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## Quantization of Drinfeld Zastavain type A

In press

algebra $\hat{sl}_n$. We introduce an affine, reduced, irreducible, normal quiver variety $Z$ which maps to the Zastava space bijectively at the level of complex points. The natural Poisson structure on the Zastava space can be described on $Z$ in terms of Hamiltonian reduction of a certain Poisson subvariety of the dual space of a (nonsemisimple) Lie algebra. The quantum Hamiltonian reduction of the corresponding quotient of its universal enveloping algebra produces a quantization $Y$ of the coordinate ring of $Z$. The same quantization was obtained in the finite (as opposed to the affine) case generically in arXiv:math/0409031. We prove that, for generic values of quantization parameters, $Y$ is a quotient of the affine Borel Yangian.

Feigin B. L., Russian Mathematical Surveys 2017 Vol. 72 No. 4 P. 707-763

This paper discusses the main known constructions of vertex operator algebras. The starting point is the lattice algebra. Screenings distinguish subalgebras of lattice algebras. Moreover, one can construct extensions of vertex algebras. Combining these constructions gives most of the known examples. A large class of algebras with big centres is constructed. Such algebras have applications ...

Added: November 5, 2020

Sergeev A., European Mathematical Society Publishing house, 2014

This book is based on a lecture course given by the author at the Educational Center of the Steklov Mathematical Institute in 2011. It is designed for a one-semester course for undergraduate students familiar with basic differential geometry and complex and functional analysis.
The universal Teichmüller space T is the quotient of the space of quasisymmetric ...

Added: April 9, 2015

Bershtein M., Tsymbaliuk A., Homomorphisms between different quantum toroidal and affine Yangian algebras / Cornell University. Series arXiv "math". 2015. No. 1512.09109.

This paper concerns the relation between the quantum toroidal algebras and the affine Yangians of $\mathfrak{sl}_n$, denoted by $\mathcal{U}^{(n)}_{q_1,q_2,q_3}$ and $\mathcal{Y}^{(n)}_{h_1,h_2,h_3}$, respectively. Our motivation arises from the milestone work Gautam and Toledano Laredo, where a similar relation between the quantum loop algebra U_q(L\\mathfrak{g})$ and the Yangian $Y_h(\mathfrak{g})$ has been established by constructing an isomorphism of ...

Added: March 16, 2016

Vologodsky V., Finkelberg M. V., Bezrukavnikov R., Cambridge Journal of Mathematics 2014 Vol. 2 No. 2 P. 163-190

Marc Haiman has reduced Macdonald Positivity Conjecture to a statement about geometry of the Hilbert scheme of points on the plane, and formulated a generalization of the conjecture where the symmetric group is replaced by the wreath product of S_n and Z/rZ. He has proven the original conjecture by establishing the geometric statement about the ...

Added: December 17, 2015

Aleksei Ilin, Leonid Rybnikov, Transformation Groups 2021 Vol. 26 No. 2 P. 537-564

The Yangian $Y(\fg)$ of a simple Lie algebra $\fg$ can be regarded as a deformation of two different Hopf algebras: the universal enveloping algebra of the current algebra $U(\fg[t])$ and the coordinate ring of the first congruence subgroup $\mathcal{O}(G_1[[t^{-1}]])$. Both of these algebras are obtained from the Yangian by taking the associated graded with respect ...

Added: April 2, 2021

Michael Finkelberg, Kamnitzer J., Pham K. et al., Advances in Mathematics 2018 Vol. 327 P. 349-389

We study a coproduct in type A quantum open Toda lattice
in terms of a coproduct in the shifted Yangian of sl2. At
the classical level this corresponds to the multiplication of
scattering matrices of euclidean SU(2) monopoles. We also
study coproducts for shifted Yangians for any simply-laced
Lie algebra. ...

Added: February 21, 2018

Bezrukavnikov R., Finkelberg M. V., Cambridge Journal of Mathematics 2014 Vol. 2 No. 2 P. 163-190

Marc Haiman has reduced Macdonald Positivity Conjecture to a statement about geometry of the Hilbert scheme of points on the plane, and formulated a generalization of the conjecture where the symmetric group is replaced by the wreath product of S_n and Z/rZ. He has proven the original conjecture by establishing the geometric statement about the ...

