Oscillator representations of quantum affine orthosymplectic superalgebras
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866913185031782400 |
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| author | Kwon, Jae-Hoon Lee, Sin-Myung Okado, Masato |
| author_facet | Kwon, Jae-Hoon Lee, Sin-Myung Okado, Masato |
| contents | 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 $G=O_\ell, Sp_{2\ell}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2304_06215 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Oscillator representations of quantum affine orthosymplectic superalgebras Kwon, Jae-Hoon Lee, Sin-Myung Okado, Masato Representation Theory Quantum Algebra 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 $G=O_\ell, Sp_{2\ell}$. |
| title | Oscillator representations of quantum affine orthosymplectic superalgebras |
| topic | Representation Theory Quantum Algebra |
| url | https://arxiv.org/abs/2304.06215 |