Differential codimensions and exponential growth

TL;DR

This paper proves that the differential PI-exponent of a finite dimensional associative algebra with derivations equals its ordinary PI-exponent, and applies this to classify certain varieties of L-algebras with near-polynomial growth.

Contribution

It establishes the equality of differential and ordinary PI-exponents for any Lie algebra of derivations, and classifies varieties with almost polynomial growth when the Lie algebra is solvable.

Findings

Differential PI-exponent equals the ordinary PI-exponent for any Lie algebra of derivations.

The result applies to classify varieties of L-algebras with exponential growth and polynomial subvarieties.

The differential codimension sequence is exponentially bounded with an integer exponential growth rate.

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 $c_n^L(A),$ $n\geq 1,$ be its differential codimension sequence. Such sequence is exponentially bounded and $\exp^L(A) = \lim_{n\to \infty}\sqrt[n]{c_n^L(A)}$ is an integer that can be computed, called differential PI-exponent of $A$. In this paper we prove that for any Lie algebra $L$, $\exp^L(A)$ 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.