arXiv subjects

Alexander Molev

Publications and source records attributed to Alexander Molev.

At least 19 recordsLinked to original sources

arXiv2026

Harish-Chandra images of orthosymplectic Sugawara operators and Casimir elements

We consider the recently constructed Segal--Sugawara vectors for the orthosymplectic Lie superalgebras. We calculate their images with respect to the Harish-Chandra homomorphism and extend this calculation to the associated Sugawara operators and Casimir elements. We also produce higher Gaudin Hamiltonians and elements of the quantum shift-of-argument subalgebras in the orthosymplectic enveloping algebra. In the Appendix, we review analogous results for the general linear Lie superalgebras.

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arXiv2025

Universal Capelli identities and quantum immanants for the queer Lie superalgebra

We apply the recently introduced idempotents for the Sergeev superalgebra to construct quantum immanants for the queer Lie superalgebra ${\mathfrak q}_N$ as central elements of its universal enveloping algebra. We prove universal odd and even Capelli identities for ${\mathfrak q}_N$ and use them to calculate the images of the quantum immanants under the action of ${\mathfrak q}_N$ in differential operators. We show that the Harish-Chandra images of the quantum immanants coincide with the factorial Schur $Q$-polynomials.

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arXiv2025

Central elements and evaluation map for the quantum queer superalgebras

We consider the $R$-matrix presentations of the quantum queer superalgebra $U_q(q_n)$ and its affine counterpart $U_q(\widehat q_n)$. We derive crossing symmetry relations for the $R$-matrices and use them to construct central elements in both superalgebras. We also produce an epimorphism $ev:U_q(\widehat q_n)\to U_q(q_n)$ identical on the subalgebra isomorphic to $U_q(q_n)$.

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arXiv2025

Segal-Sugawara vectors for orthosymplectic Lie superalgebras

We consider the centre of the affine vertex algebra at the critical level associated with the orthosymplectic Lie superalgebra. It is well-known that the centre is a commutative superalgebra, and we construct a family of its elements in an explicit form. In particular, this gives a new proof of the formulas for the central elements for the orthogonal and symplectic Lie algebras. Our arguments rely on the properties of a new extended Brauer-type algebra.

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arXiv2025

On the Jucys-Murphy method and fusion procedure for the Sergeev superalgebra

We use the Jucys-Murphy elements to construct a complete set of primitive idempotents for the Sergeev superalgebra ${\mathcal S}_n$. We produce seminormal forms for the simple modules over ${\mathcal S}_n$ and over the spin symmetric group algebra with explicit constructions of basis vectors. We show that the idempotents can also be obtained from a new version of the fusion procedure.

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arXiv2024

Generalized finite and affine $W$-algebras in type $A$

We construct a new family of affine $W$-algebras $W^k(\lambda,\mu)$ parameterized by partitions $\lambda$ and $\mu$ associated with the centralizers of nilpotent elements in $\mathfrak{gl}_N$. The new family unifies a few known classes of $W$-algebras. In particular, for the column-partition $\lambda$ we recover the affine $W$-algebras $W^k(\mathfrak{gl}_N,f)$ of Kac, Roan and Wakimoto, associated with nilpotent elements $f\in\mathfrak{gl}_N$ of type $\mu$. Our construction is based on a version of the BRST complex of the quantum Drinfeld-Sokolov reduction. We show that the application of the Zhu functor to the vertex algebras $W^k(\lambda,\mu)$ yields a family of generalized finite $W$-algebras $U(\lambda,\mu)$ which we also describe independently as associative algebras.

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arXiv2024

The $q$-immanants and higher quantum Capelli identities

We construct polynomials ${\mathbb{S}}_{\mu}(z)$ parameterized by Young diagrams $\mu$, whose coefficients are central elements of the quantized enveloping algebra ${\rm U}_q({\mathfrak{gl}}_n)$. Their constant terms coincide with the central elements provided by the general construction of Drinfeld and Reshetikhin. For another special value of $z$, we get $q$-analogues of Okounkov's quantum immanants for ${\mathfrak{gl}}_n$. We show that the Harish-Chandra image of ${\mathbb{S}}_{\mu}(z)$ is a factorial Schur polynomial. We also prove quantum analogues of the higher Capelli identities and derive Newton-type identities.

