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.

arXiv comment20 pages

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Cited by in corpus (4)

• The $L_{\infty }$-deformations of associative Rota-Baxter algebras and homotopy Rota-Baxter operators
• Maurer-Cartan characterizations and cohomologies of compatible Lie algebras
• Review of deformation theory II: a homotopical approach
• A survey on deformations, cohomologies and homotopies of relative Rota-Baxter Lie algebras