Ribbon braided module categories, quantum symmetric pairs and Knizhnik-Zamolodchikov equations

Let $\mathfrak u$ be a compact semisimple Lie algebra, and $\sigma$ be a Lie algebra involution of $\mathfrak u$. Let Rep$_q(\mathfrak u)$ be the ribbon braided tensor C*-category of $U_q(\mathfrak u)$-representations for $0<q<1$. We introduce three module C*-categories over Rep$_q(\mathfrak u)$ starting from the input data $(\mathfrak u,\sigma)$. The first construction is based on the theory of cyclotomic KZ-equations. The second construction uses the notion of quantum symmetric pair as developed by G. Letzter. The third construction uses a variation of Drinfeld twisting. In all three cases the module C*-category is ribbon twist-braided in the sense of A. Brochier---this is essentially due to B. Enriquez in the first case, is proved by S. Kolb in the second case, and is closely related to work of J. Donin, P. Kulish, and A. Mudrov in the third case. We formulate a conjecture concerning equivalence of these ribbon twist-braided module C*-categories, and confirm it in the rank one case.