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

Abstract: Let $\mathcal{V}<sup>c(\mathfrak{gl}_N)$ be Etingof--Kazhdan's quantum affine vertex algebra associated with the trigonometric $R$-matrix. We establish a connection between suitably generalized deformed $φ$-coordinated $\mathcal{V}<sup>c(\mathfrak{gl}_N)$-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 $\mathcal{V}<sup>c(\mathfrak{gl}_N)$ 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.

• The paper establishes a correspondence between evaluation-type deformed modules and representations of Uₕ(glₙ) and Oₕ(Matₙ) using an extended φ‐coordinated module theory.
• It generalizes the φ‐coordinated module framework by introducing weakly compatible pairs that control nonassociativity and braided commutativity in a quantum setting.
• At the critical level, the construction yields q-analogues of quantum immanants, unveiling new algebraic symmetries and deep links to quantum invariant theory.

Evaluation-Type Deformed Modules over Quantum Affine Vertex Algebras of Type $A$

Introduction and Objectives

The paper "Evaluation-type deformed modules over the quantum affine vertex algebras of type $A$" (2604.12842) aims to establish a rigorous connection between Etingof–Kazhdan's quantum affine vertex algebra $\mathcal{V}^c(\mathfrak{gl}_N)$—associated with the trigonometric $R$-matrix—and representations of both the quantized enveloping algebra $U_h(\mathfrak{gl}_N)$ and the reflection equation algebra $\mathcal{O}_h(Mat_N)$. The central technical innovation is a generalization of $\phi$-coordinated module theory to accommodate deformation and non-associativity governed by weakly compatible pairs. Furthermore, the authors show that the center of $\mathcal{V}^c(\mathfrak{gl}_N)$ at critical level $c=-N$ facilitates the construction of $q$-analogues of quantum immanants for $A$0. These results are also extended to the rational case via the normalized Yang $A$1-matrix and double Yangian modules.

Structural Foundations

Quantum Affine Vertex Algebra and the Trigonometric $A$2-Matrix

The construction begins with a detailed presentation of $A$3 as an $A$4-adic quantized enveloping algebra via the RTT framework with the trigonometric $A$5-matrix. The Faddeev–Reshetikhin–Takhtajan (FRT) presentation is used, parameterizing the algebra by two sets of generators $A$6 and organizing relations through tensorial $A$7-type commutation relations.

The reflection equation algebra $A$8 is constructed analogously, expressed in terms of the $A$9-matrix and permutation operators, incorporating the structure required for braided commutativity and covariance with respect to $\mathcal{V}^c(\mathfrak{gl}_N)$0.

Quantum Vertex Algebra Structure and Braiding

Employing Etingof-Kazhdan's theory, the quantum affine vertex algebra $\mathcal{V}^c(\mathfrak{gl}_N)$1 is defined over the $\mathcal{V}^c(\mathfrak{gl}_N)$2-adic topology, with its field assignments and operator product encapsulated via explicit RTT-type relations for the generator matrices $\mathcal{V}^c(\mathfrak{gl}_N)$3. The vacuum module structure is specified, and the quantum vertex algebra structure is made explicit, with the vertex operator

$\mathcal{V}^c(\mathfrak{gl}_N)$4

and an $\mathcal{V}^c(\mathfrak{gl}_N)$5-matrix-based braiding map $\mathcal{V}^c(\mathfrak{gl}_N)$6 satisfying the additive quantum Yang-Baxter equation.

Weakly Compatible Pairs and Deformed $\mathcal{V}^c(\mathfrak{gl}_N)$7-Coordinated Modules

Weakly Compatible Pairs

A core contribution is the definition and construction of a multiplicative weakly compatible pair $\mathcal{V}^c(\mathfrak{gl}_N)$8 encoding, respectively, a multiplicative YBE solution and an intertwiner, both regular at $\mathcal{V}^c(\mathfrak{gl}_N)$9 and related via a controlled "nonassociativity" operator. The pair provides the correct formalism to connect the additive structure of the quantum vertex algebra with multiplicative structures that naturally arise in modules for $R$0 and $R$1.

