The category and operad of matching dialgebras

TL;DR

This paper systematically studies matching dialgebras through operad theory, establishing their Koszul property, connections to other algebraic structures, and explicit homology computations.

Contribution

It introduces the operad of matching dialgebras, proves its Koszulity, and constructs free objects and homology explicitly.

Findings

Matching dialgebras are Koszul operads.

Connections to semi-homomorphisms and matched pairs are established.

Explicit homology complex for matching dialgebras is provided.

Abstract

This paper gives a systematic study of matching dialgebras corresponding to the operad $As^{(2)}$ in \cite{Zi} as the only Koszul self dual operad there other than the operads of associative algebras and Poisson algebras. The close relationship of matching dialgebras with semi-homomorphisms and matched pairs of associative algebras are established. By anti-symmetrizing, matching dialgerbas are also shown to give compatible Lie algebras, pre-Lie algebras and PostLie algebras. By the rewriting method, the operad of matching dialgebras is shown to be Koszul and the free objects are constructed in terms of tensor algebras. The operadic complex computing the homology of the matching dialgebras is made explicit.