Tensor K-matrices for quantum symmetric pairs

Andrea Appel, Bart Vlaar

Published 2024-02-26, updated 2024-09-04Version 2

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 upon 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 $V\in \mathcal{O}_\theta$, that is $V$ is a $U_q(\mathfrak{g})$-module in category $\mathcal{O}$ satisfying an integrability property determined by $\theta$. $W_\theta$ is equipped with a structure of a bimodule category over $\mathcal{O}_\theta$ 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 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 $W_\theta$ and any finite-dimensional module over the corresponding quantum affine algebra. This formal series is normalized to a trigonometric K-matrix if the factors in the tensor product are both finite-dimensional irreducible modules over the quantum affine algebra.