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Universal quantizations of nilpotent Slodowy slices

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Finite dimensional complex simple Lie algebras are classified by Dynkin diagrams. The simply laced diagrams also classify the finite subgroups of SL2 which, in turn, give rise to a nice family of singular algebraic varieties: the Kleinian singularities. A conjecture of Grothendieck, proven by Brieskorn, states that the Kleinian singularity of Dynkin type X can be constructed as the subregular transverse slice to the nilpotent cone in the corresponding Lie algebra. In fact the transverse slice gives rise to a versal deformation of the Kleinian singularity. At the same time the transverse slices are quantized by finite W-algebras. Recently, in a joint work with Ambrosio, Carnovale, Esposito, we have proven a quantum analogue of Brieskorn’s theorem, proving a universal property for the finite W-algebra. In this talk I will give a gentle introduction to the theory and explain our main result and applications.

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