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Oscillator representations of quantum affine orthosymplectic superalgebras

Jae-Hoon Kwon, Sin-Myung Lee, Masato Okado

Abstract

We introduce a category of $q$-oscillator representations over the quantum affine superalgebras of type $D$ and construct a new family of its irreducible representations. Motivated by the theory of super duality, we show that these irreducible representations naturally interpolate the irreducible $q$-oscillator representations of type $X_n^{(1)}$ and the finite-dimensional irreducible representations of type $Y_n^{(1)}$ for $(X,Y)=(C,D),(D,C)$ under exact monoidal functors. This can be viewed as a quantum (untwisted) affine analogue of the correspondence between irreducible oscillator and irreducible finite-dimensional representations of classical Lie algebras arising from Howe's reductive dual pairs $(\mathfrak{g},G)$, where $\mathfrak{g}=\mathfrak{sp}_{2n}, \mathfrak{so}_{2n}$ and $G=O_\ell, Sp_{2\ell}$.

Oscillator representations of quantum affine orthosymplectic superalgebras

Abstract

We introduce a category of -oscillator representations over the quantum affine superalgebras of type and construct a new family of its irreducible representations. Motivated by the theory of super duality, we show that these irreducible representations naturally interpolate the irreducible -oscillator representations of type and the finite-dimensional irreducible representations of type for under exact monoidal functors. This can be viewed as a quantum (untwisted) affine analogue of the correspondence between irreducible oscillator and irreducible finite-dimensional representations of classical Lie algebras arising from Howe's reductive dual pairs , where and .
Paper Structure (37 sections, 197 equations)

This paper contains 37 sections, 197 equations.

Table of Contents

  1. Introduction
  2. Quantum affine superalgebras of type $D$
  3. Generalized quantum groups of affine type $D$
  4. Quantum affine superalgebras of type $D$
  5. Classical limits of $\ring{{\mathcal{U}}}_D(\epsilon)$-modules
  6. Quasi $R$ matrix for ${\mathcal{U}}_D(\epsilon)$
  7. ${\mathcal{U}}_D(\epsilon)$-modules $\mathcal{W}(x)$ and $\mathcal{W}^{\otimes 2}(x)$
  8. Truncation functors
  9. Truncation of type $\mathfrak{c}$
  10. Truncation of type $\mathfrak{d}$
  11. Truncations of $\mathcal{W}(x)$ and $\mathcal{W}^{\otimes 2}(x)$
  12. Oscillator representations of $\mathfrak{sp}_{2n}$ and $\mathfrak{so}_{2n}$
  13. Howe duality for $(\mathfrak{sp}_{2n},O_\ell)$ and $(\mathfrak{so}_{2n},O_\ell)$
  14. Howe duality for $(\mathfrak{so}_{2n},{Sp}_{2\ell})$ and $(\mathfrak{sp}_{2n},{Sp}_{2\ell})$
  15. Categories ${O}^{\mathfrak c}_{\rm osc}$ and ${O}^{\mathfrak d}_{\rm osc}$
  16. ...and 22 more sections