Tensor K-matrices for quantum symmetric pairs

Let g be a symmetrizable Kac-Moody algebra, Uq (g) its quantum group, and Uq (k) Uq (g) a quantum symmetric pair subalgebra determined by a distinguished Lie algebra automorphism. We introduce a category W_ of weight Uq (k) -modules, which is acted upon by the category of weight Uq (g) -modules via tensor products. We construct a universal tensor K-matrix K (that is, a solution of a reflection equation) in a completion of Uq (k) Uq (g). This yields a natural operator on any tensor product M V, where M W_ and V O_, V is a Uq (g) -module in category O satisfying an integrability property determined by. W_ is equipped with a canonical structure of a bimodule category over O_ and the action of K is encoded by a new categorical structure, which we refer to as a boundary structure on the bimodule category W_. This provides the most comprehensive algebraic framework to date for quantum integrability in the presence of boundaries. In particular, it generalizes a theorem of Kolb which describes a braided module structure on finite-dimensional Uq (k) -modules when g is finite-dimensional. We apply our construction to the case of an affine Lie algebra, where it yields a formal tensor K-matrix valued in the endomorphisms of the tensor product of any module in W_ and any finite-dimensional module over the corresponding quantum affine algebra. We prove that this formal series can be normalized to a trigonometric K-matrix if the factors in the tensor product are both finite-dimensional irreducible modules over the quantum affine algebra.