Review of deformation theory I: Concrete formulas for deformations of algebraic structures

TL;DR

This review provides explicit formulas for representations and cohomologies of various algebraic structures, and explains how their deformations are characterized via Maurer-Cartan elements in differential graded algebras.

Contribution

It offers concrete formulas and a unified framework for understanding deformations of multiple algebraic structures through Maurer-Cartan elements.

Findings

Explicit formulas for representations and cohomologies of algebraic structures

Characterization of deformations via Maurer-Cartan elements

Connection between cohomologies and differential graded algebras

Abstract

In this review article, first we give the concrete formulas of representations and cohomologies of associative algebras, Lie algebras, pre-Lie algebras, Leibniz algebras and 3-Lie algebras and some of their strong homotopy analogues. Then we recall the graded Lie algebras and graded associative algebras that characterize these algebraic structures as Maurer-Cartan elements. The corresponding Maurer-Cartan element equips the graded Lie or associative algebra with a differential. Then the deformations of the given algebraic structures are characterized as the Maurer-Cartan elements of the resulting differential graded Lie or associative algebras. We also recall the relation between the cohomologies and the differential graded Lie and associative algebras that control the deformations.