arXiv subjects

Vyacheslav Futorny

Publications and source records attributed to Vyacheslav Futorny.

At least 19 recordsLinked to original sources

arXiv2026

Categorical Lie-Rinehart modules and Shen-Larsson functors

We develop a categorical framework for Lie-Rinehart monoids and their weak modules in a symmetric monoidal category. Using crossed homomorphisms, we construct a natural action of the monoidal category of modules over a Lie monoid on the category of weak Lie-Rinehart modules, thereby obtaining categorical versions of the Shen-Larsson functors. We further characterize the conditions under which the category of weak modules admits a monoidal structure and identify the corresponding condition for the associated functors to be strict monoidal. A dual theory for Lie- Rinehart comonoids and weak comodules is developed using cocrossed homomorphisms. Combining the module and comodule constructions, we obtain a bimodule category structure on the category of weak modules. Finally, we specialize the general framework to the symmetric monoidal category of super vector spaces, recovering Lie-Rinehart superalgebras and their associated Shen-Larsson-type constructions.

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arXiv2026

Galois Rings: Ring-Theoretic Properties and Applications to Coulomb Branches and Affine Hecke Algebras

Galois rings and Galois orders, introduced by Futorny and Ovsienko, are realized as subrings of fixed subrings of skew group (or monoid) rings and have numerous applications in the structure and representation theory of associative algebras. This paper consists of two parts. The first parte investigates ring-theoretic properties that follow from the Galois ring structure alone. In particular, we estabilish natural conditions under which Galois are Ore domains or (semi)prime Goldie rings. We also study several ring-theoretic dimensions and combine the theories of Galois rings and PI-algebras to obtain new structural results. In the second part, we apply these results, together with general techniques from ring theory, to affine Hecke algebras in the sense of Ginzburg, Kapranov, and Vasserot, as well as to spherical Coulomb branch algebras. In particular, we prove that these algebras are Jacobson semiprimitive, satisfy the Nullstellensatz, and determine several of their ring-theoretic dimensions. For affine Hecke algebras, we further prove that they satisfy the maximal Nullstellensatz, are integral over their centers, and determine their T-ideals of polynomial identities, PI-degree, and PI-exponents. For spherical Coulomb branch algebras, we compute the Krull dimension, estabilish that they satisfy the Gelfand-Kirillov conjecture, and prove that they are not PI-algebras

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arXiv2026

Representations of Super Yangians with Gelfand-Tsetlin bases

The evaluation homomorphism from the super Yangian $\mathrm Y(\mathfrak{gl}_{m|n})$ to the universal enveloping algebra $\mathrm U(\mathfrak{gl}_{m|n})$ allows one to regard the $\mathfrak{gl}_{m|n}$-modules as $\mathrm Y(\mathfrak{gl}_{m|n})$-modules. In this paper, we establish necessary and sufficient conditions for simple tensor products of covariant evaluation modules to be tame. We also provide an explicit construction of Gelfand-Tsetlin bases for tensor products of skew modules.

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arXiv2026

Representations of twisted quantum affine algebras

We develop the representation theory of imaginary Verma modules for twisted quantum affine algebras and construct the corresponding Kashiwara algebras. The twisted case presents substantial new difficulties compared with the untwisted setting: the PBW root vectors have nontrivial orbit structure, the imaginary root spaces occur with multiplicities, and roots of unity enter essentially into the defining commutation relations. Our first main result is an explicit PBW-type basis for twisted quantum imaginary Verma modules associated with the natural imaginary partition of the affine root system. This provides a precise compatibility between the twisted quantum and classical theories that is not immediate from the standard PBW theory. We then determine the structure and irreducibility of the twisted quantum imaginary Verma modules. We prove that the Heisenberg submodule generated by the imaginary root vectors is irreducible precisely at nonzero central charge, and establish the corresponding irreducibility criterion for the reduced twisted imaginary Verma modules at zero central charge. The second main part of the paper introduces the Kashiwara algebra in the twisted setting. The required current formula for the generators differs essentially from the untwisted formula and is needed to construct the Omega operators. We derive the resulting Omega-operator commutation relations, including the root-of-unity factors specific to the twisted cases. These relations lead to a new presentation of the Kashiwara algebra associated with the reduced twisted imaginary Verma modules. We prove that the negative current algebra is a simple module over this Kashiwara algebra and construct a symmetric non-degenerate bilinear form characterized by the Omega operators.

