On the Quantum Metaplectic Howe Duality
Fuente:
arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866913414165561344 |
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| author | Brito, Matheus De Martino, Marcelo |
| author_facet | Brito, Matheus De Martino, Marcelo |
| contents | We establish a quantum analogue of the classical metaplectic Howe duality involving the pair of Lie algebras $(\mathfrak{sp}_{2n},\mathfrak{sl}_2)$ in the case when $n=1$. Our results yield commuting representations of the pair of Drinfeld-Jimbo quantum groups $(\mathcal U_{q^2}(\mathfrak{sl}_2),\mathcal U_{q}(\mathfrak{sl}_2))$ realized in a suitable algebra of $q$-differential operators acting on the space of symplectic polynomial spinors. We obtain $q$-analogues for the symplectic Dirac operator, the Fischer decomposition, the expression for the symplectic polynomial monogenics and for the projection operators onto the monogenics. We also discuss $q$-analogues of generalized symmetries of the $q$-symplectic Dirac operator raising the homogeneous polynomial degree. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_02205 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Quantum Metaplectic Howe Duality Brito, Matheus De Martino, Marcelo Quantum Algebra Representation Theory 17B37, 20G42, 05E10 We establish a quantum analogue of the classical metaplectic Howe duality involving the pair of Lie algebras $(\mathfrak{sp}_{2n},\mathfrak{sl}_2)$ in the case when $n=1$. Our results yield commuting representations of the pair of Drinfeld-Jimbo quantum groups $(\mathcal U_{q^2}(\mathfrak{sl}_2),\mathcal U_{q}(\mathfrak{sl}_2))$ realized in a suitable algebra of $q$-differential operators acting on the space of symplectic polynomial spinors. We obtain $q$-analogues for the symplectic Dirac operator, the Fischer decomposition, the expression for the symplectic polynomial monogenics and for the projection operators onto the monogenics. We also discuss $q$-analogues of generalized symmetries of the $q$-symplectic Dirac operator raising the homogeneous polynomial degree. |
| title | On the Quantum Metaplectic Howe Duality |
| topic | Quantum Algebra Representation Theory 17B37, 20G42, 05E10 |
| url | https://arxiv.org/abs/2407.02205 |