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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 $\mathcal{A}^{ext}_{Q}$ 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 $\mathbb{C}_q^*\subset A\subset T$.