Differential codimensions and exponential growth

Abstract

Let A be a finite dimensional associative algebra with derivations over a field of characteristic zero, i.e., an algebra whose structure is enriched by the action of a Lie algebra L by derivations, and let cnL(A), n≥1, be its differential codimension sequence. Such sequence is exponentially bounded and expL⁡(A)=limn→∞⁡cnL(A)n is an integer that can be computed, called differential PI-exponent of A.In this paper we prove that for any Lie algebra L, expL⁡(A) coincides with exp⁡(A), the ordinary PI-exponent of A. Furthermore, in case L is a solvable Lie algebra, we apply such result to classify varieties of L-algebras of almost polynomial growth, i.e., varieties of exponential growth such that any proper subvariety has polynomial growth.