Quantum loop groups and $K$-theoretic stable envelopes

TL;DR

This paper builds a geometric bridge between the preprojective $K$-theoretic Hall algebra of a quiver and the quantum loop group associated to that quiver, via stable envelopes of Nakajima quiver varieties. It constructs the double Hall algebra ${\mathcal{A}}$ and realizes its action on localized $K$-theory groups ${K}(\boldsymbol{w})$ through Nakajima correspondences, identifying generators with explicit $R$-matrix coefficients. Central to the program is embedding ${\mathcal{A}}$ into the quantum group ${\mathscr{U}}_q(\widehat{{\mathfrak{g}}}_Q)$ in a way compatible with the coproduct, which hinges on a conjectural description of the coproduct for certain elements and on proving coproduct preservation for the core generators. The construction leverages $K$-theoretic stable envelopes to produce matrix coefficients obeying the quantum Yang–Baxter equation, enabling a FRT-type realization of the quantum group from geometric data. If the coproduct conjecture holds, the paper establishes, via a sequence of steps anchored in localization and fixed-point analysis, an isomorphism between ${\mathcal{A}}$ and ${\mathscr{U}}_q(\widehat{{\mathfrak{g}}}_Q)$, yielding a robust geometric model for the quantum loop group action on quiver-variety $K$-theory with potential integral-form refinements.

Abstract

We develop the connection between the preprojective $K$-theoretic Hall algebra of a quiver $Q$ and the quantum loop group associated to $Q$ via stable envelopes of Nakajima quiver varieties.

Theorems & Definitions (41)

• Remark 1.2
• Conjecture 1.5
• Remark 1.6
• Theorem 1.8
• Remark 2.4
• Theorem 2.7
• Definition 2.9
• Proposition 2.10
• proof
• Remark 2.12