Quantized Weyl algebras, the double centralizer property, and a new First Fundamental Theorem for $U_q(\mathfrak{gl}_n)$
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| author | Letzter, Gail Sahi, Siddhartha Salmasian, Hadi |
| author_facet | Letzter, Gail Sahi, Siddhartha Salmasian, Hadi |
| contents | Let $\mathcal P:=\mathcal P_{m\times n}$ denote the quantized coordinate ring of the space of $m\times n$ matrices, equipped with natural actions of the quantized enveloping algebras $U_q(\mathfrak{gl}_m)$ and $U_q(\mathfrak{gl}_n)$. Let $\mathcal L$ and $\mathcal R$ denote the images of $U_q(\mathfrak{gl}_m)$ and $U_q(\mathfrak{gl}_n)$ in $\mathrm{End}(\mathcal P)$, respectively. We define a $q$-analogue of the algebra of polynomial-coefficient differential operators inside $\mathrm{End}(\mathcal P)$, henceforth denoted by $\mathcal{PD}$, and we prove that $\mathcal L\cap \mathcal{PD}$ and $\mathcal{R}\cap \mathcal{PD}$ are mutual centralizers inside $\mathcal{PD}$. Using this, we establish a new First Fundamental Theorem of invariant theory for $U_q(\mathfrak{gl}_n)$. We also compute explicit formulas in terms of $q$-determinants for generators of the intersections with $\mathcal{PD}$ of the images of the Cartan subalgebras of $U_q(\mathfrak{gl}_m)$ and $U_q(\mathfrak{gl}_n)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_09101 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Quantized Weyl algebras, the double centralizer property, and a new First Fundamental Theorem for $U_q(\mathfrak{gl}_n)$ Letzter, Gail Sahi, Siddhartha Salmasian, Hadi Quantum Algebra Rings and Algebras Representation Theory 17B37, 20G42 Let $\mathcal P:=\mathcal P_{m\times n}$ denote the quantized coordinate ring of the space of $m\times n$ matrices, equipped with natural actions of the quantized enveloping algebras $U_q(\mathfrak{gl}_m)$ and $U_q(\mathfrak{gl}_n)$. Let $\mathcal L$ and $\mathcal R$ denote the images of $U_q(\mathfrak{gl}_m)$ and $U_q(\mathfrak{gl}_n)$ in $\mathrm{End}(\mathcal P)$, respectively. We define a $q$-analogue of the algebra of polynomial-coefficient differential operators inside $\mathrm{End}(\mathcal P)$, henceforth denoted by $\mathcal{PD}$, and we prove that $\mathcal L\cap \mathcal{PD}$ and $\mathcal{R}\cap \mathcal{PD}$ are mutual centralizers inside $\mathcal{PD}$. Using this, we establish a new First Fundamental Theorem of invariant theory for $U_q(\mathfrak{gl}_n)$. We also compute explicit formulas in terms of $q$-determinants for generators of the intersections with $\mathcal{PD}$ of the images of the Cartan subalgebras of $U_q(\mathfrak{gl}_m)$ and $U_q(\mathfrak{gl}_n)$. |
| title | Quantized Weyl algebras, the double centralizer property, and a new First Fundamental Theorem for $U_q(\mathfrak{gl}_n)$ |
| topic | Quantum Algebra Rings and Algebras Representation Theory 17B37, 20G42 |
| url | https://arxiv.org/abs/2206.09101 |