An algebra isomorphism on $U(\mathfrak{gl}_n)$

TL;DR

This paper establishes an algebra isomorphism involving localizations of universal enveloping algebras of certain Lie algebras, providing new proofs for the Gelfand-Kirillov conjecture and analyzing module categories.

Contribution

It introduces a novel algebra isomorphism for localized universal enveloping algebras of algebras alg_n, offering new proofs of the Gelfand-Kirillov conjecture and classifying related module categories.

Findings

Proves an isomorphism: localized $U(rak{s}_n)$ is isomorphic to a tensor product involving Weyl algebra and $U(rak{s}_{n-1})$.

Provides a new proof of the Gelfand-Kirillov conjecture for algebras alg_n.

Shows the category of certain Harish-Chandra modules is equivalent to the category of finite-dimensional algebras alg_{n-1} modules with wild representation type.

Abstract

For each positive integer $n$, let $\mathfrak{s}_n=\mathfrak{gl}_n\ltimes \mathbb{C}^n$. We show that $U(\mathfrak{s}_{n})_{X_{n}}\cong \mathcal{D}_{n}\otimes U(\mathfrak{s}_{n-1})$ for any $n\in\mathbb{Z}_{\geq 2}$, where $U(\mathfrak{s}_{n})_{X_{n}}$ is the localization of $U(\mathfrak{s}_{n})$ with respect to the subset $X_n:=\{e_1^{i_1}\cdots e_{n}^{i_{n}}\mid i_1,\dots,i_{n}\in \mathbb{Z}_+\}$, and $\mathcal{D}_{n}$ is the Weyl algebra $\mathbb{C}[x_1^{\pm 1}, \cdots, x_{n}^{\pm 1}, \frac{\partial}{\partial x_1},\cdots, \frac{\partial}{\partial x_{n}}]$. As an application, we give a new proof of the Gelfand-Kirillov conjecture for $\mathfrak{s}_n$ and $\mathfrak{gl}_n$. Moreover we show that the category of Harish-Chandra $U(\mathfrak{s}_{n})_{X_n}$-modules with a fixed weight support is equivalent to the category of finite dimensional $\mathfrak{s}_{n-1}$-modules whose representation type is wild, for any $n\in \mathbb{Z}_{\geq 2}$.