Relative braid group symmetries on i-quantum groups

In the theory of quantum groups, Lusztig's braid group symmetries, associated to the Weyl group of the underlying Lie algebra, have played a fundamental role. i-Quantum groups, which are quantum algebras arising from the theory of quantum symmetric pairs, can be viewed as generalizations of quantum groups.

In this dissertation, we initiate a general approach to the relative braid group symmetries, associated to relative Weyl group of the underlying symmetric pair, on (universal) i-quantum groups and their modules. We construct such symmetries for i-quantum groups of arbitrary finite type and quasi-split Kac-Moody type. Our approach is built on new intertwining properties of quasi K-matrices which we develop and braid group symmetries on (Drinfeld double) quantum groups. Explicit formulas for these new symmetries on i-quantum groups are obtained.

We establish a number of fundamental properties for these symmetries on i-quantum groups, strikingly parallel to their well-known quantum group counterparts. We apply these symmetries to fully establish rank one factorizations of quasi K-matrices, and this factorization property in turn helps to show that the new symmetries satisfy relative braid relations. As a consequence, conjectures of Kolb-Pellegrini and Dobson-Kolb are settled affirmatively.

Finally, the above approach allows us to construct compatible relative braid group actions on modules over quantum groups for the first time. Explicit formula for the relative braid group actions on modules are obtained, in terms of elements in i-quantum groups.