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

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.

Review of deformation theory II: a homotopical approach

“…More generally, deformations of A ∞ -modules over an A ∞ algebra are controlled by a certain non-unital dg algebra, cf. [9] regarding this example.…”

“…This article is the second part of a review of deformation theory, although it can be read independently. The first part [9] gives concrete formulas for dg Lie algebras controlling deformations of various algebraic structures: associative algebras, Lie algebras, pre-Lie algebras, Leibniz algebras, 3-Lie algebras and some of their ∞-variants. We refer to the first part for examples of the theory presented here.…”