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Quasi Modules for the Quantum Affine Vertex Algebra in Type A

  • Slaven KožićEmail author
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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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Notes

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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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematics and Statistics F07University of SydneySydneyAustralia
  2. 2.Department of Mathematics, Faculty of ScienceUniversity of ZagrebZagrebCroatia

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