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## Extensions of vertex algebras. Constructions and applications

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 to the geometric Langlands programme.

Finkelberg M. V., Rybnikov L. G., Journal of the European Mathematical Society 2012

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) ...

Added: February 19, 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

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

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

Cherednik I., Feigin E., Advances in Mathematics 2015 Vol. 282 P. 220-264

Given a reduced irreducible root system, the corresponding nil-DAHA is used to calculate the extremal coefficients of nonsymmetric Macdonald polynomials in the limit t→∞ and for antidominant weights, which is an important ingredient of the new theory of nonsymmetric q-Whittaker function. These coefficients are pure q-powers and their degrees are expected to coincide in the ...

Added: September 3, 2015

Feigin B. L., Hashizume K. undefined., Hoshino A. et al., Journal of Mathematical Physics 2009 Vol. 50 No. 9 P. 095215-1-095215-42

We introduce a unital associative algebra associated with degenerate CP1. We show that is a commutative algebra and whose Poincare' series is given by the number of partitions. Thereby, we can regard as a smooth degeneration limit of the elliptic algebra introduced by Feigin and Odesskii [Int. Math. Res. Notices 11, 531 (1997)]. Then we ...

Added: January 25, 2013

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

Васильев М., Zabrodin A., Zotov A., Nuclear Physics B - Proceedings Supplements 2020 Vol. 952 No. 114931 P. 1-20

We establish a remarkable relationship between the quantum Gaudin models with boundary and the classical many-body integrable systems of Calogero-Moser type associated with the root systems of classical Lie algebras (B, C and D). We show that under identification of spectra of the Gaudin Hamiltonians HjG with particles velocities q˙j of the classical model all ...

Added: August 20, 2020

191574970, Mathematical notes 1992 Vol. 52 No. 4 P. 995-1002

Added: September 23, 2016

Aleksei Ilin, Leonid Rybnikov, Communications in Mathematical Physics 2019 Vol. 372 No. 1 P. 343-366

Let gg be a complex simple Lie algebra. We study the family of Bethe subalgebras in the Yangian Y(g)Y(g) parameterized by the corresponding adjoint Lie group G. We describe their classical limits as subalgebras in the algebra of polynomial functions on the formal Lie group G1[[t−1]]G1[[t−1]]. In particular we show that, for regular values of the parameter, these subalgebras are free ...

Added: October 10, 2019

Aleksei Ilin, Leonid Rybnikov, / Cornell University. Series arXiv:1703.04147 "Series math "arxiv.org"". 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

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

Feigin E., Kato S., Makedonskyi I., Journal fuer die reine und angewandte Mathematik 2020 Vol. 764 P. 181-216

We study the non-symmetric Macdonald polynomials specialized at infinity from various points of view. First, we define a family of modules of the Iwahori algebra whose characters are equal to the non-symmetric Macdonald polynomials specialized at infinity. Second, we show that these modules are isomorphic to the dual spaces of sections of certain sheaves on ...

Added: August 12, 2020

Bershtein M., Tsymbaliuk A., / 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

Feigin B. L., Jimbo M., Miwa T. et al., Kyoto Journal of Mathematics 2012 Vol. 52 No. 3 P. 621-659

In this, the third paper of the series, we construct a large family of representations of the quantum toroidal gl(1)-algebra whose bases are parameterized by plane partitions with various boundary conditions and restrictions. We study the corresponding formal characters. As an application We obtain a Gelfand-Zetlin-type basis for a class of irreducible lowest weight gl(infinity)-modules. ...

Added: February 5, 2013

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

Borisov A., Mamaev I., Bizyaev I. A., Regular and Chaotic Dynamics 2016 Vol. 21 No. 5 P. 556-580

In this paper, we consider in detail the 2-body problem in spaces of constant positive curvature. We perform a reduction (analogous to that in rigid body dynamics) after which the problem reduces to analysis of a two-degree-of-freedom system. In the general case, in canonical variables the Hamiltonian does not correspond to any natural mechanical system. ...

Added: April 5, 2017

Feigin B. L., Jimbo M., Miwa T. et al., Journal of Algebra 2013 Vol. 380 P. 78-108

We define and study representations of quantum toroidal gln with natural bases labeled by plane partitions with various conditions. As an application, we give an explicit description of a family of highest weight representations of quantum affine gln with generic level. ...

Added: February 28, 2013

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

Burman Y. M., / Cornell University. Series math "arxiv.org". 2013. No. 1309.4477.

Given a representation V of a group G, there are two natural ways of defining a representation of the group algebra k[G] in the external power V^{\wedge m}. The set L(V) of elements of k[G] for which these two ways give the same result is a Lie algebra and a representation of G. For the ...

Added: November 19, 2013

Bezrukavnikov R., Ivan Losev, / Cornell University. Series arXiv "math". 2017. No. 1708.01385.

Added: October 9, 2017

Pogrebkov A., Theoretical and Mathematical Physics 2016 Vol. 187 No. 3 P. 823-834

We show that the non-Abelian Hirota difference equation is directly related to a commutator identity
on an associative algebra. Evolutions generated by similarity transformations of elements of this algebra
lead to a linear difference equation. We develop a special dressing procedure that results in an integrable
non-Abelian Hirota difference equation and propose two regular reduction procedures that lead ...

Added: September 9, 2016

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

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