Quantum groups, quantum tori, and the Grothendieck-Springer resolution

TL;DR

This paper constructs an algebra embedding of quantum groups into quantum coordinate rings of certain double Bruhat cells, inspired by Poisson geometry and quantum Beilinson-Bernstein theory, revealing new structural insights.

Contribution

It introduces a novel embedding of quantum groups into quantum coordinate rings via the Heisenberg double, connecting Poisson geometry and quantum algebra structures.

Findings

Embedding factors through the Heisenberg double

Relates quantum Borel subalgebra to quantum coordinate rings

Inspired by Poisson geometry of the Grothendieck-Springer resolution

Abstract

We construct an algebra embedding of the quantum group $U_q(\mathfrak{g})$ into the quantum coordinate ring $\mathcal{O}_q[G^{w_0,w_0}/H]$ of the reduced big double Bruhat cell in $G$. This embedding factors through the Heisenberg double $\mathcal{H}_q$ of the quantum Borel subalgebra $U_{\geq0}$, which we relate to $\mathcal{O}_q[G]$ via twisting by the longest element of the quantum Weyl group. Our construction is inspired by the Poisson geometry of the Grothendieck-Springer resolution studied by Evens and Lu, and the quantum Beilinson-Bernstein theorem investigated by Backelin, Kremnitzer, and Tanisaki.