Colength of varieties of linear algebras
Let \textbf{V} be a variety of (not necessarily associative) algebras over a field \(\Phi\) of characteristic 0. The \(n\)-th cocharacter \(\chi_{n}({\mathbf V})\), \(n=1,2,\ldots\), is defined as in the case of associative PI-algebras, as the character of the symmetric group \(S_{n}\) acting on the multilinear polynomials of the absolutely free algebra modulo the polynomial identities of \textbf{V}. The colength \(l_{n}({\mathbf V})\) is defined as the sum \(\sum_{\lambda\vdash n}m_{\lambda}\) of the multiplicities \(m_{\lambda}\) in the decomposition \(\chi_{n}({\mathbf V})=\sum_{\lambda\vdash n}m_{\lambda}\chi_{\lambda}\), where \(\chi_{\lambda}\) is the irreducible \(S_{n}\)-character related to the partition \(\lambda\) of \(n\). In the case of associative PI-algebras, the colength \(l_{n}({\mathbf V})\) is polynomially bounded. In the paper under review the authors construct a family of nonassociative algebras which satisfy the fourth Capelli identity and generate varieties \textbf{V} with exponentially growing colength. The construction of the algebras is quite interesting and it seems that it may be used (or at least modified) to build other examples in the theory of nonassociative algebras with polynomial identity.
scientific article; zbMATH DE number 5155477