arXiv subjects

Leonid Rybnikov

Publications and source records attributed to Leonid Rybnikov.

At least 19 recordsLinked to original sources

arXiv2026

Athinization of irreducible $\widehat{\mathfrak{gl}}_n$-modules with dominant highest weights

We study the Gelfand-Tsetlin realization of generic Verma modules for the affine Lie algebra $\widehat{\mathfrak{gl}}_n$ by viewing them as thin modules over the affine Yangian $Y(\widehat{\mathfrak{sl}}_n)$. By results of arXiv:0812.4656, these modules admit a basis indexed by periodic Gelfand-Tsetlin patterns with explicit formulas for the Yangian action, and we identify them with the evaluation modules introduced by Kodera arXiv:1806.09884. Our main result describes the specialization from generic highest weights to dominant highest weights (not necessarily integral). We call the resulting construction athinization: an irreducible $\widehat{\mathfrak{gl}}_n$-module, which is not thin as a module over the affine Kac-Moody algebra, is realized as a thin module over the larger (and ''more affine'') algebra $Y(\widehat{\mathfrak{sl}}_n)$. Combinatorially, this realization is obtained by restricting the generic periodic Gelfand-Tsetlin basis to a distinguished subset of permitted patterns. We prove that the span of these patterns carries a well-defined affine Yangian action. In particular, this construction yields explicit Gelfand-Tsetlin-type bases for admissible representations of $\widehat{\mathfrak{gl}}_n$ in the sense of Kac-Wakimoto, providing a new combinatorial realization of these modules. We compare the formulas for characters coming from this combinatorics with those for minimal models of $W$-algebras of the type $A_n$ via the principal specialization. Further, we obtain analogous results for representations of $U_q\widehat{\mathfrak{gl}}_n$ via their realization as thin modules over the quantum toroidal algebra of $\mathfrak{gl}_n$.

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arXiv2026

Bethe subalgebras in antidominantly shifted Yangians

The loop group $G((z^{-1}))$ of a simple complex Lie group $G$ has a natural Poisson structure. We introduce a natural family of Poisson commutative subalgebras $\overline{\mathbf{B}}(C) \subset \mathcal{O}(G((z^{-1}))$ depending on the parameter $C\in G$ called classical universal Bethe subalgebras. To every antidominant cocharacter $μ$ of the maximal torus $T \subset G$ one can associate the closed Poisson subspace $\mathcal{W}_μ$ of $G((z^{-1}))$ (the Poisson algebra $\mathcal{O}(\mathcal{W}_μ)$ is the classical limit of so-called shifted Yangian $Y_μ(\mathfrak{g})$). We consider the images of $\overline{\mathbf{B}}(C)$ in $\mathcal{O}(\mathcal{W}_μ)$, that we denote by $\overline{B}_μ(C)$, that should be considered as classical versions of (not yet defined in general) Bethe subalgebras in shifted Yangians. For regular $C$ centralizing $μ$, we compute the Poincaré series of these subalgebras. For $\mathfrak{g}=\mathfrak{gl}_n$, we define the natural quantization ${\mathbf{Y}}^{\mathrm{rtt}}(\mathfrak{gl}_n)$ of $\mathcal{O}(\operatorname{Mat}_n((z^{-1}))))$ and universal Bethe subalgebras ${\mathbf{B}}(C) \subset {\mathbf{Y}}^{\mathrm{rtt}}(\mathfrak{gl}_n)$. Using the RTT realization of $Y_μ(\mathfrak{gl}_n)$ (invented by Frassek, Pestun, and Tsymbaliuk), we obtain the natural surjections ${\mathbf{Y}}^{\mathrm{rtt}}(\mathfrak{gl}_n) \twoheadrightarrow Y_μ(\mathfrak{gl}_n)$ which quantize the embedding $\mathcal{W}_μ\subset \operatorname{Mat}_n((z^{-1}))$). Taking the images of ${\mathbf{B}}(C)$ in $Y_μ(\mathfrak{gl}_n)$ we recover Bethe subalgebras $B_μ(C) \subset Y_μ(\mathfrak{gl}_n)$ proposed by Frassek, Pestun and Tsymbaliuk.

