Harish-Chandra bimodules for quantized Slodowy slices

By V. Ginzburg. 2008

90 citations recorded by the source corpus.

The Slodowy slice is an especially nice slice to a given nilpotent conjugacy class in a semisimple Lie algebra. Premet introduced noncommutative quantizaions of the Poisson algebra of polynomial functions on the Slodowy slice. In this paper, we define and study Harish-Chandra bimodules over Premet's algebras. We apply the technique of Harish-Chandra bimodules to prove a conjecture of Premet concerning primitive ideals, and to construct `noncommutative resolutions' of Slodowy slices via translation functors.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/171a2991a73bceee0f9956a5de1e85262df2ae18

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In Algebra and Number Theory

1. I. Losev
2. T. Arakawa
3. A. Premet
4. V. Ginzburg
5. Alexander S. Kleshchev
6. Jonathan Brundan
7. I. Gordon
8. D. Testerman

In Geometry and Topology

1. I. Losev
2. T. Arakawa
3. A. Premet
4. V. Ginzburg
5. Alexander S. Kleshchev
6. Jonathan Brundan
7. T. Creutzig
8. I. Gordon

In Mathematical Physics

1. I. Losev
2. T. Arakawa
3. A. Premet
4. V. Ginzburg
5. Alexander S. Kleshchev
6. Jonathan Brundan
7. I. Gordon
8. D. Testerman

Citations in the corpus

• Commutative quotients of finite W-algebras (2008, background): “The group Mk is a characteristic p analogue of the unipotent group M from [16] which, in turn, is a special instance of a group Nl for l = g(−1) 0 (the group Nl can be defined for any totally isotropic subspace l ⊂ g(−1); see [13]).”
• Isomorphisms of quantizations via quantization of resolutions (2010, background): “der the quotient with respect to the action of P from the left). From here it is easy to see S(e,B) is the resolution of singularities of the intersection S∩Nof S with the nil-cone N⊂g∗, compare with [Gi]. For a general parabolic subgroup P ⊂G, there are projective morphisms S(e,P) →S∩N,X///P →S res”
• Rationality of W-algebras: principal nilpotent cases (2012, methodology): “We shall now give yet another description of the functor (12) and show that (12) is equivalent to the functor constructed by Ginzburg [Gin09] and Losev [Los11], independently.”
• Associated Varieties of Modules Over Kac–Moody Algebras and C2-Cofiniteness of W-Algebras (2010, background): “exceptional, see Tables 2, 4, 6, 8, 9, 10. At this moment we do not know whether these not-exceptional C2-cofinite W-algebras are non-rational. Our strategy to prove (1) is based on Ginzburg’s reproof [Gin] of Premet’s conjecture [Pre, Conjecture 3.2] (proved by Losev [Los]) on finite W-algebras. A “c”
• Quantized symplectic actions and W -algebras (2007, methodology): “To this end one uses techniques analogous to those developed in [GG]; see also [ Gi2 ].”