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Slaven Kožić

Publications and source records attributed to Slaven Kožić.

At least 19 recordsLinked to original sources
arXiv2026

Evaluation-type deformed modules over the quantum affine vertex algebras of type $A$

Let $\mathcal{V}^c(\mathfrak{gl}_N)$ be Etingof-Kazhdan's quantum affine vertex algebra associated with the trigonometric $R$-matrix. We establish a connection between suitably generalized deformed $ϕ$-coordinated $\mathcal{V}^c(\mathfrak{gl}_N)$-modules and the representations of quantized enveloping algebra $U_h(\mathfrak{gl}_N)$ and reflection equation algebra $\mathcal{O}_h(Mat_N)$. As an application, we demonstrate how the elements of the center of $\mathcal{V}^c(\mathfrak{gl}_N)$ at the critical level $c=-N$ give rise to the $q$-analogues of quantum immanants for $U_h(\mathfrak{gl}_N)$, which were recently found by Jing, Liu and Molev. Finally, we derive the analogues of these results for the quantum affine vertex algebra associated with the normalized Yang $R$-matrix.

math.QA↗
arXiv2026

Double Yangian and reflection algebras of the Lie superalgebra $\mathfrak{gl}_{M|N}$, II: Quantum currents

In this paper, we continue our research on the double Yangian and reflection algebras of the Lie superalgebra $\mathfrak{gl}_{M|N}$. Extending the Etingof-Kazhdan construction, we introduce the structure of $h$-adic quantum vertex superalgebra on the vacuum module $V^c(\mathfrak{gl}_{M|N})$ over the double Yangian of level $c\in\mathbb{C}$. Next, we construct families of central elements in the completed double Yangian for $\mathfrak{gl}_{M|N}$ at the critical level. Finally, we show that the level $c$ restricted modules over the reflection algebra of $\mathfrak{gl}_{M|N}$ are naturally equipped with the structure of quasi $V^{2c}(\mathfrak{gl}_{M|N})$-module. By using this structure, we find explicit formulas for families of central elements in the completed reflection algebra of $\mathfrak{gl}_{M|N}$ at the critical level.

math.QA↗
arXiv2026

Deformed quantum vertex algebra modules associated with braidings

We introduce the notion of deformed quantum vertex algebra module associated with a braiding map. We construct two families of braiding maps over the Etingof-Kazhdan quantum vertex algebras associated with the rational $R$-matrices of classical types. We investigate their properties and demonstrate the applications of the corresponding deformed modules to the (generalized) Yangians and reflection algebras.

math.QA↗
arXiv2026

Invariants of the extended twisted $h$-Yangian

We investigate the extended twisted $h$-Yangian ${\rm Y}_{N,h}^{\text{tw}}$, a certain algebra which admits both the orthogonal and the symplectic $h$-Yangian as its quotients. We show that ${\rm Y}_{N,h}^{\text{tw}}$ is naturally equipped with the structure of restricted module for a certain algebra ${\rm DY}_{N,h,c}^{tw}$, which resembles the quantum double, as well as with the structure of $ϕ$-coordinated quasi module for the Etingof-Kazhdan quantum affine vertex algebra associated with the trigonometric $R$-matrix of type $A$. Finally, we demonstrate how the elements of the quantum Feigin--Frenkel center give rise to explicit formulas for central elements of ${\rm DY}_{N,h,c}^{tw}$ and invariants of ${\rm Y}_{N,h}^{\text{tw}}$ at the critical level, as well as to commutative families in the orthogonal and symplectic $h$-Yangians.

math.QA↗
arXiv2026

Quasi-particles and the Kanade-Russell and Kurşungöz formula for Capparelli's identity

We construct a quasi-particle basis of the integrable highest weight module of highest weight $3Λ_0$ for the twisted affine Lie algebra of type $A_2^{(2)}$ in the principal realization. More specifically, by introducing the concept of polychromatic quasi-particle and finding relations among quasi-particles, we construct the spanning set of the standard module. Finally, its linear independence is proved by using Kanade-Russell and Kurşungöz's Andrews-Gordon type series of Capparelli's identities.