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arXiv2026

Irreducibility of the tensor product of Yangian $ \mathrm{Y}(\mathfrak{gl}_{m|n}) $ evaluation modules

The evaluation homomorphism from the super Yangian $ \mathrm{Y}(\mathfrak{gl}_{m|n}) $ to $ \mathrm{U}(\mathfrak{gl}_{m|n}) $ induces a $ \mathrm{Y}(\mathfrak{gl}_{m|n}) $-module structure on any finite dimensional simple $ \mathrm{U}(\mathfrak{gl}_{m|n}) $-module $ L(λ) $. In this paper, we give necessary and sufficient conditions for the tensor product of such evaluation $ \mathrm{Y}(\mathfrak{gl}_{m|n}) $-modules, $ L_g(λ) \otimes L_h(γ) $, to be simple, provided each of $ λ$ and $ γ$ is either covariant tensor or essentially typical. Our proof is based on the existence of a Gelfand--Tsetlin basis for finite dimensional simple $ \mathrm{U}(\mathfrak{gl}_{m|n}) $-modules with highest weights that belong to these two families: covariant tensor and essentially typical. The obtained result is a super analogue of the Molev's result for the classical Yangian $ \mathrm{Y}(\mathfrak{gl}_n) $. Combining this with the binary property of tensor products of covariant evaluation modules, we obtain an irreducibility criterion for arbitrary tensor products of covariant evaluation modules.

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arXiv2026

Drinfeld Realization for Quantum Affine Orthosymplectic Superalgebras

A well-defined braid groupoid action is an essential tool for constructing the new Drinfeld realization of a quantum affine superalgebra. For quantum affine orthosymplectic superalgebras (types B, C, and D), this action was not fully defined, as the braid operators $T_i$ were known only up to normalization factors. In this paper, we solve this problem by providing the explicit formulas for these operators for any choice of parity. This yields a well-defined braid group action on the direct sum of these superalgebras. As a consequence, we use this action to formally introduce the new Drinfeld realization $U_q^D(\widehat{\mathfrak{g}}_s)$ for these types and prove that the corresponding Drinfeld-Jimbo quantum group $U_q(\widehat{\mathfrak{g}}_s)$ is its surjective homomorphic image. We conjecture that this map is an isomorphism.

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arXiv2026

Whittaker Modules for W type Cartan Lie superalgebras

We consider the category of Whittaker modules for the Lie superalgebra $W_{m,n}$ of vector fields on $\mathbb{C}^{(m|n)}$. For any $\mathbf{a}\in \mathbb{C}^m$ we show the equivalence between the blocks $Ω_{\mathbf a}^{\widetilde{W}_{m,n}}$ of the category of $(AW)_{m,n}$-Whittaker modules with finite-dimensional Whittaker vector spaces and the category of finite-dimensional modules over certain Lie subsuperalgebra $T_{m,n}$ of $(AW)_{m,n}$ (and also of $\mathfrak{gl}{(m,n)})$. Then we apply the covering technique to study Whittaker $W_{m,n}$-modules and describe simple modules in the category $Ω_{\mathbf a}^{{W}_{m,n}}$ of such modules with finite-dimensional Whittaker vector spaces and with non-singular ${\mathbf a}$.

