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

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 $\mathcal{V}_c (R)$.