Deformation Theory

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 6 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. Table of content: 1. Algebras and modules – p. 2 2. Cohomology – p. 8 3. Classical deformation theory – p. 9 4. Structures of (co)associative (co)algebras – p. 16 5. dg-Lie algebras and the Maurer-Cartan equation – p. 22 6. L∞-algebras and the Maurer-Cartan equation – p. 28 7. Homotopy invariance of the Maurer-Cartan equation – p. 34 8. Deformation quantization of Poisson manifolds – p. 37 Conventions. All algebraic objects will be considered over a fixed field k of characteristic zero. The symbol ⊗ will denote the tensor product over k. We will sometimes use the same symbol for both an algebra and its underlying space. Acknowledgement. We would like to thank Dietrich Burde for useful comments on a preliminary version of this paper. We are also indebted to Ezra Getzler for turning our attention to a remarkable paper [7]. Also suggestions of M. Goze and E. Remm were very helpful.