Operads of compatible structures and weighted partitions

TL;DR

This paper introduces operads that encode compatibility conditions for algebraic structures, demonstrating their decompositions and Koszul properties for various types like Lie, associative, and pre-Lie algebras.

Contribution

It presents new operads for compatible algebraic structures, proves their decompositions via black and white products, and establishes their Koszulity for several important classes.

Findings

Operads for compatible Lie, associative, and pre-Lie algebras are Koszul.

Decompositions of these operads are achieved through black and white products.

The poset method of B. Vallette is used to prove Koszulity.

Abstract

In this paper we describe operads encoding two different kinds of compatibility of algebraic structures. We show that there exist decompositions of these in terms of black and white products and we prove that they are Koszul for a large class of algebraic structures by using the poset method of B. Vallette. In particular we show that this is true for the operads of compatible Lie, associative and pre-Lie algebras.