Quantized symplectic actions and W -algebras

By I. Losev. 2007

120 citations recorded by the source corpus.

With a nilpotent element in a semisimple Lie algebra g one associates a finitely generated associative algebra W called a W-algebra of finite type. This algebra is obtained from the universal enveloping algebra U(g) by a certain Hamiltonian reduction. We observe that W is the invariant algebra for an action of a reductive group G with Lie algebra g on a quantized symplectic affine variety and use this observation to study W. Our results include an alternative definition of W, a relation between the sets of prime ideals of W and of the corresponding universal enveloping algebra, the existence of a one-dimensional representation of W in the case of classical g and the separation of elements of W by finite dimensional representations.

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

View Paper: https://www.semanticscholar.org/paper/bdfe69f101bf3bb3bf97f1d12d50778f6c4a31a5

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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. Pavel Etingof

In Geometry and Topology

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

In Mathematical Physics

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

Citations in the corpus

• Isomorphisms of quantizations via quantization of resolutions (2010, methodology): “a different (Walgebra or symplectic reflection algebra) construction. In Section 5 we establish results involving W-algebras. In Subsection 5.1 we recall the definitions of W-algebras due to the author, [L2] and Premet, [P]. Then in Subsection 5.2 we introduce certain ramifications of W-algebras, the pa”
• Harish-Chandra bimodules for quantized Slodowy slices (2008, methodology): “ results of this paper is Theorem 4.1.5 that describes crucial properties of that functor. We use the above theorem to provide an alternate proof of a conjecture of Premet that was proved by Losev in [Lo1]. The result of Losev relates finite dimensional Ac-modules to primitive ideals I⊂ Ug such that ”
• Associated Varieties of Modules Over Kac–Moody Algebras and C2-Cofiniteness of W-Algebras (2010, methodology): “ure that all the C2-cofinite W-algebras appearing in this way are 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 “chiralization” of the argument of [Gin] proves the vanishing of the B”
• Commutative quotients of finite W-algebras (2008, methodology): “d very few changes to the original proof in [34] are actually required. In the meantime two different proofs of [33, Conjecture 3.2] have appeared in the literature; the first one was found by Losev in [24] and the second one by Ginzburg in [16]. Our proof relies on the method developed in [34], the o”
• Completions of symplectic reflection algebras (2010, methodology): “f “reduction theorems” for ideals and Harish-Chandra bimodules of SRA’s. Our results regarding ideals are described in Subsection 1.3. This part is mostly inspired by the author’s work on W-algebras, [Lo1],[Lo2]. In Section 4 we give some miscellaneous applications of results of the first two section”
• Deformations of symplectic singularities and orbit method for semisimple Lie algebras (2016, methodology): “n Section 4.5. In Section 5 we establish our version of the orbit method. We start by recalling results and constructions related to W-algebras. In Section 5.1 we present their construction following [Lo1, Lo2]. Then in Section 5.2 we recall the restriction functor for HarishChandra bimodules from [”