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

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## Notes

Equivalently, \(\mathop {\underset{\text {LR}}{\cdot }}\) is the standard multiplication in the algebra \({\mathrm {End}}\,{\mathbb {C}}^N \otimes ({\mathrm {End}}\,{\mathbb {C}}^N )^{op}\) and \(\mathop {\underset{\text {RL}}{\cdot }}\) is the product in \(({\mathrm {End}}\,{\mathbb {C}}^N)^{op}\otimes {\mathrm {End}}\,{\mathbb {C}}^N \), where \(A^{op}\) denotes the opposite algebra of

*A*.

## References

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]Drinfeld, V.G.: Hopf algebras and the quantum Yang-Baxter equation. Soviet Math. Dokl.

**32**, 254–258 (1985)Drinfeld, V.G.: Quantum groups, Proceedings of the International Congress of Mathematicians

**1**(Berkeley, Calif., 1986), 798–820, Amer. Math. Soc., Providence, RI (1987)Etingof, 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]Frenkel, E., Ben-Zvi, D.: Vertex Algebras, Algebraic Curves, Mathematical Surveys and Monographs, vol. 88, Second ed., American Mathematical Society, Providence, RI (2004)

Frenkel, I., Lepowsky, J., Meurman, A.: Vertex operator algebras and the Monster. Pure and Applied Mathematics, 134. Academic Press Inc, Boston, MA (1988)

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

**146**, 1–60 (1992)Gardini, M.: Quantum vertex algebras, Ph.D. thesis, Sapienza – University of Rome (2018)

Gurevich, D.: Algebraic aspects of quantum Yang-Baxter equation. Algebra i Analiz

**2**, 119–148 (1990) (in Russian). translation in Leningr. Math. J.**2**(1991), 801–828.Gurevich, D.: Hecke Symmetries and Braided Lie Algebras, in Spinors, Twistors, Clifford Algebras and Quantum Deformation, pp. 317–326. Kluwer Acad. Publ, Dordrecht (1993)

Gurevich, D., Pyatov, P., Saponov, P.: Representation theory of (modified) Reflection Equation Algebra of the \(GL(m|n)\) type. Algebra i Analiz

**20**, 70–133 (2008) (in Russian). translation in St. Petersburg Math. J.**20**(2009), 213–253. arXiv:math/0612815 [math.QA]Gurevich, D., Saponov, P.: Braided Yangians. J. Geom. Phys.

**138**, 124–143 (2019). arXiv:1612.05929 [math.QA]Gurevich, D., Saponov, P., Slinkin, A.: Bethe subalgebras in braided Yangians and Gaudin-type models. Comm. Math. Phys.

**374**, 689–704 (2020). arXiv:1810.03126 [math.QA]Hadjiivanov, L.K., Isaev, A.P., Ogievetsky, O.V., Pyatov, P.N., Todorov, I.T.: Hecke algebraic properties of dynamical \(R\)-matrices. Application to related quantum matrix algebras. J. Math. Phys.

**40**, 427–448 (1999). arXiv:q-alg/9712026Iohara, K.: Bosonic representations of Yangian double \(DY_{\hbar }({\mathfrak{g} })\) with \({\mathfrak{g} }={\mathfrak{g} \mathfrak{l} }_N,{\mathfrak{sl} }_N\). J. Phys. A

**29**, 4593–4621 (1996). arXiv:q-alg/9603033Jimbo, M.: A \(q\)-difference analogue of \(U (g)\) and the Yang-Baxter equation. Lett. Math. Phys.

**10**, 63–69 (1985)Jing, N., Kong, F., Li, H.-S., Tan, S.: \((G,\chi _\phi )\)-equivariant \(\phi \)-coordinated quasi modules for nonlocal vertex algebras. J. Algebra

**570**, 24–74 (2021). arXiv:2008.05982 [math.QA]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.: Vertex algebras for beginners. University Lecture Series, 10. American Mathematical Society, Providence, RI (1997)

Kassel, C.: Quantum Groups. Graduate texts in mathematics; vol.

**155**, Springer-Verlag (1995)Khoroshkin, S.M.: Central Extension of the Yangian Double, arXiv:q-alg/9602031

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.: \(h\)-adic quantum vertex algebras in types B, C, D and their \(\phi \)-coordinated modules. J. Phys. A: Math. Theor.

**54**, 485202 (2021) (27pp). arXiv:2107.10184 [math.QA]Kulish, P.P., Sklyanin, E.K.: Quantum spectral transform method: recent developments. In: Integrable Quantum Field Theories, Lecture Notes in Phys.

**151**Springer, Berlin-Heidelberg, pp. 61–119 (1982)Li, H.-S.: \(\hbar \)-adic quantum vertex algebras and their modules. Comm. Math. Phys.

**296**, 475–523 (2010). arXiv:0812.3156 [math.QA]Li, H.-S.: \(\phi \)-Coordinated Quasi-Modules for Quantum Vertex Algebras. Comm. Math. Phys.

**308**, 703–741 (2011). arXiv:0906.2710 [math.QA]Molev, A.: Yangians and classical Lie algebras. Mathematical Surveys and Monographs, 143. American Mathematical Society, Providence, RI (2007)

Molev, A.I., Ragoucy, E.: Representations of reflection algebras. Rev. Math. Phys.

**14**, 317–342 (2002). arXiv:math/0107213 [math.QA]Ogievetsky, O.: Uses of quantum spaces. Contemp. Math.

**294**, 161–232 (2002)Reshetikhin, NYu., Semenov-Tian-Shansky, M.A.: Central extensions of quantum current groups. Lett. Math. Phys.

**19**, 133–142 (1990)Sklyanin, E.K.: Boundary conditions for integrable quantum systems. J. Phys. A

**21**, 2375–2389 (1988)Takhtajan, L.A., Faddeev, L.D.: Quantum inverse scattering method and the Heisenberg XYZ-model. Russian Math. Surv.

**34**(5), 11–68 (1979)Tarasov, V.O.: Structure of quantum \(L\)-operators for the \(R\)-matrix of of the XXZ-model. Theor. Math. Phys.

**61**, 1065–1071 (1984)Tarasov, V.O.: Irreducible monodromy matrices for the \(R\)-matrix of the XXZ-model and lattice local quantum Hamiltonians. Theor. Math. Phys.

**63**, 440–454 (1985)

## Acknowledgements

The author would like to thank the anonymous referees for careful reading and many valuable comments and suggestions which helped him to improve the manuscript. This work has been supported in part by Croatian Science Foundation under the project UIP-2019-04-8488.

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Kožić, S. On the *h*-adic Quantum Vertex Algebras Associated with Hecke Symmetries.
*Commun. Math. Phys.* **397**, 607–634 (2023). https://doi.org/10.1007/s00220-022-04498-4

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DOI: https://doi.org/10.1007/s00220-022-04498-4