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arXiv2026

Whittaker constructions for quantum affine algebras

The goals of the paper are 3-fold. First, we revisit the construction of imaginary Whittaker modules over untwisted affine Kac-Moody Lie algebras. These modules are obtained using the parabolic induction from irreducible Whittaker modules over the associated Heisenberg Lie algebras. We show that the infinite support condition for Whittaker functions on Heisenberg Lie algebras is essential for irreducibility: when the support is finite the modules becomes reducible, yielding infinite chains of submodules. We establish the irreducibility criterion for the induced modules over affine Lie algebras and construct a large family of such modules. In particular, we obtain a class of irreducible modules on which the derivation acts neither semisimply nor freely. Second, we consider quantum analogs of imaginary Whittaker modules and establish irreducibility for a family of such modules. Finally, we prove the irreducibility of a certain class of modules over $\mathcal{U}_q(A_1^{(1)})$, which are not quantum deformations of irreducible modules for the affine Kac-Moody Lie algebra $A_1^{(1)}$. Our results can be potentially extended to all types of untwisted quantum affine algebras, providing a pathway toward their classification.

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arXiv2026

Superelliptic Affine Lie algebras and orthogonal polynomials II

Let $\mathfrak{g}$ be a finite-dimensional complex simple Lie algebra and $r,m\ge 2$. The universal central extension of the superelliptic current algebra $\mathfrak{g}\otimes A$ is $\widehat{\mathfrak{g}\otimes A}\cong\mathfrak{g}\otimes A \oplus(Ω^1_A/dA)$, where $A=\mathbb{C}[t,t^{-1},u]/\langle u^m-(1-2ct^r+t^{2r})\rangle$. We compute the recursion relations governing a natural cocycle basis in $Ω^1_A/dA$ and encode them by generating functions admitting closed integral expressions of superelliptic type. The $2r$ possible choices of initial conditions are classified into four structural types; two canonical choices (types~1 and~2) produce two distinguished polynomial families. We prove that these polynomials satisfy fourth-order linear ordinary differential equations in~$c$, valid for all integers $r,m\ge 2$. For the type~2 family the proof combines the Picard-Fuchs theory of the superelliptic curve $u^m=1-2ct^r+t^{2r}$ with an algebraic identification of the explicit coefficient formulas via a rational-function identity argument. After a parity restriction and a reindexing, the resulting sequences are identified with associated ultraspherical polynomials. We show that, for each admissible~$n$ and $m\ge4$, the corresponding fourth-order equations admit a unique polynomial solution up to scalar multiples.

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arXiv2025

Representations of Hamiltonian Lie algebras

We consider the Shen-Larsson functor from the category of modules for the symplectic Lie algebra $\s$ to the category of modules for the Hamiltonian Lie algebra and show that it preserves the irreducibility except in the finite number of cases. The obtained irreducible modules for the Hamiltonian Lie algebra are cuspidal, whose weight multiplicities equal the dimension of the corresponding module of the symplectic Lie algebra. This extends well-known results for other Cartan type Lie algebras to the Hamiltonian case.

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arXiv2025

Dimension growth and Gelfand-Kirillov dimension of representations of quantum groups

We consider two algebraic invariants in the representation theory of quantized enveloping algebras: the dimension growth of simple modules for the De Concini-Kac quantum group at roots of unity, and the Gelfand-Kirillov dimension of simple highest weight modules for the quantum group at generic $q$. In spite of being defined for different values of the parameter $q$, these invariants reflect closely related features in the respective contexts. We show that several new phenomena appear in the quantum case and the representations with non-integral weights contribute to both invariants in a way that cannot be ignored. Building on this, we determine the minimal non-zero value of these invariants for each Lie type. As an application we show that quantum cuspidal modules at generic $q$ can occur only when the underlying semisimple Lie algebra has simple components of type $A$, $B$, or $C$, providing a more explicit representation-theoretic distinction with the classical case.