math.QA↗
arXiv2026

Associating modules for the $h$-Yangian and quantum elliptic algebra in type $A$ with $h$-adic quantum vertex algebras

We consider the Etingof-Kazhdan quantum vertex algebra $\mathcal{V}^c(R)$ associated with the trigonometric and elliptic $R$-matrix of type $A.$ We establish a connection between (restricted) modules for the $h$-Yangian $\textrm{Y}_h(\mathfrak{gl}_N)$ and the elliptic quantum algebra $\mathcal{A}_{h,p}(\widehat{\mathfrak{gl}}_2)$ of level zero, and deformed (twisted) $ϕ$-coordinated $\mathcal{V}^c(R)$-modules. As its application, in the trigonometric case, we construct new families of central elements of $\mathcal{V}^c(R)$ at the critical level $c=-N,$ which we then use to derive commutative families in the $h$-Yangian $\textrm{Y}_h(\mathfrak{gl}_N).$

math.QA↗
arXiv2025

On two families of quantum vertex algebras of FRT-type

We consider two new families of quantum vertex algebras which are associated with the type $A$ trigonometric $R$-matrix and elliptic $R$-matrix of the eight-vertex model. We show that their $ϕ$-coordinated representation theory is governed by the so-called FRT-operator, $h$-adically restricted operator satisfying the FRT-relation, and we demonstrate some applications of this result. Finally, in the elliptic case, we investigate the properties of the quantum determinant associated with the corresponding quantum vertex algebra.

math.QA↗
arXiv2024

Associating deformed $ϕ$-coordinated modules for the quantum affine vertex algebra with orthogonal twisted $h$-Yangians

We consider the Etingof-Kazhdan quantum vertex algebra $\mathcal{V}^c(\mathfrak{gl}_N)$ associated with the trigonometric $R$-matrix of type $A$. By combining Li's theory of $ϕ$-coordinated modules and the ideas from our previous paper, we introduce the notion of deformed $ϕ$-coordinated quantum vertex algebra module. We show that the orthogonal twisted $h$-Yangians and restricted modules for the generalized orthogonal twisted $h$-Yangians can be equipped with the structure of (truncated) deformed $ϕ$-coordinated $\mathcal{V}^c(\mathfrak{gl}_N)$-module and demonstrate its applications.

math.QA↗
arXiv2024

Lepowsky's and Wakimoto's product formulas for the affine Lie algebras $C_l^{(1)}$

In this paper, we recall Lepowsky's and Wakimoto's product character formulas formulated in a new way by using arrays of specialized weighted crystals of negative roots for affine Lie algebras of type $C_l^{(1)}$, $D_{l+1}^{(2)}$ and $A_{2l}^{(2)}$. Lepowsky-Wakimoto's infinite periodic products appear as one side of (conjectured) Rogers-Ramanujan-type combinatorial identities for affine Lie algebras of type $C_l^{(1)}$.

math.RT↗
arXiv2023

Double Yangian and reflection algebras of the Lie superalgebra $\mathfrak{gl}_{m|n}$

We study the double Yangian associated with the Lie superalgebra $\mathfrak{gl}_{m|n}$. Our main focus is on establishing the Poincaré-Birkhoff-Witt Theorem for the double Yangian and constructing its central elements in the form of coefficients of the quantum contraction. Next, as an application, we introduce reflection algebras, certain left coideal subalgebras of the level 0 double Yangian, and find their presentations by generators and relations.

math.QA↗
arXiv2023

Yangian deformations of $\mathcal{S}$-commutative quantum vertex algebras and Bethe subalgebras

We construct a new class of quantum vertex algebras associated with the normalized Yang $R$-matrix. They are obtained as Yangian deformations of certain $\mathcal{S}$-commutative quantum vertex algebras and their $\mathcal{S}$-locality takes the form of a single $RTT$-relation. We establish some preliminary results on their representation theory and then further investigate their braiding map. In particular, we show that its fixed points are closely related with Bethe subalgebras in the Yangian quantization of the Poisson algebra $\mathcal{O}(\mathfrak{gl}_N((z^{-1})))$, which were recently introduced by Krylov and Rybnikov. Finally, we extend this construction of commutative families to the case of trigonometric $R$-matrix of type $A$.

