Derived bracket construction and Manin products

TL;DR

This paper extends the derived bracket construction to algebras over binary quadratic operads, linking it to Manin products, and explores dualities between differential and Rota-Baxter operators.

Contribution

It introduces a functorial derived product construction via Manin white products and identifies the operad of prePoisson algebras as a Manin black product, revealing new operadic relationships.

Findings

Derived product construction is a functor via Manin white product.

Operad of prePoisson algebras is isomorphic to a Manin black product.

Differential and Rota-Baxter operators are Koszul dual.

Abstract

We will extend the classical derived bracket construction to any algebra over a binary quadratic operad. We will show that the derived product construction is a functor given by the Manin white product with the operad of permutation algebras. As an application, we will show that the operad of prePoisson algebras is isomorphic to Manin black product of the Poisson operad with the preLie operad. We will show that differential operators and Rota-Baxter operators are, in a sense, Koszul dual to each other.