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arXiv2025

Orthogonalization and polarization of Yangians

For every family of orthogonal polynomials, we define a new realization of the Yangian of ${\mathfrak{gl}}_n$. Except in the case of Dickson polynomials, the new realizations do not satisfy the RTT relation. We obtain an analogue of the Christoffel-Darboux formula. Similar construction can be made for any family of functions satisfying certain recurrence relations, for example, $q$-Pochhhammer symbols and Bessel functions. Furthermore, using an analogue of the Jordan-Schwinger map, we define the ternary Yangian for a Lie algebra as a flat deformation of the current algebra of certain ternary extension of the given Lie algebra.

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arXiv2025

Harish-Chandra modules for map and affine Lie superalgebras

We obtain a classification of simple modules with finite weight multiplicities over basic classical map superalgebras. Any such module is parabolic induced from a simple cuspidal bounded module over a cuspidal map superalgebra. Further on, any simple cuspidal bounded module is isomorphic to an evaluation module. As an application, we obtain a classification of all simple Harish-Chandra modules for basic classical loop superalgebras. Extending these results to affine Kac-Moody Lie superalgebras obtained by adding the degree derivation, we construct a family of bounded simple modules of level zero, and conjecture that all bounded simple cuspidal modules belong to this family. Finally, we show that for affine Kac-Moody Lie superalgebras of type I the Kac induction functor reduces the classification of all simple bounded modules to the classification of the same class of modules over the even part, whose classification is claimed by Dimitrov and Grantcharov.

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arXiv2025

Gelfand-Tsetlin modules for Lie algebras of rank $2$

We explicitly construct families of simple modules for Lie algebras of rank $2$, on which certain commutative subalgebra acts diagonally and has a simple spectrum. In type $A$ these modules are well known generic Gelfand-Tsetlin modules and they can be viewed as such for other rank $2$ Lie algebras.

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arXiv2025

Explicit realization of bounded modules for symplectic Lie algebras: spinor versus oscillator

We provide an explicit combinatorial realization of all simple and injective (hence, and projective) modules in the category of bounded $\mathfrak{sp}(2n)$-modules. This realization is defined via a natural tableaux correspondence between spinor-type modules of $\mathfrak{so}(2n)$ and oscillator-type modules of $\mathfrak{sp}(2n)$. In particular, we show that, in contrast with the $A$-type case, the generic and bounded $\mathfrak{sp}(2n)$-modules admit an analog of the Gelfand-Graev continuation from finite-dimensional representations.

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arXiv2024

Superelliptic Affine Lie algebras and orthogonal polynomials

We construct two families of orthogonal polynomials associated with the universal central extensions of the superelliptic Lie algebras. These polynomials satisfy certain fourth order linear differential equations, and one of the families is a particular collection of associated ultraspherical polynomials. We show that the generating functions of the polynomials satisfy fourth order linear PDEs. Since these generating function can be represented by superelliptic integrals, we have examples of linear PDEs of fourth order with explicit solutions without complete integrability.

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arXiv2024

Generalized Grothendieck's simultaneous resolution and associated varieties of simple affine vertex algebras

The closure of a Diximier sheet is the image of a generalized Grothendieck's simultaneous resolution. We show that the associated variety of simple affine vertex algebras is contained in the closure of the Diximier sheet when a chiralization of generalized Grothendieck's simultaneous resolution exists. This generalizes in a conceptual manner the results obtained by the first named author and Anne Moreau and, in particular, allows to extend them to the types A_2, C_n, E_6, E_7.

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arXiv2024

Smooth representations of affine Kac-Moody algebras

Smooth modules for affine Kac-Moody algebras have a prime importance for the quantum field theory as they correspond to the representations of the universal affine vertex algebras. But, very little is known about such modules beyond the category of positive energy representations. We construct a new class of smooth modules over affine Kac-Moody algebras. In a particular case, these modules are isomorphic to those induced from generalized Whittaker modules for Takiff Lie algebras. We establish the irreducibility criterion for constructed modules in the case of the affine sl(2) Lie algebra.

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