# Derived identities of differential algebras

## TL;DR

This paper explores how identities in differential algebras translate into identities for associated binary operations, revealing a connection with operad compositions and Novikov algebras.

## Contribution

It establishes a method to derive identities for operations defined via derivations in differential algebras, linking them to operad compositions and Novikov algebra structures.

## Key findings

- Identities for operations $ riangleright$ and $ riangleleft$ are derived from differential algebra identities.
- The operations satisfy relations of the operad Var∘Nov, connecting to Novikov algebras.
- No additional identities hold beyond those derived from the operad composition.

## 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.