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B. Feigin

Publications and source records attributed to B. Feigin.

At least 19 recordsLinked to original sources
arXiv2026

$(\mathfrak{gl}_{n},\mathfrak{gl}_{m})$-duality and Olshanski homomorphism

We show that the images of the Bethe subalgebras of the Yangians $Y(\mathfrak{gl}_{n})$ and $Y(\mathfrak{gl}_{m})$ under the homomorphisms to $U(\mathfrak{gl}_{n+m})$ given by the Olshanski centralizer construction coincide. We use this result to obtain the $(\mathfrak{gl}_{n},\mathfrak{gl}_{m})$-duality of the trigonometric Gaudin model and the XXX-spin chain. The duality is obtained in an explicit way relating the generating differential operator on one side and the generating difference operator on the other, thus agreeing with the result of Mukhin, Tarasov and Varchenko arXiv:math/0605172.

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arXiv2025

Extensions of a commuting pair of quantum toroidal $\mathfrak{gl}_1$

We introduce a family of algebras $\mathcal{A}_{M,N}$, $M,N\in\mathbb{Z}$, as an extension of a pair of commuting quantum toroidal $\mathfrak{gl}_1$ subalgebras $\mathcal{E}_1,\check{\mathcal{E}}_1$, wherein the parameters are tuned in a specific way according to $M,N$. In the case $M=\pm 1$, algebra $\mathcal{A}_{\pm1,N}$ is a shifted quantum toroidal $\mathfrak{gl}_2$ algebra introduced in [FJM2]. Conjecturally there is a coproduct homomorphism $\mathcal{A}_{M,N_1+N_2}\to\mathcal{A}_{M,N_1}\hat\otimes\mathcal{A}_{M,N_2}$ to a completed tensor product, whose restriction to the subalgebras $\mathcal{E}_1,\check{\mathcal{E}}_1$ coincides with the standard Drinfeld coproduct. We give examples of $\mathcal{A}_{M,N}$ modules constructed on certain direct sums of tensor products of Fock modules of $\mathcal{E}_1\otimes\check{\mathcal{E}}_1$.

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arXiv2025

Affinization of shifted quantum affine $\mathfrak{gl}_2$

We give a realization $\mathcal{A}_0$ of quantum toroidal algebra associated to $\mathfrak{gl}_2$ which can be viewed as an affinization of the Drinfeld new realization of quantum affine $\mathfrak{gl}_2$. We use this realization to define an affinization $\mathcal{A}_N$, $N\in{\mathbb Z}$, of shifted quantum affine $\mathfrak{gl}_2$. We construct a large family of representations of dominantly shifted algebra $\mathcal A_N$, $N>0$. The examples of representations with even positive $N$ appear in the study of extensions of deformed $W$-algebras of type $\mathfrak{gl}(N+2|1)$.

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arXiv2024

Remarks on $q$-difference opers arising from quantum toroidal algebras

We propose a conjectural correspondence between the spectra of the Bethe algebra for the quantum toroidal $\mathfrak{gl}_2$ algebra on relaxed Verma modules, and $q$-hypergeometric opers with apparent singularities. We introduce alongside the notion of apparent singularities for linear $q$-difference operators and discuss some of their properties. We also touch on a generalization to $\mathfrak{gl}_n$.

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arXiv2023

Commutative subalgebra of a shuffle algebra associated with quantum toroidal $\mathfrak{gl}_{m|n}$

We define and study the shuffle algebra $Sh_{m|n}$ of the quantum toroidal algebra $\mathcal E_{m|n}$ associated to Lie superalgebra $\mathfrak{gl}_{m|n}$. We show that $Sh_{m|n}$ contains a family of commutative subalgebras $\mathcal B_{m|n}(s)$ depending on parameters $s=(s_1,\dots,s_{m+n})$, $\prod_i s_i=1$, given by appropriate regularity conditions. We show that $\mathcal B_{m|n}(s)$ is a free polynomial algebra and give explicit generators which conjecturally correspond to the traces of the $s$-weighted $R$-matrix computed on the degree zero part of $\mathcal E_{m|n}$ modules of levels $\pm 1$.

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arXiv2023

Gaudin model and Deligne's category

We show that the construction of the higher Gaudin Hamiltonians associated to the Lie algebra $\mathfrak{gl}_{n}$ admits an interpolation to any complex $n$. We do this using the Deligne's category $\mathcal{D}_{t}$, 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 $n$ 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-deifferential operator to have no monodromy. Motivated by the Bethe ansatz conjecture for the Gaudin model associated with the Lie superalgebra $\mathfrak{gl}_{n\vert n'}$, we show that a ratio of monodromy-free differential operators is a pseudo-differential operator without monodromy.

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arXiv2021

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

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.

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arXiv2021

Deformations of $\mathcal W$ algebras via quantum toroidal algebras

The deformed $\mathcal W$ algebras of type $\textsf{A}$ have a uniform description in terms of the quantum toroidal $\mathfrak{gl}_1$ algebra $\mathcal E$. We introduce a comodule algebra $\mathcal K$ over $\mathcal E$ which gives a uniform construction of basic deformed $\mathcal W$ currents and screening operators in types $\textsf{B},\textsf{C},\textsf{D}$ including twisted and supersymmetric cases. We show that a completion of algebra $\mathcal K$ contains three commutative subalgebras. In particular, it allows us to obtain a commutative family of integrals of motion associated with affine Dynkin diagrams of all non-exceptional types except $\textsf{D}^{(2)}_{\ell+1}$. We also obtain in a uniform way deformed finite and affine Cartan matrices in all classical types together with a number of new examples, and discuss the corresponding screening operators.