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arXiv2023

Eigenvalues of quantum Gelfand invariants

We consider the quantum Gelfand invariants which first appeared in a landmark paper by Reshetikhin, Takhtadzhyan and Faddeev (1989). We calculate the eigenvalues of the invariants acting in irreducible highest weight representations of the quantized enveloping algebra for ${\mathfrak {gl}}_n$. The calculation is based on Liouville-type formulas relating two families of central elements in the quantum affine algebras of type $A$.

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arXiv2023

Representations of the super-Yangian of type $B(n,m)$

We are concerned with finite-dimensional irreducible representations of the Yangians associated with the orthosymplectic Lie superalgebras ${\frak osp}_{2n+1|2m}$. Every such representation is highest weight and we use embedding theorems and odd reflections of Yangian type to derive necessary conditions for an irreducible highest weight representation to be finite-dimensional. We conjecture that these conditions are also sufficient. We prove the conjecture in the case $n=1$ and arbitrary $m\geqslant 1$.

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arXiv2023

On the quantum argument shift method

In a recent work by two of us the argument shift method was extended from the symmetric algebra ${\rm S}({\mathfrak g})$ of the general linear Lie algebra ${\mathfrak g}$ to the universal enveloping algebra ${\rm U}({\mathfrak g})$. We show in this paper that some features of this 'quantum argument shift method' can be applied to the remaining classical matrix Lie algebras ${\mathfrak g}$. We prove that a single application of the quasi-derivation to central elements of ${\rm U}({\mathfrak g})$ yields elements of the corresponding quantum Mishchenko-Fomenko subalgebra. We show that generators of this subalgebra can be obtained by iterated application of the quasi-derivation to generators of the center of ${\rm U}({\mathfrak g})$.

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arXiv2023

Gaussian generators for the Yangian associated with the Lie superalgebra $\mathfrak{osp}(1|2m)$

We give a new presentation of the Yangian for the orthosymplectic Lie superalgebra $\mathfrak{osp}_{1|2m}$. It relies on the Gauss decomposition of the generator matrix in the $R$-matrix presentation. The defining relations between the Gaussian generators are derived from a new version of the Drinfeld-type presentation of the Yangian for $\mathfrak{osp}_{1|2}$ and some additional relations in the Yangian for $\mathfrak{osp}_{1|4}$ by an application of the embedding theorem for the super-Yangians.

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arXiv2022

Quantum Sugawara operators in type $A$

We construct Sugawara operators for the quantum affine algebra of type $A$ in an explicit form. The operators are associated with primitive idempotents of the Hecke algebra and parameterized by Young diagrams. This generalizes a previous construction (2016) where one-column diagrams were considered. We calculate the Harish-Chandra images of the Sugawara operators and identify them with the eigenvalues of the operators acting in the $q$-deformed Wakimoto modules.

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arXiv2020

Representations of Quantum Affine Algebras in their $R$-Matrix Realization

We use the isomorphisms between the $R$-matrix and Drinfeld presentations of the quantum affine algebras in types $B$, $C$ and $D$ produced in our previous work to describe finite-dimensional irreducible representations in the $R$-matrix realization. We also review the isomorphisms for the Yangians of these types and use Gauss decomposition to establish an equivalence of the descriptions of the representations in the $R$-matrix and Drinfeld presentations of the Yangians.

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arXiv2020

Isomorphism between the R-matrix and Drinfeld presentations of quantum affine algebra: type C

An explicit isomorphism between the $R$-matrix and Drinfeld presentations of the quantum affine algebra in type $A$ was given by Ding and I. Frenkel (1993). We show that this result can be extended to types $B$, $C$ and $D$ and give a detailed construction for type $C$ in this paper. In all classical types the Gauss decomposition of the generator matrix in the $R$-matrix presentation yields the Drinfeld generators. To prove that the resulting map is an isomorphism we follow the work of E. Frenkel and Mukhin (2002) in type $A$ and employ the universal $R$-matrix to construct the inverse map. A key role in our construction is played by a homomorphism theorem which relates the quantum affine algebra of rank $n-1$ in the $R$-matrix presentation with a subalgebra of the corresponding algebra of rank $n$ of the same type.

