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On the quantum affine vertex algebra associated with trigonometric R-matrix


We apply the theory of \(\phi \)-coordinated modules, developed by H.-S. Li, to the Etingof–Kazhdan quantum affine vertex algebra associated with the trigonometric R-matrix of type A. We prove, for a certain associate \(\phi \) of the one-dimensional additive formal group, that any (irreducible) \(\phi \)-coordinated module for the level \(c\in {\mathbb {C}}\) quantum affine vertex algebra is naturally equipped with a structure of (irreducible) restricted level c module for the quantum affine algebra in type A and vice versa. In the end, we discuss relation between the centers of the quantum affine algebra and the quantum affine vertex algebra.

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  1. We explain the precise meaning of relations (0.1) and (0.2) in Sect. 1.2.

  2. Notice the swapped variables in this term.

  3. Note that the expression \({\iota _{x,x_0}}(1-x)^p F(x+x_0)\) is considered modulo \(U_0\) because, otherwise, the aforementioned substitution would not be well-defined (although the same substitution is well-defined when applied to \((1-x)^p F(x+x_0)\) with \(F(x+x_0)\) being regarded as a rational function with respect to the variables x and \(x_0\)).

  4. We should mention that the notation in this paper slightly differs from [26]. In particular, the algebra \(\text {U}(R)\), as defined in [26, Sect. 2], coincides with the algebra \(\overline{\mathrm{U}}_{h}^{+} ({\widehat{{\mathfrak {gl}}}}_N)\) defined in Sect. 2.1.

  5. Note that, in contrast with (2.32) and (2.34), the vectors u and v in (2.33) are swapped, so that in (3.56) we have \(T_{24}^+ (0)\) and \(T_{13}^+(0)\) instead of \(T_{23}^+ (0)\) and \(T_{14}^+(0)\).


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The author would like to thank Naihuan Jing, Mirko Primc and Martina Stojić for helpful and stimulating discussions. We would also like to thank the anonymous referee for many valuable comments and suggestions which helped us to improve the manuscript. The research reported in this paper was finalized during the author’s visit to Max Planck Institute for Mathematics in Bonn. The author is grateful to the Institute for its hospitality and financial support. This work has been supported in part by Croatian Science Foundation under the project UIP-2019-04-8488.

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Correspondence to Slaven Kožić.

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Kožić, S. On the quantum affine vertex algebra associated with trigonometric R-matrix. Sel. Math. New Ser. 27, 45 (2021).

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  • Quantum affine algebra
  • Quantum vertex algebra
  • \(\phi \)-Coordinated module
  • Quantum current

Mathematics Subject Classification

  • 17B37
  • 17B69
  • 81R50