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arXiv2026

Maximal transitivity of the cactus group on standard Young tableaux

The action of the cactus group $C_n$ on Young tableaux of a given shape $λ$ goes back to Berenstein and Kirillov and arises naturally in the study of crystal bases and quantum integrable systems. We show that this action is $2$-transitive on standard Young tableaux of the shape $λ$ if and only if $λ$ is not self-transpose and not a single hook. Moreover, we show that in these cases, the image of the cactus group in the permutation group of standard Young tableaux is either the whole permutation group or the alternating group, and prove that both cases are possible for infinitely many $λ$ (though the alternating group is more frequent). As an application, this implies that the Galois group of solutions to the Bethe ansatz in the Gaudin model attached to the Lie group $GL_d$ is, in many cases, at least the alternating group. This also extends the results of Sottile and White on the multiple transitivity of the Galois group of Schubert calculus problems in Grassmannians to many new cases.

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arXiv2025

Bethe subspaces and wonderful models for toric arrangements

We study the family of commutative subspaces in the trigonometric holonomy Lie algebra $t^{\mathrm{trig}}_Φ$, introduced by Toledano Laredo, for an arbitrary root system $Φ$. We call these subspaces \emph{Bethe subspaces} because they can be regarded as quadratic components of \emph{Bethe subalgebras} in the Yangian corresponding to the root system $Φ$, that are responsible for integrals of the generalized XXX Heisenberg spin chain. Bethe subspaces are naturally parametrized by the complement of the corresponding toric arrangement . We prove that this family extends regularly to the minimal wonderful model $X_Φ$ of the toric arrangement described by De Concini and Gaiffi, thus giving a compactification of the parameter space for Bethe subspaces. For classical types $A_n, B_n, C_n, D_n$, we show that this extension is faithful. As a special case, when $Φ$ is of type $A_n$, our construction agrees with the main result of Aguirre--Felder--Veselov on the closure of the set of quadratic Gaudin subalgebras. Our work is also closely related to, and refines in this root system setting, a parallel compactification result of J. Peters obtained for more general toric arrangements arising from quantum multiplication on hypertoric varieties. Next, we show that the Bethe subspaces assemble into a vector bundle over $X_Φ$, which we identify with the logarithmic tangent bundle of $X_Φ$. Finally, we formulate conjectures extending these results to the setting of Bethe subalgebras in Yangians and to the quantum cohomology rings of Springer resolutions. We plan to address this in our next papers.

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arXiv2025

Gaudin models and moduli space of flower curves

We introduce and study the family of trigonometric Gaudin subalgebras in $U g^{\otimes n}$ for arbitrary simple Lie algebra $g$. This is the family of commutative subalgebras of maximal possible transcendence degree that serve as a universal source for higher integrals of the trigonometric Gaudin quantum spin chain attached to $g$. We study the parameter space that indexes all possible degenerations of subalgebras from this family. In particular, we show that (rational) inhomogeneous Gaudin subalgebras of $ U g^{\otimes n}$ previously studied in \cite{ffry} arise as certain limits of trigonometric Gaudin subalgebras. Moreover, we show that both families of commutative subalgebras glue together into the one parameterized by the space $\overline{\mathcal F}_n$, which is the total space of degeneration of the Deligne-Mumford space of stable rational curves $ \overline M_{n+2} $ to the moduli space of cactus flower curves $ \overline F_n $ recently introduced in \cite{iklpr}. As an application, we show that trigonometric Gaudin subalgebras act on tensor products of irreducible finite-dimensional $g$-modules without multiplicities, under some explicit assumptions on the parameters in terms of two different real forms of $\overline M_{n+2}$. This gives rise to a monodromy action of the affine cactus group on the set of eigenstates for the trigonometric Gaudin model. We also explain the relation between the trigonometric Gaudin model and the quantum cohomology of affine Grassmannians slices.

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arXiv2025

Cactus flower spaces and monodromy of Bethe vectors

We continue the study of cactus flower moduli spaces $\overline{F}_n$ and Gaudin models started in arXiv:2308.06880, arXiv:2407.06424. We show that isomorphism classes of operadic coverings of the real form $\overline{F}_n(\mathbb{R})$ are naturally one-to-one with equivalence classes of concrete coboundary monoidal categories (i.e. coboundary monoidal categories that admit a faithful monoidal functor to sets) with certain semisimplicity and finiteness conditions. Following the strategy of arXiv:1708.05105, for any complex semisimple Lie algebra $\mathfrak{g}$, we recover Kashiwara $\mathfrak{g}$-crystals, as a concrete coboundary category, from the coverings given by Bethe eigenlines for inhomogeneous Gaudin models. Using this, we compute the monodromy of Bethe eigenlines for trigonometric Gaudin models over two different real loci. In the particular case of minuscule highest weights, this can be regarded as combinatorial version of the wall-crossing conjecture of Bezrukavnikov and Okounkov for quantum cohomology of symplectic resolutions in the case of minuscule resolutions of slices in the affine Grassmannian.

