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Lucia Bagnoli

Publications and source records attributed to Lucia Bagnoli.

14 recordsLinked to original sources
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

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

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↗
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 $A$.

math.QA↗
arXiv2022

Homology of the complexes of finite Verma modules over $CK_6$

We compute the homology of the first and third quadrants of the complexes of finite Verma modules over the annihilation superalgebra $\mathcal{A}(CK_{6})\cong E(1,6)$, associated with the conformal superalgebra $CK_6$, obtained in \cite{ck6}. This computation allows us to explicitly realize the irreducible quotients of degenerate finite Verma modules over $\mathcal{A}(CK_{6})$.

math.RT↗
arXiv2021

Computation of the homology of the complexes of finite Verma modules for $K'_4$

We compute the homology of the complexes of finite Verma modules over the annihilation superalgebra $\mathcal A(K'_{4})$, associated with the conformal superalgebra $K'_{4}$, obtained in \cite{K4}. We use the computation of the homology in order to provide an explicit realization of all the irreducible quotients of finite Verma modules over $\mathcal A(K'_{4})$.

math.RT↗
arXiv2021

An upper bound on the degree of singular vectors for $E(1,6)$

The aim of this work is to prove a technical result, that had been stated by Boyallian, Kac and Liberati \cite{ck6}, on the degree of singular vectors of finite Verma modules over the exceptional Lie superalgebra $E(1,6)$ that is isomorphic to the annihilation superalgebra associated with the conformal superalgebra $CK_{6}$.

math.RT↗
arXiv2021

Classification of finite irreducible conformal modules for $K'_4$

We classify the finite irreducible modules over the conformal superalgebra $K'_{4}$ by their correspondence with finite conformal modules over the associated annihilation superalgebra $\mathcal A(K'_{4})$. This is achieved by a complete classification of singular vectors in generalized Verma modules for $\mathcal A(K'_{4})$. We also show that morphisms between generalized Verma modules can be arranged in infinitely many bilateral complexes.

math.RT↗