Hom-associative algebras, Admissibility and Relative averaging operators

TL;DR

This paper introduces relative averaging operators on Hom-associative algebras, providing new characterizations and showing how they generate various Hom-algebra structures, advancing the understanding of Hom-algebra theory.

Contribution

It defines and characterizes relative averaging operators on Hom-associative algebras, linking them to Hom-dialgebras and other Hom-algebra structures.

Findings

Characterizations via graphs and Nijenhuis operators

Construction of Hom-associative (tri)dialgebras from operators

Derivation of Hom-Jordan and Hom-Leibniz algebras from dialgebras

Abstract

We introduce the notion of relative averaging operators on Hom-associative algebras with a representation. Relative averaging operators are twisted generalizations of relative averaging operators on associative algebras. We give two characterizations of relative averaging operators of Hom-associative algebras via graphs and Nijenhuis operators. A (homomorphic) relative averaging operator of Hom-associative algebras with respect to a given representation gives rise to Hom-associative (tri)dialgebras. By admissibility, a Hom-Jordan (tri)dialgebra and a Hom-(tri)Leibniz algebra can be obtained from Hom-associative (tri)dialgebra.