Derived identities of differential algebras

Abstract

Suppose $A$ is a not necessarily associative algebra with a derivation $d$. Then $A$ may be considered as a system with two binary operations $\succ $ and $\prec $ defined by $x\succ y = d(x)y$, $x\prec y = xd(y)$, $x,y\in A$. Suppose $A$ satisfies some multi-linear polynomial identities. We show how to find the identities that hold for operations $\prec $ and $\succ $. It turns out that if $A$ belongs to a variety governed by an operad Var then $\succ $ and $\prec $ satisfy the defining relations of the operad Var$\circ $Nov, where $\circ $ is the Manin white product of operads, Nov is the operad of Novikov algebras. Moreover, there are no other identities that hold for operations $\succ $, $\prec $ on an arbitrary differential Var-algebra.