K-theoretic stable envelope is an important tool in both K-theoretic enumerative geometry of quiver varieties and representation theory of quantum groups. One important application is that it will give the geometric realization of the quantum loop group, which is usually called the Maulik-Okounkov quantum loop group. It has been conjectured that the MO quantum loop group is isomorphic to the Drinfeld double of the K-theoretic Hall algebra of the corresponding quiver type.

In these talks we show that the Maulik-Okounkov quantum loop group of arbitrary quiver type Q is isomorphic to the Drinfeld double of the preprojective K-theoretic Hall algebras. Moreover, we show that such an isomorphism can be restricted to the quasi-triangular Hopf algebra isomorphism between the wall subalgebra of the MO quantum loop group and the slope subalgebra of the preprojective KHA. If time permits, we will talk about its applications to both enumerative geometry and geometric representation theory.