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Deformation theory (lecture notes). (English) Zbl 1199.13015
Summary: First three sections of this overview paper cover classical topics of deformation theory of associative algebras and necessary background material. We then analyze algebraic structures of the Hochschild cohomology and describe the relation between deformations and solutions of the corresponding Maurer-Cartan equation. In Section 7 we generalize the Maurer-Cartan equation to strongly homotopy Lie algebras and prove the homotopy invariance of the moduli space of solutions of this equation. In the last section we indicate the main ideas of Kontsevich’s proof of the existence of deformation quantization of Poisson manifolds.

MSC:
13D10 Deformations and infinitesimal methods in commutative ring theory
14D15 Formal methods and deformations in algebraic geometry
53D55 Deformation quantization, star products
46L65 Quantizations, deformations for selfadjoint operator algebras
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