{f(b_i)}\in\mathbb{R}^+\) for all \(i\), \(1\leq i\leq n\), and \(f(b_i)\) are called the degrees of the table algebra \((A,B)\). If all the structure constants and the degrees of the table algebra are integers, then \((A,B)\) is called an integral table algebra. For \(a\in A\), if \(a=\sum^n_{m=1}\lambda_mb_m\), then the support of \(a\) is defined to be \(\text{supp}(a)=\{b_k\mid\lambda_k>0\}\) and \(b\in B\) is called a faithful basis element if \(B=\bigcup^\infty_{i=1}\text{supp}(b^i)\). The width of \(b\in B\) is defined to be the support of \(b\overline b\) and \(b\) is said to be real if \(b=\overline b\).NEWLINENEWLINENEWLINEIn the paper under review the author classifies all the integral table algebras \((A,B)\) such that the degree of each basis element is either \(1\) or \(\lambda\), where \(\lambda>2\), with the additional condition that there is a real faithful element \(b\in B\) of width \(2\). There are five classes of integral table algebras with the above properties and in each case the author determines the structure constants of the algebra. This paper is part of the author's Ph.D. thesis which was done in 1998 at the Tarbiat Moddares University, Tehran, Iran.