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We study the differential identities of the algebra Mₖ (F) of k k matrices over a field F of characteristic zero when its full Lie algebra of derivations, L=Der (Mₖ (F) ), acts on it. We determine a set of 2 generators of the ideal of differential identities of Mₖ (F) for k 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 Mₖ (F) has almost polynomial growth for all k 2.
Brox et al. (Thu,) studied this question.
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