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Let $\mathfrak u$ be a compact semisimple Lie algebra, and $\sigma$ be a Liealgebra involution of $\mathfrak u$. Let Rep$_q(\mathfrak u)$ be the ribbonbraided 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 fromthe input data $(\mathfrak u,\sigma)$. The first construction is based on thetheory of cyclotomic KZ-equations. The second construction uses the notion ofquantum symmetric pair as developed by G. Letzter. The third construction usesa variation of Drinfeld twisting. In all three cases the module C*-category isribbon twist-braided in the sense of A. Brochier---this is essentially due toB. Enriquez in the first case, is proved by S. Kolb in the second case, and isclosely related to work of J. Donin, P. Kulish, and A. Mudrov in the thirdcase. We formulate a conjecture concerning equivalence of these ribbontwist-braided module C*-categories, and confirm it in the rank one case.

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