Universal K-matrices for quantum Kac-Moody algebras. (English) Zbl 1522.17017
In this paper the authors present a general construction of universal \(K\)-matrices for quantum groups corresponding to arbitrary symmetrizable Kac-Moody algebras. The construction is based on the notion of a cylindrical bialgebra, which is a quasitriangular bialgebra endowed with a universal solution of a generalized reflection equation, called a universal \(K\)-matrix, which yields an action of cylindrical braid groups on tensor products of representations. The relevant reflection equation here is twisted by an algebra automorphism which may not preserve the coproduct; the obstruction to being a morphism of quasitriangular bialgebras is however controlled by a Drinfeld twist. New examples of such universal \(K\)-matrices are shown to arise from quantum symmetric pairs of Kac-Moody type and depend upon the choice of a pair of generalized Satake diagrams. In the case of finite type, the construction yields a discrete family of universal \(K\)-matrices interpolating between the universal \(K\)-matrix of M. Balagović and S. Kolb [J. Reine Angew. Math. 747, 299–353 (2019; Zbl 1425.81058)] and the quasi-\(K\)-matrix of H. Bao and W. Wang [A new approach to Kazhdan-Lusztig theory of type \(B\) via quantum symmetric pairs. Paris: Société Mathématique de France (SMF) (2018; Zbl 1411.17001)]. It is also shown that quantum symmetric pairs for the quantum loop algebra \(U_q(L\mathfrak{sl}_2)\), the universal \(K\)-matrices constructed in the paper give rise to formal solutions of a generalized reflection equation with a spectral parameter.
Reviewer: Sonia Natale (Córdoba)
MSC:
| 17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
| 20F36 | Braid groups; Artin groups |
| 16T10 | Bialgebras |
| 81R10 | Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations |
| 17B67 | Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras |
| 46A32 | Spaces of linear operators; topological tensor products; approximation properties |
| 20F55 | Reflection and Coxeter groups (group-theoretic aspects) |
Keywords:
cylindrical bialgebra; generalized reflection equation; symmetrizable Kac-Moody algebra; quantum symmetric pair; generalized Satake diagramReferences:
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