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Quasi Modules for the Quantum Affine Vertex Algebra in Type A

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Abstract

We consider the quantum affine vertex algebra \({\mathcal{V}_{c}(\mathfrak{gl}_N)}\) associated with the rational R-matrix, as defined by Etingof and Kazhdan. We introduce certain subalgebras \({\textrm{A}_c (\mathfrak{gl}_N)}\) of the completed double Yangian \({\widetilde{\textrm{DY}}_{c}(\mathfrak{gl}_N)}\) at the level \({c\in\mathbb{C}}\), associated with the reflection equation, and we employ their structure to construct examples of quasi \({\mathcal{V}_{c}(\mathfrak{gl}_N)}\)-modules. Finally, we use the quasi module map, together with the explicit description of the center of \({\mathcal{V}_{c}(\mathfrak{gl}_N)}\), to obtain formulae for families of central elements in the completed algebra \({\widetilde{\textrm{A}}_c (\mathfrak{gl}_N)}\).

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Acknowledgments

The author would like to thank Alexander Molev for fruitful discussions. We would also like to thank the anonymous referee for useful comments and suggestions which helped us to improve the manuscript. The research was partially supported by the Australian Research Council and by the Croatian Science Foundation under the Project 2634.

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Authors and Affiliations

  1. School of Mathematics and Statistics F07, University of Sydney, Sydney, NSW, 2006, Australia

    Slaven Kožić

  2. Department of Mathematics, Faculty of Science, University of Zagreb, Bijenička Cesta 30, 10000, Zagreb, Croatia

    Slaven Kožić

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  1. Slaven Kožić
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Correspondence to Slaven Kožić.

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Communicated by Y. Kawahigashi

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Dedicated to Mirko Primc on the occasion of his 70th birthday

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Kožić, S. Quasi Modules for the Quantum Affine Vertex Algebra in Type A. Commun. Math. Phys. 365, 1049–1078 (2019). https://doi.org/10.1007/s00220-019-03291-0

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