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arXiv2024

The moduli space of cactus flower curves and the virtual cactus group

The space $ \ft_n = \C^n/\C $ of $n$ points on the line modulo translation has a natural compactification $ \overline \ft_n $ as a matroid Schubert variety. In this space, pairwise distances between points can be infinite; it is natural to imagine points at infinite distance from each other as living on different projective lines. We call such a configuration of points a ``flower curve'', since we picture the projective lines joined into a flower. Within $ \ft_n $, we have the space $ F_n = \C^n \setminus Δ/ \C $ of $ n$ distinct points. We introduce a natural compatification $ \overline F_n $ along with a map $ \overline F_n \rightarrow \overline \ft_n $, whose fibres are products of genus 0 Deligne-Mumford spaces. We show that both $\overline \ft_n$ and $\overline F_n$, are special fibers of $1$-parameter families whose generic fibers are, respectively, Losev-Manin and Deligne-Mumford moduli spaces of stable genus $0$ curves with $n+2$ marked points. We find combinatorial models for the real loci $ \overline \ft_n(\BR) $ and $ \overline F_n(\BR) $. Using these models, we prove that these spaces are aspherical and that their equivariant fundamental groups are the virtual symmetric group and the virtual cactus groups, respectively. The degeneration of a twisted real form of the Deligne-Mumford space to $\overline F_n(\mathbb{R})$ gives rise to a natural homomorphism from the affine cactus group to the virtual cactus group.

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arXiv2024

Bethe subalgebras in Yangians and the wonderful compactification

We study the family of Bethe subalgebras in the Yangian $Y(\mathfrak{g})$ 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 $G_1[[t^{-1}]]$. In particular we show that, for regular values of the parameter, these subalgebras are free polynomial algebras with the same Poincare series as the Cartan subalgebra of the Yangian. Next, we extend the family of Bethe subalgebras to the De Concini--Procesi wonderful compactification $\overline{G}\supset G$ and describe the subalgebras corresponding to generic points of any stratum in $\overline{G}$ as Bethe subalgebras in the Yangian of the corresponding Levi subalgebra in $\mathfrak{g}$.

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arXiv2022

Bethe subalgebras in Yangians and Kirillov-Reshetikhin crystals

Let $\mathfrak{g}$ be a complex simple finite dimensional Lie algebra and $G$ be the adjoint Lie group with the Lie algebra $\mathfrak{g}$. To every $C \in G$ one can associate a commutative subalgebra $B(C)$ in the Yangian $Y(\mathfrak{g})$, which is responsible for the integrals of the (generalized) $XXX$ Heisenberg magnet chain. Using the approach of arXiv:1708.05105, we construct a natural structure of affine crystals on spectra of $B(C)$ in Kirillov-Reshetikhin $Y(\mathfrak{g})$-modules in type $A$. We conjecture that such a construction exists for arbitrary $\mathfrak{g}$ and gives Kirillov-Reshetikhin crystals. Our main technical tool is the degeneration of Bethe subalgebras in the Yangian to commutative subalgebras $\mathcal{A}_χ^{\mathrm{u}}$ in the universal enveloping of the current Lie algebra, $U(\mathfrak{g}[t])$, which depend on the parameter $χ$ from the Lie algebra $\mathfrak{g}$ (and are of independent interest). We show that these subalgebras come from the Feigin-Frenkel center on the critical level as described by Feigin, Frenkel and Toledano Laredo in arXiv:math/0612798. This allows to prove that our affine crystals in type $A$ are indeed Kirillov-Reshetikhin by reducing to the crystal structure on the spectra of inhomogeneous Gaudin model which is already known (arXiv:1708.05105).

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arXiv2021

Spectra of Bethe subalgebras of $Y(\mathfrak{gl}_n)$ in tame representations

We study the eigenproblem for Bethe subalgebras of the Yangian $Y(\mathfrak{gl}_n)$ in tame representations, i.e. in finite dimensional representations which admit Gelfand-Tsetlin bases. Namely, we prove that for any tensor product of skew modules $V=\otimes_{i=1}^k V_{λ_i \setminus μ_i}(z_i)$ over the Yangian $Y(\mathfrak{gl}_n)$ with generic $z_i$'s, the family of Bethe subalgebras $B(X)$ with $X$ being a regular element of the maximal torus of $GL_n$ (or, more generally, with $X \in \overline{M_{0,n+2}}$) acts with a cyclic vector on $V$. Moreover, for $X$ in the real form of $\overline{M_{0,n+2}}$ which is the closure of regular unitary diagonal matrices we show, that the family of subalgebras $B(X)$ acts with simple spectrum on $\otimes_{i=1}^k V_{λ_i \setminus μ_i}(z_i)$ for generic $z_i$'s where all $V_{λ_i \setminus μ_i}(z_i)$ are Kirillov-Reshetikhin modules. In the subsequent paper we will use this to define a KR-crystal structure on the spectrum of a Bethe subalgebra on $V$.