Deformed $R$2-Coordinated Module Theory

Generalizing Li's $R$3-coordinated module framework, the paper introduces the notion of weak $R$4-deformed $R$5-coordinated modules. In contrast to standard modules, associativity now holds only up to specific controlled deformations governed by $R$6. The axioms are explicitly formulated: weak $R$7-associativity, $R$8-locality (encoding braided commutativity), and a vacuum property for the vertex operator map.

Main Results and Their Implications

Connection to $R$9 and $U_h(\mathfrak{gl}_N)$0

A primary result is that for any topologically free $U_h(\mathfrak{gl}_N)$1-module $U_h(\mathfrak{gl}_N)$2, there exists a unique deformed $U_h(\mathfrak{gl}_N)$3-coordinated $U_h(\mathfrak{gl}_N)$4-module structure with vertex operator map acting via the algebra's evaluation representation. Explicitly,

$U_h(\mathfrak{gl}_N)$5

where $U_h(\mathfrak{gl}_N)$6 are polynomials in $U_h(\mathfrak{gl}_N)$7 constructed from the quantized algebra generators. This intertwining precisely encodes representation data lifted to the vertex algebra context.

Furthermore, modules satisfying mild regularity conditions yield, via evaluation at suitable points, representations of the reflection equation algebra $U_h(\mathfrak{gl}_N)$8. The reflection equation—an $U_h(\mathfrak{gl}_N)$9-type relation for the images of the quantum vertex algebra generators—naturally holds in this setting, confirming that these modules encompass solutions to the reflection equation in the $\mathcal{O}_h(Mat_N)$0-adic framework.

Central Elements and Quantum Immanants

At the critical level $\mathcal{O}_h(Mat_N)$1, the explicit construction of families of central elements in $\mathcal{O}_h(Mat_N)$2 is aligned with the classical Feigin-Frenkel center in the $\mathcal{O}_h(Mat_N)$3 limit. Under the deformed $\mathcal{O}_h(Mat_N)$4-coordinated module functor, these central elements are mapped to $\mathcal{O}_h(Mat_N)$5-analogues of quantum immanants for $\mathcal{O}_h(Mat_N)$6. The construction leverages the fusion procedure for the Hecke algebra and combinatorics of standard tableaux, producing symmetrized elements characterized as idempotents in the Hecke algebra which project to central polynomials—explicitly matching the $\mathcal{O}_h(Mat_N)$7-immanants recently studied in the quantum Capelli context by Jing, Liu, and Molev.

Extension to Double Yangian and Rational $\mathcal{O}_h(Mat_N)$8-Matrix

The formalism extends to the quantum affine vertex algebra associated with the normalized Yang $\mathcal{O}_h(Mat_N)$9-matrix, realized in the double Yangian context. Here, the compatible pairs are formulated additively, and the module theory specializes to deformed modules over $\phi$0, with resulting central elements providing a homomorphic image of Okounkov's quantum immanants in the classical universal enveloping algebra framework.

Theoretical and Practical Implications

The development of deformed $\phi$1-coordinated module theory enables a formalism that consistently relates representations of quantum vertex algebras and quantum groups—even in the presence of non-standard braidings (e.g., trigonometric and rational $\phi$2-matrices) and with $\phi$3-adic completions. The connections to quantum immanants and higher Capelli elements signal new algebraic symmetries, potentially impacting the classification of quantum invariants and the explicit realization of quantum transfer matrices in integrable systems. On the theoretical side, the work provides structural tools for investigating centers and module categories for quantum affine and double Yangian algebras.

Conclusion

This paper systematically develops the theory of evaluation-type deformed $\phi$4-coordinated modules for quantum affine vertex algebras of type $\phi$5, forging a robust connection to both $\phi$6 and $\phi$7 representations. The treatment of weak compatibility, deformed associativity, and extension to rational cases significantly broadens the landscape of quantum vertex module theory. The realization of quantum immanants via the vertex algebraic centers not only solidifies structural analogues with classical objects but also bridges contemporary developments in quantum symmetric functions and quantum invariant theory. Potential future directions include extension to other types or generalized quantum symmetric pairs, further exploration of categorification aspects, or applications to representation theory in the context of boundary integrable models.