Universal K-matrices for quantum Kac-Moody algebras

Abstract

We introduce the notion of a cylindrical bialgebra, which is a quasitriangular bialgebra $H$ endowed with a universal K-matrix, i.e., a universal solution of a generalized reflection equation, yielding an action of cylindrical braid groups on tensor products of its representations. We prove that new examples of such universal K-matrices arise from quantum symmetric pairs of Kac-Moody type and depend upon the choice of a pair of generalized Satake diagrams. In finite type, this yields a refinement of a result obtained by Balagović and Kolb, producing a family of non-equivalent solutions interpolating between the quasi-K-matrix originally due to Bao and Wang and the full universal K-matrix. Finally, we prove that this construction yields formal solutions of the generalized reflection equation with a spectral parameter in the case of finite-dimensional representations over the quantum affine algebra $U_{q}L{\mathfrak{sl}}_2$.

Minor edits. Statement of Lemma 7.11 (iii) corrected. 57 pages

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• The alternating presentation of $U_q(\widehat{g{l}_2})$ from Freidel-Maillet algebras
• Integrable boundary conditions for staggered vertex models
• Monodromy of the Casimir connection of a symmetrisable Kac-Moody algebra
• Relative braid group symmetries on $ı$quantum groups of Kac-Moody type
• Quantum supersymmetric pairs of basic types
• Tensor K-matrices for quantum symmetric pairs
• Reflective centers of module categories and quantum K-matrices
• Generalized Schur-Weyl dualities for quantum affine symmetric pairs and orientifold KLR algebras
• Reflection Maps Associated with Involutions and Factorization Problems, and Their Poisson Geometry
• Boundary transfer matrices arising from quantum symmetric pairs
• Compatibility of Drinfeld presentations for split affine Kac-Moody quantum symmetric pairs
• Defining relations of quantum symmetric pair coideal subalgebras
• A new PBW basis for the alternating central extension of the $q$-Onsager algebra
• Drinfeld rational fractions for affine Kac-Moody quantum symmetric pairs
• Quantum Howe duality and Schur duality of type AIII
• Freidel-Maillet type presentations of $U_q(s{l}_2)$