Added: December 20, 2014

Karasev M., Russian Journal of Mathematical Physics 2016 Vol. 23 No. 4 P. 483-489

We discuss two examples of classical mechanical systems which can become quantum either because of degeneracy of an integral of motion or because of tuning parameters at resonance. In both examples, the commutativity of the symmetry algebra is breaking, and noncommutative symmetries arise. Over the new noncommutative algebra, the system can reveal its quantum behavior ...

Added: October 22, 2016

Finkelberg M. V., Rybnikov L. G., Journal of the European Mathematical Society 2014 Vol. 16 No. 2 P. 235-271

Drinfeld Zastava is a certain closure of the moduli space of maps from the projective line to the Kashiwara flag scheme of the affine Lie algebra of the special linear group. We introduce an affine, reduced, irreducible, normal quiver variety Z isomorphic to the zastava space. The natural Poisson structure on the zastava space can ...

Added: January 16, 2014

Cherednik I., Feigin B. L., Advances in Mathematics 2013 Vol. 248 No. 25 P. 1050-1088

Using the DAHA-Fourier transform of q-Hermite polynomials multiplied by level-one theta functions, we obtain expansions of products of any number of such theta functions in terms of the q-Hermite polynomials. An ample family of modular functions satisfying Rogers-Ramanujan type identities for arbitrary (reduced, twisted) affine root systems is obtained as an application. A relation to ...

Added: September 30, 2013

Takasaki K., Takebe T., Теоретическая и математическая физика (Российская Федерация) 2012 Vol. 171 No. 2 P. 683-690

We briefly review a recursive construction of hbar-dependent solutions of the Kadomtsev-Petviashvili hierarchy. We give recurrence relations for the coefficients X_n of an ħ-expansion of the operator X = X_0 + hbar X_1 + hbar^2 X_2 + ... for which the dressing operator W is expressed in the exponential form W = exp(X/hbar). The wave ...

Added: June 22, 2012

Finkelberg M. V., Rybnikov L. G., Algebraic Geometry 2014 Vol. 1 No. 2 P. 166-180

Drinfeld zastava is a certain closure of the moduli space of maps from the projective line to the Kashiwara flag scheme of an affine Lie algebra g^. In case g is the symplectic Lie algebra spN, we introduce an affine, reduced, irreducible, normal quiver variety Z which maps to the zastava space isomorphically in characteristic 0. The natural Poisson structure on ...

Added: October 25, 2013

Khoroshkin S. M., Nazarov M., Shapiro A., Journal of Algebra 2014 Vol. 418 P. 265-291

We define natural classes of rational and polynomial representations of the Yangian of the general linear Lie algebra. We also present the classification and explicit realizations of all irreducible rational representations of the Yangian. ...

Added: December 8, 2014

Gritsenko V., Nikulin V., Lorentzian Kac-Moody algebras with Weyl groups of 2-reflections / Cornell University. Series arXiv "math". 2016.

We describe a new large class of Lorentzian Kac–Moody algebras. For all ranks, we classify 2-reflective hyperbolic lattices S with the group of 2-reflections of finite volume and with a lattice Weyl vector. They define the corresponding hyperbolic Kac–Moody algebras of restricted arithmetic type which are graded by S. For most of them, we construct ...

Added: March 17, 2016

Feigin B. L., Finkelberg M. V., Rybnikov L. G. et al., Selecta Mathematica, New Series 2011 Vol. 17 No. 3 P. 573-607

Laumon moduli spaces are certain smooth closures of the moduli spaces of maps from the projective line to the flag variety of GLn. We construct the action of the Yangian of sln in the cohomology of Laumon spaces by certain natural correspondences. We construct the action of the affine Yangian (two-parametric deformation of the universal ...

Added: October 9, 2012

Khoroshkin S. M., Nazarov M., Papi P., Journal of Algebra 2011 Vol. 346 P. 189-226

We give explicit realizations of irreducible representations of the Yangian of the general linear Lie algebra and of its twisted analogues, corresponding to symplectic and orthogonal Lie algebras. In particular, we develop the fusion procedure for twisted Yangians. For the non-twisted Yangian, this procedure goes back to the works of Cherednik. ...

