arXiv: Rings and Algebras | 2019
Cocharacters for the weak polynomial identities of the Lie algebra of $3\\times 3$ skew-symmetric matrices.
Abstract
Let $so_3(K)$ be the Lie algebra of $3\\times 3$ skew-symmetric matrices over a field $K$ of characteristic 0. The ideal $I(M_3(K),so_3(K))$ of the weak polynomial identities of the pair $(M_3(K),so_3(K))$ consists of the elements $f(x_1,\\ldots,x_n)$ of the free associative algebra $K\\langle X\\rangle$ with the property that $f(a_1,\\ldots,a_n)=0$ in the algebra $M_3(K)$ of all $3\\times 3$ matrices for all $a_1,\\ldots,a_n\\in so_3(K)$. The generators of $I(M_3(K),so_3(K))$ were found by Razmyslov in the 1980 s. In this paper the cocharacter sequence of $I(M_3(K),so_3(K))$ is computed. In other words, the ${\\mathrm{GL}}_p(K)$-module structure of the algebra generated by $p$ generic skew-symmetric matrices is determined. Moreover, the same is done for the closely related algebra of $\\mathrm{SO}_3(K)$-equivariant polynomial maps from the space of $p$-tuples of $3\\times 3$ skew-symmetric matrices into $M_3(K)$ (endowed with the conjugation action). In the special case $p=3$ the latter algebra is a module over a $6$-variable polynomial subring in the algebra of $\\mathrm{SO}_3(K)$-invariants of triples of $3\\times 3$ skew-symmetric matrices, and a free resolution of this module is found. The proofs involve methods and results of classical invariant theory, representation theory of the general linear group and explicit computations with matrices.