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arXiv2021

On classical limits of Bethe subalgebras in Yangians

The Yangian $Y(\mathfrak{g})$ of a simple Lie algebra $\mathfrak{g}$ can be regarded as a deformation of two different Hopf algebras: the universal enveloping algebra $U(\mathfrak{g}[t])$ 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 to an appropriate filtration on Yangian. Bethe subalgebras form a natural family of commutative subalgebras depending on a group element $C$ of the adjoint group $G$. The images of these algebras in tensor products of fundamental representations give all integrals of the quantum XXX Heisenberg magnet chain. We describe the associated graded of Bethe subalgebras as subalgebras in $U(\mathfrak{g}[t])$ and in $\mathcal{O}(G_1[[t^{-1}]])$ for all semisimple $C\in G$. We show that associated graded in $U(\mathfrak{g}[t])$ of the Bethe subalgebra assigned to the identity of $G$ is the universal Gaudin subalgebra of $U(\mathfrak{g}[t])$ obtained from the center of the corresponding affine Kac-Moody algebra at the critical level. This generalizes Talalaev's formula for generators of the universal Gaudin subalgebra to $\mathfrak{g}$ of any type. In particular, this shows that higher Hamiltonians of the Gaudin magnet chain can be quantized without referring to the Feigin-Frenkel center at the critical level. Using our general result on associated graded of Bethe subalgebras, we compute some limits of Bethe subalgebras corresponding to regular semisimple $C\in G$ as $C$ goes to an irregular semisimple group element $C_0$. We show that this limit is the product of the smaller Bethe subalgebra and a quantum shift of argument subalgebra in the universal enveloping algebra of the centralizer of $C_0$ in $\mathfrak{g}$. This generalizes the Nazarov-Olshansky solution of Vinberg's problem on quantization of shift of argument subalgebras.

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arXiv2020

Crystals and monodromy of Bethe vectors

Fix a semisimple Lie algebra g. Gaudin algebras are commutative algebras acting on tensor product multiplicity spaces for g-representations. These algebras depend on a parameter which is a point in the Deligne-Mumford moduli space of marked stable genus 0 curves. When the parameter is real, then the Gaudin algebra acts with simple spectrum on the tensor product multiplicity space and gives us a basis of eigenvectors. In this paper, we study the monodromy of these eigenvectors as the parameter varies within the real locus; this gives an action of the fundamental group of this moduli space, which is called the cactus group. We prove a conjecture of Etingof which states that the monodromy of eigenvectors for Gaudin algebras agrees with the action of the cactus group on tensor products of g-crystals. In fact, we prove that the coboundary category of normal g-crystals can be reconstructed using the coverings of the moduli spaces. Our main tool is the construction of a crystal structure on the set of eigenvectors for shift of argument algebras, another family of commutative algebras which act on any irreducible g-representation. We also prove that the monodromy of such eigenvectors is given by the internal cactus group action on g-crystals.

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arXiv2018

Gelfand-Tsetlin degeneration of shift of argument subalgebras in type D

The universal enveloping algebra of any semisimple Lie algebra $\mathfrak{g}$ contains a family of maximal commutative subalgebras, called shift of argument subalgebras, parametrized by regular Cartan elements of $\mathfrak{g}$. For $\mathfrak{g}=\mathfrak{gl}_n$ the Gelfand-Tsetlin commutative subalgebra in $U(\mathfrak{g})$ arises as some limit of subalgebras from this family. In our previous work (arXiv:1807.11126) we studied the analogous limit of shift of argument subalgebras for the Lie algebras $\mathfrak{g}=\mathfrak{sp}_{2n}$ and $\mathfrak{g}=\mathfrak{o}_{2n+1}$. We described the limit subalgebras in terms of Bethe subalgebras of twisted Yangians $Y^-(2)$ and $Y^+(2)$, respectively, and parametrized the eigenbases of these limit subalgebras in the finite dimensional irreducible highest weight representations by Gelfand-Tsetlin patterns of types C and B. In this note we state and prove similar results for the last case of classical Lie algebras, $\mathfrak{g}=\mathfrak{o}_{2n}$. We describe the limit shift of argument subalgebra in terms of the Bethe subalgebra in the twisted Yangian $Y^+(2)$ and give a natural indexing of its eigenbasis in any finite dimensional irreducible highest weight $\mathfrak{g}$-module by type D Gelfand-Tsetlin patterns.

