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 \(\mathcal {W}_{\theta }\) 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 \(M\in \mathcal {W}_{\theta }\) and \(V\in {{\mathcal {O}}}_\theta \), i.e., V is a \(U_q({{\mathfrak {g}}})\)-module in category \({{\mathcal {O}}}\) satisfying an integrability property determined by \(\theta \). Canonically, \(\mathcal {W}_{\theta }\) 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 \(\mathcal {W}_{\theta }\). 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 \({{\mathcal {C}}}\) of finite-dimensional modules of 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 \(\mathcal {W}_{\theta }\) and any module in \({{\mathcal {C}}}\). 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 \({{\mathcal {C}}}\).