Maulik-Okounkov quantum loop groups and Drinfeld double of preprojective $K$-theoretic Hall algebras

TL;DR

This paper establishes an isomorphism between Drinfeld doubles of preprojective K-theoretic Hall algebras and Maulik-Okounkov quantum loop groups for quivers, revealing deep structural connections and applications to wall subalgebras.

Contribution

It proves a Hopf algebra isomorphism linking preprojective K-theoretic Hall algebras with Maulik-Okounkov quantum loop groups, and demonstrates the freeness of these algebras for arbitrary tori.

Findings

Isomorphism between Drinfeld double of Hall algebra and quantum loop group.

Identification of wall subalgebras with root subalgebras in quantum groups.

Freeness of preprojective K-theoretic Hall algebra for arbitrary tori.

Abstract

In this paper we prove the following results: Given the Drinfeld double $\mathcal{A}^{ext}_{Q}$ of the localised preprojective $K$-theoretic Hall algebra $\mathcal{A}^{+}_{Q}$ of quiver type $Q$ with the Cartan elements, there is a $\mathbb{Q}(q,t_e)_{e\in E}$-Hopf algebra isomorphism between $\mathcal{A}^{ext}_{Q}$ and the localised Maulik-Okounkov quantum loop group $U^{MO}_{q}(\hat{\mathfrak{g}}_{Q})$ of quiver type $Q$. Moreover, we prove the isomorphism of $\mathbb{Z}[q^{\pm1},t_{e}^{\pm1}]_{e\in E}$-algebras between the positive/negative half of the integral Maulik-Okounkov quantum loop group $U_{q}^{MO,\pm,\mathbb{Z}}(\hat{\mathfrak{g}}_{Q})$ with the (opposite) algebra of the integral preprojective (nilpotent) $K$-theoretic Hall algebra $\mathcal{A}^{+,\mathbb{Z}}_{Q}$ ($(\mathcal{A}^{+,nilp,\mathbb{Z}}_{Q})^{op}$) of the same quiver type $Q$. As the application, we prove that one can identify the wall subalgebra $U_{q}^{MO,\mathbb{Z}}(\mathfrak{g}_{w})$ as the root subalgebra $\mathcal{B}_{\mathbf{m},w}^{\mathbb{Z}}$ in the slope subalgebra $\mathcal{B}_{\mathbf{m}}^{\mathbb{Z}}$ as the quasitriangular Hopf $\mathbb{Z}[q^{\pm1},t_e^{\pm1}]_{e\in E}$-algebras. Moreover we use the freeness of the wall subalgebra in MO quantum loop groups to prove the freeness of the preprojective $K$-theoretic Hall algebra for arbitrary torus $\mathbb{C}_q^*\subset A\subset T$.