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arXiv2018

Gelfand-Tsetlin degeneration of shift of argument subalgebras in types B and C

The universal enveloping algebra of any semisimple Lie algebra $\mathfrak{g}$ contains a family of maximal commutative subalgebras, called shift of argument subalgebras, parametrized by regular Cartan elements of $\mathfrak{g}$. For $\mathfrak{g}=\mathfrak{gl}_n$ the Gelfand-Tsetlin commutative subalgebra in $U(\mathfrak{g})$ arises as some limit of subalgebras from this family. We study the analogous limit of shift of argument subalgebras for the Lie algebras $\mathfrak{g}=\mathfrak{sp}_{2n}$ and $\mathfrak{g}=\mathfrak{so}_{2n+1}$. The limit subalgebra is described explicitly in terms of Bethe subalgebras in twisted Yangians $Y^-(2)$ and $Y^+(2)$, respectively. We index the eigenbasis of such limit subalgebra in any irreducible finite-dimensional representation of $\mathfrak{g}$ by Gelfand-Tsetlin patterns of the corresponding type, and conjecture that this indexing is, in appropriate sense, natural. According to arXiv:1708.05105 such eigenbasis has a natural $\mathfrak{g}$-crystal structure. We conjecture that this crystal structure coincides with that on Gelfand-Tsetlin patterns defined by Littelmann in https://doi.org/10.1007/BF01236431 .

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arXiv2018

Towards a cluster structure on trigonometric zastava

We study a moduli problem on a nodal curve of arithmetic genus 1, whose solution is an open subscheme in the zastava space for projective line. This moduli space is equipped with a natural Poisson structure, and we compute it in a natural coordinate system. We compare this Poisson structure with the trigonometric Poisson structure on the transversal slices in an affine flag variety. We conjecture that certain generalized minors give rise to a cluster structure on the trigonometric zastava.

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arXiv2017

Degeneration of Bethe subalgebras in the Yangian of $\mathfrak{gl}_n$

We study degenerations of Bethe subalgebras $B(C)$ in the Yangian $Y(\mathfrak{gl}_n)$, where $C$ is a regular diagonal matrix. We show that closure of the parameter space of the family of Bethe subalgebras, which parametrizes all possible degenerations, is the Deligne-Mumford moduli space of stable rational curves $\overline{M_{0,n+2}}$. All subalgebras corresponding to the points of $\overline{M_{0,n+2}}$ are free and maximal commutative. We describe explicitly the "simplest" degenerations and show that every degeneration is the composition of the simplest ones. The Deligne-Mumford space $\overline{M_{0,n+2}}$ generalizes to other root systems as some De Concini-Procesi resolution of some toric variety. We state a conjecture generalizing our results to Bethe subalgebras in the Yangian of arbitrary simple Lie algebra in terms of this De Concini-Procesi resolution.

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arXiv2017

Comultiplication for shifted Yangians and quantum open Toda lattice

We study a coproduct in type A quantum open Toda lattice in terms of a coproduct in the shifted Yangian of sl_2. 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.

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arXiv2016

On families of differential equations on two-torus with all phase-lock areas

We consider two-parametric families of non-autonomous ordinary differential equations on the two-torus with the coordinates $(x,t)$ of the type $\dot x=v(x)+A+Bf(t)$. We study its rotation number as a function of the parameters $(A,B)$. The {\it phase-lock areas} are those level sets of the rotation number function $ρ=ρ(A,B)$ that have non-empty interiors. V.M.Buchstaber, O.V.Karpov, S.I.Tertychnyi have studied the case, when $v(x)=\sin x$ in their joint paper. They have observed the quantization effect: for every smooth periodic function $f(t)$ the family of equations may have phase-lock areas only for integer rotation numbers. Another proof of this quantization statement was later obtained in a joint paper by Yu.S.Ilyashenko, D.A.Filimonov, D.A.Ryzhov. This implies the similar quantization effect for every $v(x)=a\sin(mx)+b\cos(mx)+c$ and rotation numbers that are multiples of $\frac 1m$. We show that for every other analytic vector field $v(x)$ (i.e., having at least two Fourier harmonics with non-zero non-opposite degrees and nonzero coefficients) there exists an analytic periodic function $f(t)$ such that the corresponding family of equations has phase-lock areas for all the rational values of the rotation number.

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