\(*\)-polynomial identities of a nonsymmetric \(*\)-minimal algebra.
Historically polynomial identities with involution (or \(*\)-polynomial identities) are the first ``nonordinary'' polynomial identities to study, and they continue to be an object of intensive investigation. One of the main trends in the theory of \(*\)-polynomial identities is to follow the development of the theory of ordinary polynomial identities. The ``extremal'' algebras have the property that the \(T_*\)-ideals of their \(*\)-polynomial identities are essentially more complicated than the larger \(T_*\)-ideals. In particular, for finite dimensional algebras with involution in characteristic 0, the \(*\)-codimension sequence grows as in the ordinary case and its exponent is an integer. The related extremal algebras are subalgebras of block triangular matrices built on \(*\)-simple algebras, with some kind of symmetry of the blocks. Up to equivalence, there is a finite number of extremal algebras of a given exponent. The \(*\)-polynomial identities of the smallest algebra which is not so symmetric are the main subject of the paper under review. Namely, the authors consider the algebra \(UT_*(F\oplus F,F)\) over an infinite field \(F\) of characteristic different from 2. Here \(F\oplus F\) is equipped with the exchange involution \((a,b)^*=(b,a)\), \(a,b\in F\), and \(UT_*(F\oplus F,F)\) consists of \(4\times 4\) matrices of the form \[ \alpha e_{11}+ae_{12}+\beta(e_{22}+e_{33})+be_{34}+\gamma e_{44},\quad\alpha,\beta,\gamma,a,b\in F, \] with involution which is the flip along the ``other'' diagonal. The authors give a complete description of the polynomial identities of the algebra. They have found the basis of the identities (i.e., generators of the \(T_*\)-ideal) and a basis of the relatively free algebra. In characteristic 0, they describe also the \(GL\times GL\)-module structure of the relatively free algebra and compute the \(*\)-codimensions.
scientific article; zbMATH DE number 6306460