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arXiv2021

Evaluation modules for quantum toroidal ${\mathfrak{gl}}_n$ algebras

The affine evaluation map is a surjective homomorphism from the quantum toroidal ${\mathfrak {gl}}_n$ algebra ${\mathcal E}'_n(q_1,q_2,q_3)$ to the quantum affine algebra $U'_q\widehat{\mathfrak {gl}}_n$ at level $κ$ completed with respect to the homogeneous grading, where $q_2=q^2$ and $q_3^n=κ^2$. We discuss ${\mathcal E}'_n(q_1,q_2,q_3)$ evaluation modules. We give highest weights of evaluation highest weight modules. We also obtain the decomposition of the evaluation Wakimoto module with respect to a Gelfand-Zeitlin type subalgebra of a completion of ${\mathcal E}'_n(q_1,q_2,q_3)$, which describes a deformation of the coset theory $\widehat{\mathfrak {gl}}_n/\widehat{\mathfrak {gl}}_{n-1}$.

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arXiv2018

Towards trigonometric deformation of $\widehat{\mathfrak{sl}}_2$ coset VOA

We discuss the quantization of the $\widehat{\mathfrak{sl}}_2$ coset vertex operator algebra $\mathcal{W}D(2,1;α)$ 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 to ${\mathfrak{gl}}_2$.

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arXiv2018

Plane partitions with a "pit": generating functions and representation theory

We study plane partitions satisfying condition $a_{n+1,m+1}=0$ (this condition is called "pit") and asymptotic conditions along three coordinate axes. We find the formulas for generating function of such plane partitions. Such plane partitions label the basis vectors in certain representations of quantum toroidal $\mathfrak{gl}_1$ algebra, therefore our formulas can be interpreted as the characters of these representations. The resulting formulas resemble formulas for characters of tensor representations of Lie superalgebra $\mathfrak{gl}_{m|n}$. We discuss representation theoretic interpretation of our formulas using $q$-deformed $W$-algebra $\mathfrak{gl}_{m|n}$.

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arXiv2018

The $(\mathfrak{gl}_m,\mathfrak{gl}_n$) duality in the quantum toroidal setting

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

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arXiv2018

Fermionic formulas for eigenfunctions of the difference Toda Hamiltonian

We use the Whittaker vectors and the Drinfeld Casimir element to show that eigenfunctions of the difference Toda Hamiltonian can be expressed via fermionic formulas. Motivated by the combinatorics of the fermionic formulas we use the representation theory of the quantum groups to prove a number of identities for the coefficients of the eigenfunctions.

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arXiv2018

Branching rules for quantum toroidal gl(n)

We construct an analog of the subalgebra $Ugl(n)\otimes Ugl(m)$ of $Ugl(m+n)$ in the setting of quantum toroidal algebras and study the restrictions of various representations to this subalgebra.

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arXiv2017

Integrals of motion from quantum toroidal algebras

We identify the Taylor coefficients of the transfer matrices corresponding to quantum toroidal algebras with the elliptic local and non-local integrals of motion introduced by Kojima, Shiraishi, Watanabe, and one of the authors. That allows us to prove the Litvinov conjectures on the Intermediate Long Wave model. We also discuss the (gl(m),gl(n)) duality of XXZ models in quantum toroidal setting and the implications for the quantum KdV model. In particular, we conjecture that the spectrum of non-local integrals of motion of Bazhanov, Lukyanov, and Zamolodchikov is described by Gaudin Bethe ansatz equations associated to affine sl(2).

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arXiv2016

Finite type modules and Bethe ansatz equations

We introduce and study a category $\text{Fin}$ of modules of the Borel subalgebra of a quantum affine algebra $U_q\mathfrak{g}$, where the commutative algebra of Drinfeld generators $h_{i,r}$, corresponding to Cartan currents, has finitely many characteristic values. This category is a natural extension of the category of finite-dimensional $U_q\mathfrak{g}$ modules. In particular, we classify the irreducible objects, discuss their properties, and describe the combinatorics of the q-characters. We study transfer matrices corresponding to modules in $\text{Fin}$. Among them we find the Baxter $Q_i$ operators and $T_i$ operators satisfying relations of the form $T_iQ_i=\prod_j Q_j+ \prod_k Q_k$. We show that these operators are polynomials of the spectral parameter after a suitable normalization. This allows us to prove the Bethe ansatz equations for the zeroes of the eigenvalues of the $Q_i$ operators acting in an arbitrary finite-dimensional representation of $U_q\mathfrak{g}$.

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arXiv2016

Finite type modules and Bethe Ansatz for quantum toroidal gl(1)

We study highest weight representations of the Borel subalgebra of the quantum toroidal gl(1) algebra with finite-dimensional weight spaces. In particular, we develop the q-character theory for such modules. We introduce and study the subcategory of `finite type' modules. By definition, a module over the Borel subalgebra is finite type if the Cartan like current ψ^+(z) has a finite number of eigenvalues, even though the module itself can be infinite dimensional. We use our results to diagonalize the transfer matrix T_{V,W}(u;p) analogous to those of the six vertex model. In our setting T_{V,W}(u;p) acts in a tensor product W of Fock spaces and V is a highest weight module over the Borel subalgebra of quantum toroidal gl(1) with finite-dimensional weight spaces. Namely we show that for a special choice of finite type modules $V$ the corresponding transfer matrices, Q(u;p) and T(u;p), are polynomials in u and satisfy a two-term TQ relation. We use this relation to prove the Bethe Ansatz equation for the zeroes of the eigenvalues of Q(u;p). Then we show that the eigenvalues of T_{V,W}(u;p) are given by an appropriate substitution of eigenvalues of Q(u;p) into the q-character of V.

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