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

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 $Q$.