# Differential identities and polynomial growth of the codimensions

## TL;DR

This paper characterizes the differential identities and polynomial growth of codimensions in finite-dimensional algebras with derivations, providing new insights into their structure and bounded multiplicities of differential cocharacters.

## Contribution

It offers a characterization of the differential identities and codimension growth in finite-dimensional $L$-algebras with derivations, especially when codimensions are polynomially bounded.

## Key findings

- Characterization of differential identities for polynomially bounded codimensions
- Description of $L$-algebras with bounded differential cocharacter multiplicities
- Establishment of conditions for polynomial growth of differential codimensions

## Abstract

Let $A$ be an associative algebra over a field $F$ of characteristic zero and let $L$ be a Lie algebra over $F$. If $L$ acts on $A$ by derivations, then such an action determines an action of its universal enveloping algebra $U(L)$ and in this case we refer to $A$ as algebra with derivations or $L$-algebra. Here we give a characterization of the ideal of differential identities of finite dimensional $L$-algebras $A$ in case the corresponding sequence of differential codimensions $c_n^L (A)$, $n\geq 1$, is polynomially bounded. As a consequence, we also characterize $L$-algebras with multiplicities of the differential cocharacter bounded by a constant.