Classical integrable systems and Knizhnik-Zamolodchikov-Bernard equations

Aminov G.; Levin Andrey; Olshanetsky M.; Zotov A.

Article

Classical integrable systems and Knizhnik-Zamolodchikov-Bernard equations

JETP Letters. 2015. Vol. 101. No. 9. P. 648-655.

Aminov G., Levin A., Olshanetsky M., Zotov A.

The results obtained in the works supported in part by the Russian Foundation for Basic Research (project 12-02-00594) are briefly reviewed. We mainly focus on interrelations between classical integrable systems, Painlevé-Schlesinger equations and related algebraic structures such as classical and quantum R-matrices. The constructions are explained in terms of simplest examples. © 2015, Pleiades Publishing, Inc.

Priority areas: mathematics

Language: English

Keywords: classical integrable systems

Publication based on the results of: Алгебраическая геометрия и ее приложения(2015)

Обобщенные модели Калоджеро и Тоды

Черняков Ю. Б., Харчев С. М., Левин А.М. и др. Письма в Журнал экспериментальной и теоретической физики. 2019. Т. 109. № 2. С. 131-138.

A Calogero–Sutherland system with two types of interacting spin variables has been described using the Hitchin approach and quasicompact structure. Complete integrability has been established by means of the Lax equation specified on a singular curve and the classical r-matrix depending on the spectral parameter. Generalized Toda systems have also been considered. Their phase portraits have been described.

Added: May 24, 2019

Absolutely convergent Fourier series. An improvement of the Beurling-Helson theorem

Vladimir Lebedev. arxiv.org. math. Cornell University, 2011. No. 1112.4892v1.

We obtain a partial solution of the problem on the growth of the norms of exponential functions with a continuous phase in the Wiener algebra. The problem was posed by J.-P. Kahane at the International Congress of Mathematicians in Stockholm in 1962. He conjectured that (for a nonlinear phase) one can not achieve the growth slower than the logarithm of the frequency. Though the conjecture is still not confirmed, the author obtained first nontrivial results.

Added: Apr 12, 2012

Новые информационные технологии. Тезисы докладов XVIII международной студенческой конференции-школы-семинара

М.: МИЭМ, 2010.

Added: Apr 12, 2012

Обоснование адиабатического предела для гиперболических уравнений Гинзбурга-Ландау

Пальвелев Р., Сергеев А. Г. Труды Математического института им. В.А. Стеклова РАН. 2012. Т. 277. С. 199-214.

Added: Feb 23, 2013

Метод параметрикса для диффузий и цепей Маркова

Конаков В. Д. STI. WP BRP. Издательство попечительского совета механико-математического факультета МГУ, 2012. № 2012.

Added: Dec 5, 2012

Hypercommutative operad as a homotopy quotient of BV

Khoroshkin A., Markaryan N. S., Shadrin S. arxiv.org. math. Cornell University, 2012. No. 1206.3749.

We give an explicit formula for a quasi-isomorphism between the operads Hycomm (the homology of the moduli space of stable genus 0 curves) and BV/Δ (the homotopy quotient of Batalin-Vilkovisky operad by the BV-operator). In other words we derive an equivalence of Hycomm-algebras and BV-algebras enhanced with a homotopy that trivializes the BV-operator. These formulas are given in terms of the Givental graphs, and are proved in two different ways. One proof uses the Givental group action, and the other proof goes through a chain of explicit formulas on resolutions of Hycomm and BV. The second approach gives, in particular, a homological explanation of the Givental group action on Hycomm-algebras.

Added: Aug 29, 2012

Is the function field of a reductive Lie algebra purely transcendental over the field of invariants for the adjoint action?

Colliot-Thélène J., Kunyavskiĭ B., Vladimir L. Popov et al. Compositio Mathematica. 2011. Vol. 147. No. 2. P. 428-466.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Added: Mar 17, 2013

Cross-sections, quotients, and representation rings of semisimple algebraic groups

V. L. Popov. Transformation Groups. 2011. Vol. 16. No. 3. P. 827-856.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.

Added: Mar 16, 2013

Введение в математическую статистику

Ивченко Г. И., Медведев Ю. И. М.: ЛКИ, 2010.

Added: Apr 12, 2012

Новые информационные технологии. Тезисы докладов XIX международной студенческой конференции-школы-семинара

М.: МИЭМ, 2011.

Added: Apr 12, 2012