Tensor K-matrices for quantum symmetric pairs

Abstract

Let $\mathfrak{g}$ be a symmetrizable Kac-Moody algebra, $U_q(\mathfrak{g})$ its quantum group, and $U_q(\mathfrak{k}) \subset U_q(\mathfrak{g})$ a quantum symmetric pair subalgebra determined by a Lie algebra automorphism $\theta$. We introduce a category $W_\theta$ of weight $U_q(\mathfrak{k})$-modules, which is acted on by the category of weight $U_q(\mathfrak{g})$-modules via tensor products. We construct a universal tensor K-matrix $\mathbb{K}$ (that is, a solution of a reflection equation) in a completion of $U_q(\mathfrak{k}) \otimes U_q(\mathfrak{g})$. This yields a natural operator on any tensor product $M \otimes V$, where $M\in W_\theta$ and $V\in {O}_\theta$, that is $V$ is a $U_q(\mathfrak{g})$-module in category ${O}$ satisfying an integrability property determined by $\theta$. Canonically, $W_\theta$ is equipped with a structure of a bimodule category over ${O}_\theta$ and the action of $\mathbb{K}$ is encoded by a new categorical structure, which we call a boundary structure on $W_\theta$. This generalizes a result of Kolb which describes a braided module structure on finite-dimensional $U_q(\mathfrak{k})$-modules when $\mathfrak{g}$ is finite-dimensional. We also consider our construction in the case of the category ${C}$ of finite-dimensional modules over a quantum affine algebra, providing the most comprehensive universal framework to date for large families of solutions of parameter-dependent reflection equations. In this case the tensor K-matrix gives rise to a formal Laurent series with a well-defined action on tensor products of any module in $W_\theta$ and any module in ${C}$. This series can be normalized to an operator-valued rational function, which we call trigonometric tensor K-matrix, if both factors in the tensor product are in ${C}$.