Identities in 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 $a^{\prime }=d(a)$. 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 $a_1^{\prime }{a}_2^{\prime }\cdots a_m^{\prime }=0$ for some positive integer $m$. We then study the subalgebras of the free differential perm algebra generated by $X$ under $⧫$ 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.

25 p