Quantized Weyl algebras, the double centralizer property, and a new First Fundamental Theorem for $U_q(\mathfrak{gl}_n)$

TL;DR

This paper develops a quantum analogue of polynomial differential operators for matrix coordinate rings, establishing a new invariant theory fundamental theorem for quantum groups and explicit formulas for certain algebra generators.

Contribution

It introduces a $q$-analogue of differential operators, proves mutual centralizer properties, and establishes a new First Fundamental Theorem for $U_q(rak{gl}_n)$.

Findings

Mutual centralizer properties of quantum algebras within the differential operator algebra.

A new invariant theory fundamental theorem for quantum groups.

Explicit formulas for generators using $q$-determinants.

Abstract

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 $U_q(\mathfrak{gl}_n)$.