Isomorphisms of quantizations via quantization of resolutions

By I. Losev. 2010

103 citations recorded by the source corpus.

Abstract In this paper, we prove the existence of isomorphisms between certain non-commutative algebras that are interesting from the representation theoretic perspective and arise as quantizations of certain Poisson algebras. We show that quantizations of Kleinian singularities obtained by three different constructions are isomorphic to each other. The constructions are via symplectic reflection algebras, quantum Hamiltonian reduction, and W -algebras. Next, we prove that parabolic W -algebras in type A are isomorphic to quantum Hamiltonian reductions associated to quivers of type A. Finally, we show that the symplectic reflection algebras for wreath-products of the symmetric group and a Kleinian group are isomorphic to certain quantum Hamiltonian reductions. Our results involving W -algebras are new, while for those dealing with symplectic reflection algebras we just find new proofs. A key ingredient in our proofs is the study of quantizations of symplectic resolutions of appropriate Poisson varieties.

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

View Paper: https://www.semanticscholar.org/paper/1979549aac085ecfa1cfd0e451dc759c3bd2cc7e

How researchers see this paper

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In Algebra and Number Theory

1. I. Gordon
2. I. Losev
3. E. Gorsky
4. Ben Webster
5. Pavel Etingof
6. Nicholas Proudfoot
7. Tom Braden
8. Tudor Dimofte

In Geometry and Topology

1. I. Losev
2. I. Gordon
3. E. Gorsky
4. Pavel Etingof
5. Ben Webster
6. Tudor Dimofte
7. D. Gaiotto
8. V. Ginzburg

In Mathematical Physics

1. I. Losev
2. I. Gordon
3. E. Gorsky
4. Pavel Etingof
5. Ben Webster
6. Tudor Dimofte
7. D. Gaiotto
8. V. Ginzburg

Citations in the corpus

• Representations of Rational Cherednik algebras with minimal support and torus knots (2013, methodology): “n. Set Dh := DC h, where C stands for the trivial G-module. This is a quantization of the structure sheaf OXe. We remark that Dh is almost the same as the canonical quantization ofXestudied in[BK] and[L6]. The only difference isthatthestructure of the C[[z∗,h]]-algebra is changed, since here we have ”
• The Coulomb Branch of 3d N = 4 Theories (2015, background): “) More so, if one requires that quantization be equivariant with respect to the U(1)H , U(1)C actions on the rings of functions then the period must simply lie in H2(M,C) itself [40].”
• Quantizations of conical symplectic resolutions I: local and global structure (2012, background): “xed point of −φand take ξ= 0, the quantization Qred is isomorphic to its own opposite, and therefore to the canonical quantization. ✷ The following proposition is implicit in the principal results of [Los12], but does not seem to be explicitly stated in the generality that we need. Our proof is simi”
• On category O for cyclotomic rational Cherednik algebras (2011, background): “In these cases the symplectic reflection algebras are related to differential operators on the quiver representation spaces and to symplectic resolutions of quotient singularities (see [44] for instance).”