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arXiv2023

Semi-infinite construction for the double Yangian of type $A_1^{(1)}$

We consider certain infinite dimensional modules of level 1 for the double Yangian $\text{DY}(\mathfrak{gl}_2)$ which are based on the Iohara-Kohno realization. We show that they possess topological bases of Feigin-Stoyanovsky-type, i.e. the bases expressed in terms of semi-infinite monomials of certain integrable operators which stabilize and satisfy the difference two condition. Finally, we give some applications of these bases to the representation theory of the corresponding quantum affine vertex algebra.

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arXiv2022

Combinatorial bases of standard modules of twisted affine Lie algebras in types $A_{2l-1}^{(2)}$ and $D_{l+1}^{(2)}$: rectangular highest weights

We consider the standard modules of rectangular highest weights of affine Lie algebras in types $A_{2l-1}^{(2)}$ and $D_{l+1}^{(2)}$. By using vertex algebraic techniques we construct the combinatorial bases for standard modules and their principal subspaces and parafermionic spaces. Finally, we compute the corresponding character formulae and, as an application, we obtain two new families of combinatorial identities.

math.RT↗
arXiv2022

A note on constructing quasi modules for quantum vertex algebras from twisted Yangians

In this note, we consider the twisted Yangians $\text{Y}(\mathfrak{g}_N)$ associated with the orthogonal and symplectic Lie algebras $\mathfrak{g}_N=\mathfrak{o}_N,\mathfrak{sp}_N$. First, we introduce a certain subalgebra $\text{A}_c(\mathfrak{g}_N)$ of the double Yangian for $\mathfrak{gl}_N$ at the level $c\in\mathbb{C}$, which contains the centrally extended $\text{Y}(\mathfrak{g}_N)$ at the level $c$ as well as its vacuum module $\mathcal{M}_c(\mathfrak{g}_N)$. Next, we employ its structure to construct examples of quasi modules for the quantum affine vertex algebra $\mathcal{V}_c(\mathfrak{gl}_N)$ associated with the Yang $R$-matrix. Finally, we use the description of the center of $\mathcal{V}_c(\mathfrak{gl}_N)$ to obtain explicit formulae for families of central elements for a certain completion of $\text{A}_c(\mathfrak{g}_N)$ and invariants of $\mathcal{M}_c(\mathfrak{g}_N)$.

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arXiv2022

On the $h$-adic quantum vertex algebras associated with Hecke symmetries

We study the quantum vertex algebraic framework for the Yangians of RTT-type and the braided Yangians associated with Hecke symmetries, introduced by Gurevich and Saponov. First, we construct several families of modules for the aforementioned Yangian-like algebras which, in the RTT-type case, lead to a certain $h$-adic quantum vertex algebra $\mathcal{V}_c (R)$ via the Etingof-Kazhdan construction, while, in the braided case, they produce ($ϕ$-coordinated) $\mathcal{V}_c (R)$-modules. Next, we show that the coefficients of suitably defined quantum determinant can be used to obtain central elements of $\mathcal{V}_c (R)$, as well as the invariants of such ($ϕ$-coordinated) $\mathcal{V}_c (R)$-modules. Finally, we investigate a certain algebra which is closely connected with the representation theory of $\mathcal{V}_c (R)$.

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arXiv2021

$h$-adic quantum vertex algebras in types $B$, $C$, $D$ and their $ϕ$-coordinated modules

We introduce the $h$-adic quantum vertex algebras associated with the trigonometric $R$-matrices in types $B$, $C$ and $D$, thus generalizing the well-known Etingof-Kazhdan construction in type $A$. We show that restricted modules for quantum affine algebras in types $B$, $C$ and $D$ are naturally equipped with the structure of $ϕ$-coordinated module for the aforementioned $h$-adic quantum vertex algebras.

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arXiv2021

On the Heisenberg algebra associated with the rational $R$-matrix

We associate a deformation of Heisenberg algebra to the suitably normalized Yang $R$-matrix and we investigate its properties. Moreover, we construct new examples of quantum vertex algebras which possess the same representation theory as the aforementioned deformed Heisenberg algebra.

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