On two families of quantum vertex algebras of FRT-type

TL;DR

This paper introduces two new families of quantum vertex algebras linked to type A trigonometric and elliptic R-matrices, exploring their representation theory and properties of quantum determinants.

Contribution

It develops the $$-coordinated representation theory for these algebras using the FRT-operator and investigates quantum determinants in the elliptic case.

Findings

Representation theory governed by FRT-operator

Applications demonstrated for the FRT-operator approach

Properties of quantum determinants analyzed in elliptic case

Abstract

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 $\phi$-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.