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arXiv · 2403.09337

Differential identities of matrix algebras

Jose Brox·Carla Rizzo
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Abstract

We study the differential identities of the algebra $M_k(F)$ of $k\times k$ matrices over a field $F$ of characteristic zero when its full Lie algebra of derivations, $L=\mbox{Der}(M_k(F))$, acts on it. We determine a set of 2 generators of the ideal of differential identities of $M_k(F)$ for $k\geq 2$. Moreover, we obtain the exact values of the corresponding differential codimensions and differential cocharacters. Finally we prove that, unlike the ordinary case, the variety of differential algebras with $L$-action generated by $M_k(F)$ has almost polynomial growth for all $k\geq 2$.

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Jose Brox, Carla Rizzo. 2024-03-14. Differential identities of matrix algebras. https://arxiv.org/abs/2403.09337

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