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.

References in corpus (2)

• Splitting of operations, Manin products and Rota-Baxter operators
• On embedding of dendriform algebras into Rota---Baxter algebras