Maulik-Okounkov quantum loop groups and Drinfeld double of preprojective $K$-theoretic Hall algebras
Tianqing Zhu
TL;DR
This work establishes a deep link between Maulik-Okounkov quantum loop groups and the Drinfeld doubles of localised preprojective K-theoretic Hall algebras for quivers. It develops a comprehensive framework: (i) constructing preprojective and nilpotent KHAs, their localised/integral extensions, and their geometric actions on Nakajima varieties; (ii) formulating slope filtrations and shuffle realizations that yield Drinfeld doubles and explicit R-matrix factorizations; (iii) proving isomorphisms between localised MO quantum loop groups and the extended KHA, with wall subalgebras and integral forms aligning under a conjectured integrality condition; and (iv) detailing the factorisation, freeness, and Hopf structures of wall and slope subalgebras, including nilpotent variants, culminating in an integrated picture where each side realizes the same quantum group structure through geometric and algebraic data.
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 negative half of the integral Maulik-Okounkov quantum loop group $U_{q}^{MO,-,\mathbb{Z}}(\hat{\mathfrak{g}}_{Q})$ with the opposite algebra of the integral nilpotent $K$-theoretic Hall algebra $\mathcal{A}^{+,nilp,\mathbb{Z}}_{Q}$ of the same quiver type $Q$. As a result, one can identify the universal $R$-matrix for the root subalgebra $\mathcal{B}_{\mathbf{m},w}$ of the slope subalgebra $\mathcal{B}_{\mathbf{m}}$ in $\mathcal{A}^{ext}_{Q}$ with the wall $R$-matrix of the wall subalgebra $U_{q}^{MO}(\mathfrak{g}_{w})$ in $U^{MO}_{q}(\hat{\mathfrak{g}}_{Q})$. Moreover, under the integrality conjecture for the integral preprojective $K$-theoretic Hall algebra $\mathcal{A}^{+,\mathbb{Z}}_{Q}$, we prove the isomorphism of $\mathbb{Z}[q^{\pm1},t_{e}^{\pm1}]_{e\in E}$-algebras between the positive half of the integral Maulik-Okounkov quantum loop group $U_{q}^{MO,+,\mathbb{Z}}(\hat{\mathfrak{g}}_{Q})$ with the integral preprojective $K$-theoretic Hall algebra $\mathcal{A}^{+,\mathbb{Z}}_{Q}$ of the same quiver type $Q$.