Quantized Weyl algebras, the double centralizer property, and a new First Fundamental Theorem for $U_q(\mathfrak{gl}_n)$
| Autor: | Letzter, Gail, Sahi, Siddhartha, Salmasian, Hadi |
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| Rok vydání: | 2022 |
| Předmět: | |
| Zdroj: | J. Phys A.: Math. Theor. 2024, Special Issue on Dualities and Symmetries in Integrable Systems |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1088/1751-8121/ad3ef1 |
| Popis: | 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 $U_q(\mathfrak{gl}_n)$. Comment: The original submission has been thoroughly revised. An error in the proof of Theorem C (which is Theorem B in the revised version) was corrected |
| Databáze: | arXiv |
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