Quantization of Slodowy slices

By Wee Liang Gan, V. Ginzburg. 2001

211 citations recorded by the source corpus.

We give a direct proof of (a slight generalization of) the recent result of A. Premet related to generalized Gelfand-Graev representations and of an equivalence due to Skryabin.

Subfields: Algebra and Number Theory, Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d30661050fc42e3a9f5b0a867ef65783c8bee1b9

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In Algebra and Number Theory

In Applied Mathematics

In Mathematical Physics

Citations in the corpus

Shifted Yangians and finite W-algebras (2004, methodology): “Let us just state briefly here the analogue of [P, Proposition 6.3] which we regard as the fundamental structure theorem for W(χ); see also [ GG , Theorem 4.1] and [L, Theorem 2.3].”
Rationality of W-algebras: principal nilpotent cases (2012, background): “It follows that VarH0 fq (A(L(kΛ0))) is a nonzero C∗-invariant subvariety of Sf q . As VarH0 fq (A(L(kΛ0))) = VarA(L(kΛ0))∩ Sfq by Theorem 3.2 (i) and the C ∗-action of S fq is contracting to fq (see [GG]), VarA(L(kΛ0)) must contain the point fq as required. Conjecture 1. For a ﬁnitely strongly gene”
Quantized symplectic actions and W -algebras (2007, methodology): “Gχ. Moreover, the description of the Poisson structure on S given in [ GG ], Section 3, implies that any Poisson subvariety S0 of S is the union of irreducible components of the intersections of S with G-stable subvarieties of g∗. Therefore for any such S0 we have the equality dim GS0 =d imS0 +d imG”
Representation theory of W-algebras (2005, background): “considered in this article are the W -algebras associated with principal nilpotent orbits of ¯g. There has been rapidly growing interest in W -algebras associated with other nilpotent orbits (see eg. [57, 30, 12])7. The method developed in the present paper works for general W -algebras as well (cf.”
Isomorphisms of quantizations via quantization of resolutions (2010, methodology): “rresponding to λ and h = 1. It is easy to see that the natural homomorphism Wh →Wλ is an epimorphism. Now let us explain the approach to W-algebras of Premet, [P], in the version of Gan and Ginzburg, [GG1]. Recall the grading g= L i∈Zg(i), the subalgebra m⊂g, the element χ ∈g∗ from Subsection 4.3. C”
On the structure of the category O for W-algebras (2008, background): “) and deﬁne the subalgebra m := l⊕ L i6−2 g(i) ⊂g. According to Premet, [Pr1], a W-algebra Wassociated with e is, by deﬁnition, (U/Umχ)adm, where mχ := {ξ −hχ,ξi,ξ ∈m}. As Gan and Ginzburg checked in [GG], Wdoes not depend on the choice of l up to some natural isomorphism. In particular, there is a ”
Schur–Weyl duality for higher levels (2006, methodology): “In our ltered deformation of this picture, we replace Cl[x1;:::;xd] o CSd with the degenerate cyclotomic Hecke algebra Hd(), and U(ge) with the nite W -algebra W ( ) associated to the partition . Let us recall the denition of the latter algebra following [BK3]; see also [P1, P2, GG , BGK].”
The Gelfand-Kirillov conjecture and Gelfand-Tsetlin modules for finite W-algebras (2010, background): “The growing interest in finite W -algebras theory is due, on the one hand, to their geometric realizations as quantizations of Slodowy slices (Premet [34] and Gan and Ginzburg [19]) and, on the other hand, to their close connections with Yangian theory, which was initially proposed by Ragoucy and So”
Generic singularities of nilpotent orbit closures (2015, background): “Its smooth locus admits a symplectic form (see [GG02]), which is this restriction of the symplectic form on O.”
Enveloping algebras of Slodowy slices and the Joseph ideal (2005, methodology): “We recall that the Poisson structure on grH is determined in [10].”
The Coulomb Branch of 3d N = 4 Theories (2015, background): “One may, for example, take Sρ to be the affine subspace ρ(e) + ker ad(ρ(f)) inside slp ; this is the standard Slodowy slice, with a manifest Poisson structure [71].”
Primitive ideals, non-restricted representations and finite W-algebras (2006, methodology): “Recall that W (g, e) is a BRST quantisation of the Poisson algebra grHχ, and it is also isomorphic to the Zhu algebra of the vertex (affine) W -algebra associated with g and e; see [8], [7], [11], [18] for more detail.”
Representation theory of symmetric groups and related Hecke algebras (2009, unknown): “We now define the finite W -algebra W (λ) associated to the partition λ, following [36]; see also [165, 71].”
Generalised Gelfand-Graev Representations in Small Characteristics (2014, background): “Achar, Olivier Dudas, Simon Goodwin, Sebastian Herpel, Gunter Malle, Iulian Simion and Britta Spa¨th for useful conversations/correspondence. We especially thank Simon Goodwin for pointing us towards [GG02] and [BR85], which formed important ingredients in this paper. Finally the author gratefully a”