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Фейгин Борис Львович

Факультет математики

Профиль на hse.ru ↗ тел.: +7 (495) 772-95-90 | 15320
Публикаций
147
Языков
1
Наград
11
Конференций
1
Профиль Публикации (147) Курсы (4)

Профессиональные интересы

представления тороидальных алгебркомбинторика плоских разбиений

Должности

  • Главный научный сотрудник — Факультет математики
  • профессор — Факультет математики
  • Руководитель научного коллектива — Факультет математики

Био

  • · Начал работать в НИУ ВШЭ в 2009 году.
  • · Научно-педагогический стаж: 42 года.

Образование

  • 2022 · Член-корреспондент РАН
  • 1996 · Доктор физико-математических наук: Институт теоретической физики им. Л.Д. Ландау РАН, специальность 01.00.00 «Физико-математические науки»
  • 1983 · Кандидат физико-математических наук: Санкт-Петербургское отделение математического института им. В.А. Стеклова РАН, специальность 01.00.00 «Физико-математические науки»
  • 1979 · Аспирантура: Ярославский государственный университет им. П.Г. Демидова, специальность «Математика»
  • 1974 · Специалитет: Московский государственный университет им. М.В. Ломоносова, специальность «Математика», квалификация «Математик»

Опыт работы

  • · 2009: Работает в НИУ ВШЭ с года

Награды и поощрения

  • · Медаль "Признание - 15 лет успешной работы" НИУ ВШЭ (март 2024)
  • · Благодарность НИУ ВШЭ (декабрь 2023)
  • · Благодарность Высшей школы экономики (сентябрь 2021)
  • · Благодарность Министра экономического развития Российской Федерации (сентябрь 2017)
  • · Почетная грамота Высшей школы экономики (январь 2014)
  • · Надбавка за публикацию в международном рецензируемом научном издании (2020–2022, 2018–2020)
  • · Надбавка за регулярные публикации в международных рецензируемых научных изданиях (2021–2026)
  • · Надбавка за статью в зарубежном рецензируемом журнале (2014–2016, 2012–2014)
  • · Надбавка за статью в зарубежном рецензируемом научном издании (2016–2018)
  • · Лучший преподаватель — 2012
  • · Лауреат премии "Золотая Вышка" 2014 в номинации Достижения в науке

Гранты и проекты

  • — · на соискание учёной степени кандидата наук

Конференции (1)

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  • · 2017: Transformation groups 2017 (Москва). Доклад: Shifted toroidal algebras and corresponding vertex operator algebra

Идентификаторы исследователя

Публикации (147)

Gaudin model and Deligne’s category

2024 · ARTICLE · en

We show that the construction of the higher Gaudin Hamiltonians associated with the Lie algebra gl(n) admits an interpolation to any complex number n. We do this using the Deligne’s category D(t), which is a formal way to define the category of finite-dimensional representations of the group GL(n), when n is not necessarily a natural number. We also obtain interpolations to any complex number n of the no-monodromy conditions on a space of differential operators of order n, which are considered to be a modern form of the Bethe ansatz equations. We prove that the relations in the algebra of higher Gaudin Hamiltonians for complex n are generated by our interpolations of the no-monodromy conditions. Our constructions allow us to define what it means for a pseudo-differential operator to have no monodromy. Motivated by the Bethe ansatz conjecture for the Gaudin model associated with the Lie superalgebra gl(n|n'), we show that a ratio of monodromy-free differential operators is a pseudo-differential operator without monodromy.

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Combinatorics of vertex operators and deformed W-algebra of type D(2, 1; α)

2022 · ARTICLE · en

We consider sets of screening operators with fermionic screening currents. We study sums of vertex operators which formally commute with the screening operators assuming that each vertex operator has rational contractions with all screening currents with only simple poles. We develop and use the method of qq-characters which are combinatorial objects described in terms of deformed Cartan matrix. We show that each qq-character gives rise to a sum of vertex operators commuting with screening operators and describe ways to understand the sum in the case it is infinite. We discuss combinatorics of the qq-characters and their relation to the q-characters of representations of quantum groups. We provide a number of explicit examples of the qq-characters with the emphasis on the case of D(2,1;α). We describe a relationship of the examples to various integrals of motion.

