Differential codimensions and exponential growth

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