## Abstract

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.

This is a preview of subscription content, access via your institution.

## Notes

Notice the swapped variables in this term.

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 \(x_0\)).

## References

Bakalov, B., Kac, V.G.: Field algebras. Int. Math. Res. Not. (3), 123–159 (2003). arXiv:math/0204282 [math.QA]

Borcherds, R.: Vertex algebras, Kac–Moody algebras, and the Monster. Proc. Natl. Acad. Sci. USA

**83**, 3068–3071 (1986)Butorac, M., Jing, N., Kožić, S.: \(h\)-Adic quantum vertex algebras associated with rational \(R\)-matrix in types \(B\), \(C\) and \(D\). Lett. Math. Phys.

**109**, 2439–2471 (2019). arXiv:1904.03771 [math.QA]Cherednik, I.V.: A new interpretation of Gelfand–Tzetlin bases. Duke Math. J.

**54**, 563–577 (1987)De. Sole, A., Gardini, M., Kac, V.G.: On the structure of quantum vertex algebras. J. Math. Phys.

**61**, 011701 (2020) (29pp). arXiv:1906.05051 [math.QA]Ding, J.: Spinor representations of \(U_q({\hat{\mathfrak{gl}}} (n))\) and quantum Boson–Fermion correspondence. Commun. Math. Phys.

**200**, 399–420 (1999). arXiv:q-alg/9510014Ding, J., Frenkel, I.B.: Isomorphism of two realizations of quantum affine algebra \(U_q(\widehat{\mathfrak{gl}} (n))\). Commun. Math. Phys.

**156**, 277–300 (1993)Ding, J., Iohara, K.: Generalization of Drinfeld quantum affine algebras. Lett. Math. Phys.

**41**, 181–193 (1997). arXiv:q-alg/9608002Etingof, P., Kazhdan, D.: Quantization of Lie bialgebras, III. Selecta Math. (N.S)

**4**, 233–269 (1998). arXiv:q-alg/9610030Etingof, P., Kazhdan, D.: Quantization of Lie bialgebras, IV. Selecta Math. (N.S)

**6**, 79–104 (2000). arXiv:math/9801043 [math.QA]Etingof, P., Kazhdan, D.: Quantization of Lie bialgebras, V. Selecta Math. (N.S.)

**6**, 105–130 (2000). arXiv:math/9808121 [math.QA]Reshetikhin, NY., Takhtadzhyan, L.A., Faddeev, L.D.: Quantization of Lie groups and Lie algebras. Algebra i Analiz

**1**(1), 178–206 (1989). (Russian). English transl., Leningrad Math. J.**1**(1990), no. 1, 193–225Frappat, L., Jing, N., Molev, A., Ragoucy, E.: Higher Sugawara operators for the quantum affine algebras of type \(A\). Commun. Math. Phys.

**345**, 631–657 (2016). arXiv:1505.03667 [math.QA]Frenkel, E.: Langlands Correspondence for Loop Groups, Cambridge Studies in Advanced Mathematics, 103. Cambridge University Press, Cambridge (2007)

Frenkel, E., Ben-Zvi, D.: Vertex Algebras, Algebraic Curves, Mathematical Surveys and Monographs, vol. 88, Second ed., American Mathematical Society, Providence, RI (2004)

Frenkel, E., Reshetikhin, N.: Towards deformed chiral algebras, preprint arXiv:q-alg/9706023

Frenkel, I.B., Jing, N.: Vertex representations of quantum affine algebras. Proc. Natl. Acad. Sci. USA

**85**, 9373–9377 (1988)Frenkel, I., Lepowsky, J., Meurman, A.: Vertex Operator Algebras and the Monster, Pure and Applied Mathematics, 134. Academic Press Inc, Boston (1988)

Frenkel, I.B., Reshetikhin, NYu.: Quantum affine algebras and holonomic difference equations. Commun. Math. Phys.

**146**, 1–60 (1992)Frenkel, I.B., Zhu, Y.-C.: Vertex operator algebras associated to representations of affine and Virasoro algebras. Duke Math. J.

**66**, 123–168 (1992)Jimbo, M.: A \(q\)-difference analogue of U(G) and the Yang–Baxter equation. Lett. Math. Phys.

**10**, 63–69 (1985)Jing, N., Kožić, S., Molev, A., Yang, F.: Center of the quantum affine vertex algebra in type \(A\). J. Algebra

**496**, 138–186 (2018). arXiv:1603.00237 [math.QA]Kac, V.G.: Infinite-Dimensional Lie Algebras, 3rd edn. Cambridge University Press, Cambridge (1990)

Kac, V.: Vertex Algebras for Beginners, University Lecture Series, 10. American Mathematical Society, Providence (1997)

Kassel, C.: Quantum Groups, Graduate texts in Mathematics, vol. 155. Springer, Berlin (1995)

Kožić, S., Molev, A.: Center of the quantum affine vertex algebra associated with trigonometric \(R\)-matrix. J. Phys. A: Math. Theor.

**50**, 325201 (2017) (21pp). arXiv:1611.06700 [math.QA]Kožić, S.: Quantum current algebras associated with rational \(R\)-matrix. Adv. Math.

**351**, 1072–1104 (2019). arXiv:1801.03543 [math.QA]Lepowsky, J., Li, H.-S.: Introduction to Vertex Operator Algebras and Their Representations, Progress in Math, vol. 227. Birkhauser, Boston (2004)

Li, H.-S.: Axiomatic \(G_1\)-vertex algebras. Commun. Contemp. Math.

**5**, 281–327 (2003). arXiv:math/0204308 [math.QA]Li, H.-S.: \(\hbar \)-adic quantum vertex algebras and their modules. Commun. Math. Phys.

**296**, 475–523 (2010). arXiv:0812.3156 [math.QA]Li, H.-S.: \(\phi \)-Coordinated quasi-modules for quantum vertex algebras. Commun. Math. Phys.

**308**, 703–741 (2011). arXiv:0906.2710 [math.QA]Li, H.-S., Tan, S., Wang, Q.: Ding–Iohara algebras and quantum vertex algebras. J. Algebra

**511**, 182–214 (2018). arXiv:1706.03636 [math.QA]Lian, B.-H.: On the classification of simple vertex operator algebras. Commun. Math. Phys.

**163**, 307–357 (1994)Perk, J.H.H., Schultz, C.L.: New families of commuting transfer matrices in \(q\)-state vertex models. Phys. Lett. A

**84**, 407–410 (1981)Reshetikhin, NYu., Semenov-Tian-Shansky, M.A.: Central extensions of quantum current groups. Lett. Math. Phys.

**19**, 133–142 (1990)Stojić, M.: Construction of algebras given by generators and infinite sum relations, in preparation

## Acknowledgements

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.

## Author information

### Authors and Affiliations

### Corresponding author

## Additional information

### Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Rights and permissions

## About this article

### Cite this article

Kožić, S. On the quantum affine vertex algebra associated with trigonometric *R*-matrix.
*Sel. Math. New Ser.* **27**, 45 (2021). https://doi.org/10.1007/s00029-021-00666-x

Accepted:

Published:

DOI: https://doi.org/10.1007/s00029-021-00666-x

### Keywords

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

### Mathematics Subject Classification

- 17B37
- 17B69
- 81R50