Rashkova, T. The Robson cubics for matrix algebras with involution (Acta Univ. Apulensis Math. Inform.).

Abstract

Let R be the free associative algebra over a field K on $n^2$ generators $a_{ij}$ and let $R\langle x\rangle$ be the free associative $K$-algebra in one further indeterminate $x.$ Consider the set of polynomials in $R\langle x\rangle$ which are satisfied by the $n\times n$ matrix $\alpha=(a_{ij}).$ Such polynomials are called laws over $R$ of the matrix $\alpha.$ Robson in [Robson, J. C. Polynomials satisfied by matrices. J. Algebra 55 (1978), no. 2, 509--520; MR523471 (80j:15012)] proved that such laws are a ``consequence" of a finite set of laws and for $n=2$ he exhibited $4$ generators called Robson cubics. Here the author considers the special case when $\alpha$ is a symmetric or skew-symmetric $2\times 2$ matrix under the transpose or symplectic involution and gives an explicit form of the Robson cubics. Some other results are also given in case $n=3.$