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

In this paper we prove the following results: Given the Drinfeld double ${\mathcal{A}}_Q^{ext}$ 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 ${\mathcal{A}}_Q^{ext}$ and the localised Maulik-Okounkov quantum loop group $U_q^{MO}({\widehat{\mathfrak{g}}}_Q)$ of quiver type $Q$. Moreover, we prove the isomorphism of $\mathbb{Z}[q^{\pm 1},t_e^{\pm 1}{]}_{e\in E}$-algebras between the positive/negative half of the integral Maulik-Okounkov quantum loop group $U_q^{MO,\pm ,\mathbb{Z}}({\widehat{\mathfrak{g}}}_Q)$ with the (opposite) algebra of the integral preprojective (nilpotent) $K$-theoretic Hall algebra ${\mathcal{A}}_Q^{+,\mathbb{Z}}$ ($({\mathcal{A}}_Q^{+,nilp,\mathbb{Z}}{)}^{op}$) 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 $\mathbb{Z}[q^{\pm 1},t_e^{\pm 1}{]}_{e\in E}$-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 ${\mathbb{C}}_q^{\ast }\subset A\subset T$.