Differential identities and varieties of almost polynomial growth

Abstract

Let V be an L-variety of associative L-algebras, i.e., algebras where a Lie algebra L acts on them by derivations, and let c L n (V), n ≥ 1, be its L-codimension sequence. If V is generated by a finite dimensional L-algebra, then such a sequence is polynomially bounded if and only if V does not contain U T 2 , the 2 × 2 upper triangular matrix algebra with trivial L-action, and U T ε 2 where L acts on U T 2 as the 1-dimensional Lie algebra spanned by the inner derivation ε induced by e 11 . In this paper we completely classify all the L-subvarieties of var L (U T 2 ) and var L (U T ε2 ) by giving a complete list of finite dimensional L-algebras generating them.