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

TL;DR

This paper develops the quantum vertex algebra framework for Yangians related to Hecke symmetries, constructing modules, identifying central elements, and exploring connections with representation theory.

Contribution

It introduces new module constructions for Yangian-like algebras and links them to $h$-adic quantum vertex algebras, advancing understanding of their structure and invariants.

Findings

Constructed modules for Yangian-like algebras leading to quantum vertex algebras.

Identified quantum determinants as sources of central elements and invariants.

Explored algebraic connections with the representation theory of these quantum structures.

Abstract

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 ($\phi$-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 ($\phi$-coordinated) $\mathcal{V}_c (R)$-modules. Finally, we investigate a certain algebra which is closely connected with the representation theory of $\mathcal{V}_c (R)$.