Special transverse slices and their enveloping algebras

By A. Premet. 2002

273 citations recorded by the source corpus.

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

View Paper: https://www.semanticscholar.org/paper/47d9d11922cb14c4a6f84303e8e247736cc7b5d7

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Citations in the corpus

Commutative quotients of finite W-algebras (2008, methodology): “In proving [33, Conjecture 3.2] under this assumption the author relied almost entirely on characteristic p methods.”
Enveloping algebras of Slodowy slices and the Joseph ideal (2005, background): “It is proved in [30] that the algebra Hχ is a filtered deformation of the graded coordinate ring k[Se] .”
Finite vs affine W-algebras (2005, background): “rature that we know of are [BT1] and [DV]. More recently there has been a revival of interest in them in connection to geometry and representation theory of simple ﬁnite-dimensional Lie algebras, see [P1], [P2], [GG], [BrK]. The earliest work is [Ko], where it was shown that the algebra Wﬁn(g,x,f) f”
Representations of Shifted Yangians and Finite W-algebras (2005, background): “here we did not include the shift by the automorphism ηat this point. The deﬁnition of W(π) originates in work of Kostant [Ko2] and Lynch [Ly], and is a special case of the construction due to Premet [P1] and then Gan and Ginzburg [GG] of non-commutative ﬁltered deformations of the coordinate algebr”
Quantized symplectic actions and W -algebras (2007, background): “The definition of U (g ,e ) is due to Premet, [ Pr1 ], (for Lagrangian y) and Gan-”
Rationality of W-algebras: principal nilpotent cases (2012, background): “ﬀerential graded algebraH• f(U(g)) is naturally a Z-graded superalgebra and H• f(M) is naturally a bimodule over H• f(U(g)). RATIONALITY OF W-ALGEBRAS; PRINCIPAL NILPOTENT CASES 9 The ﬁnite W-algebra [Pre] associatedwith (g,f)maybe deﬁned astheassociative algebra U(g,f) := H0 f(U(g)) ([D3HK], see (1”
Isomorphisms of quantizations via quantization of resolutions (2010, methodology): “ e. Recall the Slodowy slice S and the equivariant Slodowy slice X = G×S. A (ﬁnite) W-algebra is a quantization of the graded Poisson algebra C[S]. In full generality, it was ﬁrst deﬁned by Premet in [P]. In this subsection we will recall the deﬁnitions of a W-algebra following [L2] and [GG1]. For d”
Centres of Centralizers of Unipotent Elements in Simple Algebraic Groups (2011, background): “Let Qχ be the generalized Gelfand-Graev module for the universal enveloping algebra U(L(G)) associated with the triple (e, h, f) (see [31]); set Hχ = EndL(G)(Qχ) .”
Representations of Lie superalgebras in prime characteristic I (2008, methodology): “In Section 4, we construct a p-subalgebra m of g from such a Z-grading associated to a nilpotent p-character χ, following Premet [12] (the cases with d1 odd offer some new perspectives).”
Deformations of symplectic singularities and orbit method for semisimple Lie algebras (2016, methodology): “s G-modules, then O1,O2 lie in the same birational sheet. 5. W-algebras and Orbitmethod 24 IVANLOSEV 5.1. W-algebras. We start by recalling (ﬁnite) W-algebras that were originally deﬁned by Premet in [P1], although we will follow an approach from [Lo1]. Pick a nilpotent orbit O⊂ g. Choose an element”
Quantization of Slodowy slices (2001, result): “-Whittaker Dmodule with respect to the character χ : m → C if, for any x ∈ m and v ∈ V, we have (x D −x M )v = χ(x)· v. Note that the natural map Ug → Q maps the center of Ug injectively into H (c.f. [Pr] §6.2). Denote by Z + the augmentation ideal of the center of Ug. Skryabin’s result combined wit”
Primitive ideals, non-restricted representations and finite W-algebras (2006, methodology): “Let Se = e + Ker ad f be the Slodowy slice at e through the adjoint orbit of e and let Hχ be the enveloping algebra of Se; see [18], [11], [6].”
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”
On the structure of the category O for W-algebras (2008, methodology): “egory of Whittaker modules and, secondly, the categories studied by Milicic-Soergel and Backelin. Then we state the category equivalence theorem 4.1 generalizing the Skryabin equivalence theorem from [Pr1] and proving the conjecture of Brundan, Goodwin and Kleshchev. The proof of Theorem 4.1 is give”
Schur–Weyl duality for higher levels (2006, methodology): “In our filtered deformation of this picture, we replace Cl[x1,...,xd] o CSd with the degenerate cyclotomic Hecke algebra Hd() and U(ge) with the finite W-algebra W( ) associated to the partition . Let us recall the definition of the latter algebra following [BK3]; see also [ P1 , P2, GG, BGK].”