On the quantum affine vertex algebra associated with trigonometric $R$matrix
Abstract
We apply the theory of $\phi$coordinated modules, developed by H.S. Li, to the EtingofKazhdan quantum affine vertex algebra associated with the trigonometric $R$matrix of type $A$. We prove, for a certain associate $\phi$ of the onedimensional additive formal group, that any $\phi$coordinated module for the level $c\in\mathbb{C}$ quantum affine vertex algebra is naturally equipped with a structure of restricted level $c$ module for the quantum affine algebra in type $A$ and vice versa. Moreover, we show that any $\phi$coordinated module is irreducible with respect to the action of the quantum affine vertex algebra if and only if it is irreducible with respect to the corresponding action of the quantum affine algebra. In the end, we discuss relation between the centers of the quantum affine algebra and the quantum affine vertex algebra.
 Publication:

arXiv eprints
 Pub Date:
 August 2019
 arXiv:
 arXiv:1908.06517
 Bibcode:
 2019arXiv190806517K
 Keywords:

 Mathematics  Quantum Algebra;
 Mathematical Physics;
 Mathematics  Representation Theory;
 17B37;
 17B69;
 81R50
 EPrint:
 34 pages. Main Theorem extended to $\mathfrak{sl}_N$. Subsect.3.4 and Sect.4 added. Other minor changes. Comments are welcome