Lie semisimple algebras of derivations and varieties of pi-algebras with almost polynomial growth

We consider associative algebras with an action by derivations by some finite dimensional and semisimple Lie algebra. We prove that if a differential variety has almost polynomial growth, then it is generated by one of the algebras U T 2 ( W λ ) UT_2(W_λ ) or E n d ( W μ ) End(W_μ ) for some integral dominant weight λ , μ λ ,μ with μ ≠ 0 μ ≠ 0 . In the special case L = s l 2 L=sl_2 we prove that this is a sufficient condition too.