Identities in differential perm algebras

TL;DR

This paper explores polynomial identities in differential perm algebras, revealing that nontrivial identities imply differential consequences and constructing related Lie and Leibniz algebras.

Contribution

It characterizes identities in differential perm algebras, describes subalgebras generated by specific operations, and introduces perm-Witt type Lie and Leibniz algebras.

Findings

Nontrivial identities imply differential consequences of the form $a_1'a_2'\cdots a_m'=0$.

Explicit generators and dimensions of subalgebras under specific operations are provided.

Construction of perm-Witt type Lie and Leibniz algebras from differential perm algebras.

Abstract

Let $(P,\cdot,d)$ be a differential perm algebra over a field of characteristic $0$, i.e. an associative algebra satisfying $(ab)c=(ba)c$ equipped with a derivation $d$. We investigate polynomial identities in the algebras obtained from $d$ by the derived operations \[ a\prec b=ab',\quad a\succ b=a'b,\quad a\blacklozenge b=ab'+ba',\quad a\bullet b=a'b+ab',\quad a\Diamond b=ab'-ba',\quad a\circ b=a'b-ab', \] where $a'=d(a)$. Our first result shows that any nontrivial differential polynomial identity (not supported by the right annihilator forced by the perm law) implies a purely differential consequence of the form $a_1'a_2'\cdots a_m'=0$ for some positive integer $m$. We then study the subalgebras of the free differential perm algebra generated by $X$ under $\blacklozenge$ and under $\bullet$, giving explicit generating sets and computing the multilinear dimensions of their homogeneous components. Finally, we construct perm-Witt type Lie and Leibniz algebras arising naturally from differential perm algebras.