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Deformation Theory (Lecture Notes)

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M. Doubek, M. Markl, P. Zima
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** Address.**
Martin Markl, Math. Inst. of the Academy

Zitna 25, 115 67 Prague1, Czech Republic

** E-mail:**
martindoubek@seznam.cz

markl@math.cas.cz

**Abstract.**
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~\ref{s7} 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.

**AMSclassification. ** Primary 13D10, Secondary 14D15, 53D55, 46L65.

**Keywords. ** Deformation, Maurer-Cartan equation, strongly homotopy Lie algebra, deformation quantization.