Tensor K-Matrices for Quantum Symmetric Pairs

Abstract Let $${{\mathfrak {g}}}$$ g be a symmetrizable Kac–Moody algebra, $$U_q({{\mathfrak {g}}})$$ U q ( g ) its quantum group, and $$U_q({\mathfrak {k}})\subset U_q({{\mathfrak {g}}})$$ U q ( k ) ⊂ U q ( g ) a quantum symmetric pair subalgebra determined by a Lie algebra automorphism $$\theta $$ θ . We introduce a category $$\mathcal {W}_{\theta }$$ W θ of weight $$U_q({\mathfrak {k}})$$ U q ( k ) -modules, which is acted on by the category of weight $$U_q({{\mathfrak {g}}})$$ U q ( g ) -modules via tensor products. We construct a universal tensor K-matrix $${{\mathbb {K}}} $$ K (that is, a solution of a reflection equation) in a completion of $$U_q({\mathfrak {k}})\otimes U_q({{\mathfrak {g}}})$$ U q ( k ) ⊗ U q ( g ) . This yields a natural operator on any tensor product $$M\otimes V$$ M ⊗ V , where $$M\in \mathcal {W}_{\theta }$$ M ∈ W θ and $$V\in {{\mathcal {O}}}_\theta $$ V ∈ O θ , i.e., V is a $$U_q({{\mathfrak {g}}})$$ U q ( g ) -module in category $${{\mathcal {O}}}$$ O satisfying an integrability property determined by $$\theta $$ θ . Canonically, $$\mathcal {W}_{\theta }$$ W θ is equipped with a structure of a bimodule category over $${{\mathcal {O}}}_\theta $$ O θ and the action of $${{\mathbb {K}}} $$ K is encoded by a new categorical structure, which we call a boundary structure on $$\mathcal {W}_{\theta }$$ W θ . This generalizes a result of Kolb which describes a braided module structure on finite-dimensional $$U_q({\mathfrak {k}})$$ U q ( k ) -modules when $${{\mathfrak {g}}}$$ g is finite-dimensional. We also consider our construction in the case of the category $${{\mathcal {C}}}$$ 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 $$\mathcal {W}_{\theta }$$ W θ and any module in $${{\mathcal {C}}}$$ 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 $${{\mathcal {C}}}$$ C .