DOI ↗ PDF ↗

Quantum Toroidal Comodule Algebra of Type An−1 and Integrals of Motion

2022 · ARTICLE · en

We introduce an algebra Kn which has a structure of a left comodule over the quantum toroidal algebra of type An−1. Algebra Kn is a higher rank generalization of K1, which provides a uniform description of deformed W algebras associated with Lie (super)al-gebras of types BCD. We show that Kn possesses a family of commutative subalgebras. © 2022, Institute of Mathematics. All rights reserved.

DOI ↗ PDF ↗

Urod algebras and Translation of W-algebras

2022 · ARTICLE · en

In this work, we introduce Urod algebras associated to simply laced Lie algebras as well as the concept of translation of W-algebras. Both results are achieved by showing that the quantum Hamiltonian reduction commutes with tensoring with integrable representations; that is, for V and L an affine vertex algebra and an integrable affine vertex algebra associated with, we have the vertex algebra isomorphism, where in the left-hand-side the Drinfeld-Sokolov reduction is taken with respect to the diagonal action of on. The proof is based on some new construction of automorphisms of vertex algebras, which may be of independent interest. As corollaries, we get fusion categories of modules of many exceptional W-algebras, and we can construct various corner vertex algebras. A major motivation for this work is that Urod algebras of type A provide a representation theoretic interpretation of the celebrated Nakajima-Yoshioka blowup equations for the moduli space of framed torsion free sheaves on of an arbitrary rank.

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Shuffle algebra realization of quantum affine superalgebra U-v ((D)over-cap(2, 1; theta))

2021 · ARTICLE · en

Inspired by [12], we give shuffle algebra realization of positive part of quantum affine superalgebra U-v((D) over cap (2, 1; theta)) associated to any simple root systems. We also determine the shuffle algebra associated to (sl) over cap((2 vertical bar 1) with odd root system when v is a primitive root of unity of even order, generalizing results in [4].

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VOA[M4]

2020 · ARTICLE · en

We take a peek at a general program that associates vertex (or chiral) algebras to smooth 4-manifolds in such a way that operations on algebras mirror gluing operations on 4-manifolds and, furthermore, equivalent constructions of 4-manifolds give rise to equivalences (dualities) of the corresponding algebras.

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Dual description of eta-deformed OSP sigma models

2020 · ARTICLE · en

We study the dual description of the eta-deformed OSP(N|2m) sigma model in the asymptotically free regime (N > 2m+2). Compared to the case of classical Lie groups, for supergroups there are inequivalent eta-deformations corresponding to different choices of simple roots. For a class of such deformations we propose the system of screening charges depending on a continuous parameter b, which defines the eta-deformed OSP(N|2m) sigma model in the limit b -> 1 and a certain Toda QFT as b -> 0. In the sigma model regime we show that the leading UV asymptotic of the eta-deformed model coincides with a perturbed Gaussian theory. In the perturbative regime b -> 0 we show that the tree-level two-particle scattering matrix matches the expansion of the trigonometric OSP(N|2m) S-matrix.

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The (gl(m) ,gl(n) ) duality in Quantum toroidal setting

2019 · ARTICLE · en

On a Fock space constructed from mn free bosons and lattice Z mn , we give a level n action of the quantum toroidal algebra E m associated to gl m , together with a level m action of the quantum toroidal algebra E n associated to gl n . We prove that the E m transfer matrices commute with the E n transfer matrices after an appropriate identification of parameters.

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Towards trigonometric deformation of 𝔰𝔩ˆ2 coset VOA

2019 · ARTICLE · en

We discuss the quantization of the ̂ sl 2 coset vertex operator algebra W D(2,1;α) using the bosonization technique. We show that after quantization, there exist three families of commuting integrals of motion coming from three copies of the quantum toroidal algebra associated with gl 2 .

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Flat deformations of algebras and functional equations.

2019 · ARTICLE · en

We study flat deformations of quotients of a polynomial algebra in a class of graded commutative associative algebras. Functional equations and their solutions in terms of theta functions play important role in these studies. An analog of this theory in a fermionic case is also briefly discussed.

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Курсы (4)