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arXiv2020

Isomorphism between the $R$-Matrix and Drinfeld Presentations of Quantum Affine Algebra: Types $B$ and $D$

Following the approach of Ding and Frenkel [Comm. Math. Phys. 156 (1993), 277-300] for type $A$, we showed in our previous work [J. Math. Phys. 61 (2020), 031701, 41 pages] that the Gauss decomposition of the generator matrix in the $R$-matrix presentation of the quantum affine algebra yields the Drinfeld generators in all classical types. Complete details for type $C$ were given therein, while the present paper deals with types $B$ and $D$. The arguments for all classical types are quite similar so we mostly concentrate on necessary additional details specific to the underlying orthogonal Lie algebras.

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arXiv2020

Isomorphism between the $R$-matrix and Drinfeld presentations of Yangian in types $B$, $C$ and $D$

It is well-known that the Gauss decomposition of the generator matrix in the $R$-matrix presentation of the Yangian in type $A$ yields generators of its Drinfeld presentation. Defining relations between these generators are known in an explicit form thus providing an isomorphism between the presentations. It has been an open problem since the pioneering work of Drinfeld to extend this result to the remaining types. We give a solution for the classical types $B$, $C$ and $D$ by constructing an explicit isomorphism between the $R$-matrix and Drinfeld presentations of the Yangian. It is based on an embedding theorem which allows us to consider the Yangian of rank $n-1$ as a subalgebra of the Yangian of rank $n$ of the same type.

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arXiv2019

Monomial bases and branching rules

Following a question of Vinberg, a general method to construct monomial bases for finite-dimensional irreducible representations of a reductive Lie algebra was developed in a series of papers by Feigin, Fourier, and Littelmann. Relying on this method, we construct monomial bases of multiplicity spaces associated with the restriction of the representation to a reductive subalgebra. As an application, we produce monomial bases for representations of the general linear and symplectic Lie algebras associated with natural chains of subalgebras. We also show that our basis in type A is related to both the Gelfand-Tsetlin basis and the Littelmann basis via triangular transition matrices which implies that the triangularity property extends to the matrix connecting the Gelfand-Tsetlin and canonical bases. A similar relationship holds between our basis in type C and a suitably modified version of the basis constructed earlier by the first author.

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arXiv2019

Quantisation and nilpotent limits of Mishchenko-Fomenko subalgebras

For any simple Lie algebra $\mathfrak{g}$ and an element $μ\in\mathfrak{g}^*$, the corresponding commutative subalgebra $\mathcal{A}_μ$ of $\mathcal{U}(\mathfrak{g})$ is defined as a homomorphic image of the Feigin-Frenkel centre associated with $\mathfrak{g}$. It is known that when $μ$ is regular this subalgebra solves Vinberg's quantisation problem, as the graded image of $\mathcal{A}_μ$ coincides with the Mishchenko-Fomenko subalgebra $\overline{\mathcal{A}}_μ$ of $\mathcal{S}(\mathfrak{g})$. By a conjecture of Feigin, Frenkel and Toledano Laredo, this property extends to an arbitrary element $μ$. We give sufficient conditions which imply the property for certain choices of $μ$. In particular, this proves the conjecture in type C and gives a new proof in type A. We show that the algebra $\mathcal{A}_μ$ is free in both cases and produce its generators in an explicit form. Moreover, we prove that in all classical types generators of $\mathcal{A}_μ$ can be obtained via the canonical symmetrisation map from certain generators of $\overline{\mathcal{A}}_μ$. The symmetrisation map is also used to produce free generators of nilpotent limits of the algebras $\mathcal{A}_μ$ and give a positive solution of Vinberg's problem for these limit subalgebras.

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