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Morita theory in deformation quantization. (English) Zbl 1256.53057
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 [H. Bursztyn, V. Dolgushev and 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 [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.
Reviewer: Hao Xu (Cambridge)
MSC:
53D55 Deformation quantization, star products
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References:
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