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

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## 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 freeassociative $K$-algebra in one further indeterminate $x.$ Considerthe set of polynomials in $R\langle x\rangle$ which are satisfied bythe $n\times n$ matrix $\alpha=(a_{ij}).$ Such polynomials arecalled 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 aconsequence" 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 asymmetric or skew-symmetric $2\times 2$ matrix under the transposeor symplectic involution and gives an explicit form of the Robsoncubics. Some other results are also given in case $n=3.$
Lingua originale English Published - 2008

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