Lie semisimple algebras of derivations and varieties of PI-algebras with almost polynomial growth (Q6602151)

Let \(L\) be a finite dimensional semisimple Lie algebra over an algebraically closed base field of characteristic zero, acting by derivations on a finite dimensional associative algebra \(A\). The author proves that the codimensions of the differential identities of \(A\) have exponential growth if and only if all the differential identities of \(A\) are satisfied by the algebra of linear transformations of a non-trivial finite dimensional simple \(L\)-module, or by an algebra of \(2\times 2\) upper triangular matrices associated to a finite dimensional simple \(L\)-module. In the special case when \(L=\mathfrak{sl}_2\), this leads to a classification of \(L\)-varieties with almost polynomial growth of the codimensions of differential identities.