Added: February 28, 2013

Losev Ivan, Compositio Mathematica 2017 Vol. 153 No. 12 P. 2445-2481

In this paper we study categories O over quantizations of symplectic resolutions admitting Hamiltonian tori actions with finitely many fixed points. In this generality, these categories were introduced by Braden, Licata, Proudfoot and Webster. We establish a family of standardly stratified structures (in the sense of the author and Webster) on these categories O. We ...

Added: October 15, 2017

Gritsenko V., Nikulin V., Proceedings of the London Mathematical Society 2018 Vol. 116 No. 3 P. 485-533

We describe a new large class of Lorentzian Kac--Moody algebras. For all ranks, we classify 2-reflective hyperbolic lattices $S$ with the group of 2-reflections of finite volume and with a lattice Weyl vector. They define the corresponding hyperbolic Kac--Moody algebras of restricted arithmetic type which are graded by S. For most of them, we construct ...

Added: October 23, 2017

Pereskokov A., Липская А. В., Вестник Московского энергетического института 2011 № 6 С. 30

Рассмотрены радиально-симметричные решения уравнения типа Хартри, содержащего интегральную нелинейность с потенциалом взаимодействия Юкавы, а также кулоновский потенциал. В квазиклассическом приближении выведена и исследована система уравнений для самосогласованного потенциала. Выписано правило квантования типа Бора-Зоммерфельда. Найдены асимптотические собственные значения и собственные функции, локализованные в шаре. ...

Added: December 16, 2012

Aleksei Ilin, Leonid Rybnikov, Degeneration of Bethe subalgebras in the Yangian of $gl_n$ / . 2017.

We study degenerations of Bethe subalgebras $B(C)$ in the Yangian $Y(\fgl_n)$, where $C$ is a regular diagonal matrix. We show that closure of the parameter space of the family of Bethe subalgebras is the Deligne-Mumford moduli space of stable rational curves $\overline{M_{0,n+2}}$ and state a conjecture generalizing this result to Bethe subalgebras in Yangians of ...

Added: March 15, 2017

Pereskokov A., Липская А. В., Вестник Московского энергетического института 2010 № 6 С. 99-109

Рассмотрены радиально-симметричные решения уравнения типа Хартри, содержащего как кулоновский потенциал, так и интегральную нелинейность с потенциалом взаимодействия Юкавы. В квазиклассическом приближении выведены и исследованы уравнения для самосогласованного потенциала. Выписано правило квантования типа Бора-Зоммерфельда. Найдены асимптотические собственные значения и собственные функции. ...

Added: December 16, 2012

Bershtein M., Tsymbaliuk A., Journal of Pure and Applied Algebra 2019 Vol. 223 No. 2 P. 867-899

This paper concerns the relation between the quantum toroidal algebras and the affine Yangians of sln, denoted by U(n)q1,q2,q3 and Y(n)h1,h2,h3, respectively. Our motivation arises from the milestone work of Gautam and Toledano Laredo, where a similar relation between the quantum loop algebra Uq(Lg) and the Yangian Yh(g) has been established by constructing an isomorphism ...

Added: November 12, 2019

Braverman A., Rybnikov L. G., Feigin B. L. et al., Communications in Mathematical Physics 2011 Vol. 308 No. 2 P. 457-478

Recently Alday, Gaiotto and Tachikawa proposed a conjecture relating 4-dimensional super-symmetric gauge theory for a gauge group G with certain 2-dimensional conformal field theory. This conjecture implies the existence of certain structures on the (equivariant) intersection cohomology of the Uhlenbeck partial compactification of the moduli space of framed G-bundles on P^2. More precisely, it predicts ...

Added: May 12, 2012

Aleksei Ilin, Rybnikov L. G., Letters in Mathematical Physics 2018 Vol. 108 No. 4 P. 1083-1107

We study degenerations of Bethe subalgebras B(C) in the Yangian Y(gln), where C is a regular diagonal matrix. We show that closure of the parameter space of the family of Bethe subalgebras, which parameterizes all possible degenerations, is the Deligne–Mumford moduli space of stable rational curves M0,n+2¯. All subalgebras corresponding to the points of M0,n+2¯are free and maximal commutative. ...

Added: December 8, 2017