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 $n=3.$
La Mattina, D. (2008). Rashkova, T. The Robson cubics for matrix algebras with involution (Acta Univ. Apulensis Math. Inform.)..
Rashkova, T. The Robson cubics for matrix algebras with involution (Acta Univ. Apulensis Math. Inform.).
LA MATTINA, Daniela
2008-01-01
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 $n=3.$I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.