Derivations, Cohomology and Deformation of BiHom-Associative Dialgebra

TL;DR

This paper introduces BiHom-associative dialgebras, classifies their derivations, extends cohomology theory to these structures, and explores their formal deformations, advancing the understanding of BiHom algebraic systems.

Contribution

It defines BiHom-associative dialgebras, classifies their derivations, and extends cohomology and deformation theory to these new algebraic structures.

Findings

Complete classification of derivations for 2- and 3-dimensional cases.

Extension of cohomology theory to BiHom-associative dialgebras.

Evaluation of formal deformations using cohomology.

Abstract

Due to the immense importance of BiHom Type algebras and cohomology of various algebraic structures, this paper is devoted to defining the BiHom-associative dialgebra, its derivation, generalized derivation, and quasi-derivation. We provided the complete classification of these derivations of $2-$ and $3$-dimensional BiHom-associative dialgebras. We further generalized the cohomology of BiHom-associative algebras to the cohomology of BiHom-associative dialgebras. As an application to cohomology, we evaluate the one-parameter formal deformation of BiHom-associative dialgebras.