Morita theory in deformation quantization (Q420567)

This is a very nice survey paper on the classification of star products up to Morita equivalence on Poisson manifolds, which was solved in a recent paper of the author [\textit{H. Bursztyn}, \textit{V. Dolgushev} and \textit{S. Waldmann}, J. Reine Angew. Math. 662, 95--163 (2012; Zbl 1237.53080)]. The paper is organized as follows. Section 2 is an introduction to several notions of Morita equivalence; Section 3 is an exposition of Kontsevich's celebrated theorem that star products (up to gauge equivalence) on a Poisson manifold are in one-to-one correspondence with the Poisson structures (modulo diffeomorphisms); Section 4 explains the classification of Morita equivalent star products on Poisson manifolds; sections 5 and 6 consider Morita equivalence under the action of a Hopf algebra, in particular, the author's work [\textit{S. Jansen} et al., Lett. Math. Phys. 100, No. 2, 203--236 (2012; Zbl 1251.53057)] on the classification of invariant star products on a symplectic manifold with respect to a Lie algebra action is outlined.