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Topics in Representation Theory
Genus two Double Affine Hecke Algebra and its Classical Limit.
Genus two Double Affine Hecke Algebra and its Classical Limit.
组织者
演讲者
时间
2024年11月01日 13:00 至 14:30
地点
A3-4-301
线上
Zoom 518 868 7656
(BIMSA)
摘要
Double Affine Hecke Algebras were originally introduced by I.Cherednik and used in his 1995 proof of Macdonald conjecture from algebraic combinatorics. These algebras come equipped with a large automorphism group SL(2,Z) which has geometric origin, namely it is the modular group of a torus. It was subsequently shown that spherical Double Affine Hecke Algebras realize universal flat deformations of the quantum chracter variety of a torus and their existence is closely related to the fact that classical SL(n,C)-character varieties admit symplectic resolution of singularities via the Hilbert Scheme Hilb_n(\mathbb C*\times\mathbb C*).
In 2019 G.Belamy and T.Schedler have shown that SL(n,C)-character varieties of closed genus g surface admit symplectic resolutions only when g=1 or (g,n)=(2,2). In my talk I will discuss our (g,n)=(2,2) generalization of Double Affine Hecke Algebra which provide a flat deformation of quantum SL(2,C)-character variety of a closed genus two surface. I will show that solution to the word problem in our algebra has striking similarity with the Poicare-Birkhoff-Witt Theorem for the basis of Universal Enveloping Algebra of a Lie algebra. This is consistent with the philosophy formulated by A.Okounkov that resolutions of symplectic singularities should be viewed as "Lie Algebras of the XXI'st century". Joint work with Sh. Shakirov.
In 2019 G.Belamy and T.Schedler have shown that SL(n,C)-character varieties of closed genus g surface admit symplectic resolutions only when g=1 or (g,n)=(2,2). In my talk I will discuss our (g,n)=(2,2) generalization of Double Affine Hecke Algebra which provide a flat deformation of quantum SL(2,C)-character variety of a closed genus two surface. I will show that solution to the word problem in our algebra has striking similarity with the Poicare-Birkhoff-Witt Theorem for the basis of Universal Enveloping Algebra of a Lie algebra. This is consistent with the philosophy formulated by A.Okounkov that resolutions of symplectic singularities should be viewed as "Lie Algebras of the XXI'st century". Joint work with Sh. Shakirov.
演讲者介绍
I studied Applied Mathematics and Physics at the Moscow Institute of Physics and Technology, where I earned both my B.Sc. and M.Sc. degrees. In 2013, I joined the graduate program in Mathematics at Rutgers, The State University of New Jersey, and completed my Ph.D. in 2018 under the guidance of Prof. V. Retakh. After earning my doctorate, I held postdoctoral positions at the University of California Berkeley, the Centre de Recherches Mathématiques in Montreal, and the University of Toronto. In July 2024, I became an Associate Professor at the Beijing Institute of Mathematical Sciences and Applications (BIMSA)