(Co)homology of compatible associative algebras

TL;DR

This paper develops (co)homology theories for compatible associative algebras, introduces a graded Lie algebra framework, and explores applications in extensions, deformations, and homology structures.

Contribution

It introduces a new graded Lie algebra for compatible associative structures and defines their (co)homology, advancing understanding of their algebraic properties and deformation theory.

Findings

Constructed a graded Lie algebra with Maurer-Cartan elements representing compatible associative structures

Defined cohomology for compatible associative algebras and studied their extensions and deformations

Explored homology via compatible presimplicial vector spaces

Abstract

In this paper, we define and study (co)homology theories of a compatible associative algebra $A$. At first, we construct a new graded Lie algebra whose Maurer-Cartan elements are given by compatible associative structures. Then we define the cohomology of a compatible associative algebra $A$ and as applications, we study extensions, deformations and extensibility of finite order deformations of $A$. We end this paper by considering compatible presimplicial vector spaces and the homology of compatible associative algebras.