  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.
