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Affinization of q-oscillator representations of \(U_q(\mathfrak {gl}_n)\)

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Abstract

We introduce a category \(\widehat{\mathcal {O}}_{\textrm{osc}}\) of q-oscillator representations of the quantum affine algebra \(U_q({\widehat{\mathfrak {gl}}}_n)\). We show that \(\widehat{\mathcal {O}}_{\textrm{osc}}\) has a family of irreducible representations, which naturally corresponds to finite-dimensional irreducible representations of quantum affine algebra of untwisted affine type A. It is done by constructing a category of q-oscillator representations of the quantum affine superalgebra of type A, which interpolates these two families of irreducible representations. The category \(\widehat{\mathcal {O}}_{\textrm{osc}}\) can be viewed as a quantum affine analogue of the semisimple tensor category generated by unitarizable highest weight representations of \(\mathfrak {gl}_{u+v}\) (\(n=u+v\)) appearing in the \((\mathfrak {gl}_{u+v},\mathfrak {gl}_\ell )\)-duality on a bosonic Fock space.

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Acknowledgements

The authors would like to thank M. Okado for his interest in this work and letting us know the reference [29] and the referees for very careful reading of the manuscript and helpful comments.

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  1. Department of Mathematical Sciences and RIM, Seoul National University, Seoul, 08826, Korea

    Jae-Hoon Kwon

  2. Department of Mathematical Sciences, Seoul National University, Seoul, 08826, Korea

    Sin-Myung Lee

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  1. Jae-Hoon Kwon
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Correspondence to Sin-Myung Lee.

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This work is supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (No. 2019R1A2C1084833 and 2020R1A5A1016126).

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Kwon, JH., Lee, SM. Affinization of q-oscillator representations of \(U_q(\mathfrak {gl}_n)\). Lett Math Phys 113, 58 (2023). https://doi.org/10.1007/s11005-023-01675-x

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