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Algebras and Representation Theory

, Volume 22, Issue 1, pp 177–209 | Cite as

Several realizations of Fock modules for toroidal \(\ddot {U}_{q,d}(\mathfrak {sl}_{n})\)

  • Alexander TsymbaliukEmail author
Article

Abstract

In this paper, we relate the well-known Fock representations of \(\ddot {U}_{q,d}(\mathfrak {sl}_{n})\) to the vertex, shuffle, and ‘L-operator’ representations of \(\ddot {U}_{q,d}(\mathfrak {sl}_{n})\). These identifications generalize those for the quantum toroidal algebra of \(\mathfrak {gl}_{1}\), which were recently established in Feigin et al. (J. Phys. A 48(24), 244001, 2015).

Keywords

Quantum toroidal algebras Shuffle algebras L operators Fock module Vertex module 

Mathematics Subject Classification (2010)

20G42 17B37 

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Notes

Acknowledgments

I would like to thank P. Etingof, B. Feigin, M. Finkelberg, and A. Negut for many stimulating discussions over the years. I am indebted to the anonymous referee for insightful comments on the first version of the paper, which led to a better exposition of the material.

I would like to thank the Max Planck Institute for Mathematics in Bonn for the hospitality and support in June 2015, where part of this project was carried out. The author also gratefully acknowledges support from the Simons Center for Geometry and Physics, Stony Brook University, at which most of the research for this paper was performed, as well as Yale University, where the final version of this paper was completed.

This work was partially supported by the NSF Grants DMS–1502497, DMS–1821185.

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Authors and Affiliations

  1. 1.Department of MathematicsYale UniversityNew HavenUSA

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