Tensor K-matrices for quantum symmetric pairs

We work with a symmetrizable Kac Moody algebra, its quantum group, and a quantum symmetric pair subalgebra determined by an involutive Lie algebra automorphism. We introduce a category of weight modules for the symmetric pair subalgebra and show that it is naturally acted on by weight modules for the full quantum group through tensor product. Within an appropriate completion of the tensor product of the symmetric pair subalgebra with the full quantum group, we construct a universal tensor K matrix, meaning a universal solution of the reflection equation. This element defines canonical operators on mixed tensor products of a symmetric pair module with a quantum group module satisfying a suitable integrability condition dictated by the automorphism. We then explain how the resulting operators are best understood categorically: the symmetric pair module category carries a canonical bimodule category structure over the relevant category of quantum group modules, and the universal tensor K matrix is encoded by an additional piece of structure that we call a boundary structure. Our construction extends earlier work of Kolb, which established a braided module type structure for finite dimensional symmetric pair modules in finite type. We also treat the setting of finite dimensional modules over quantum affine algebras, obtaining a broad universal framework for large families of parameter dependent solutions to reflection equations. In this affine case, the tensor K matrix produces a formal Laurent series acting on all relevant tensor products, and when both tensor factors are finite dimensional it can be normalized to an operator valued rational function, which we call the trigonometric tensor K matrix.