By quantum symmetric pairs one means an analogue for quantized enveloping algebras of symmetric pairs in Lie theory (related to symmetric spaces).
While the usual symmetric pairs are pairs of a semisimple Lie algebra and an involution on it and are classified by Satake diagrams, quantum symmetric pairs are pairs of a Drinfeld-Jimbo quantum group and its right (or left) coideal subalgebra which in the classical limit gives , where is a -invariant subalgebra. However, in the quantum case the right coideal subalgebra is not an invariant subalgebra under some involution in general. There is also another approach to construct -analogues of depending on solutions of the reflection equation; the two approaches are shown compatible.
The notion in the general framework of left coideal subalgebras is due to:
Gail Letzter, Symmetric pairs for quantized enveloping algebras, J. Algebra 220 (1999) 729–767 [doi:10.1006/jabr.1999.8015]
Gail Letzter, Quantum symmetric pairs and their zonal spherical functions, Transformation Groups 8 (2003) 261–292 [arXiv:math/0204103, doi:10.1007/s00031-003-0719-9]
The earlier approach via reflection equation is from
The comparison between the two approaches is made in
Further developments:
Gail Letzter, Cartan subalgebras for quantum symmetric pair coideals, Representation Theory 23 (2019) 99–153 [doi:10.1090/ert/523]
Martina Balagović, Stefan Kolb, The bar involution for quantum symmetric pairs, Representation Theory 19 (2015) 186–210 [arXiv:1409.5074, doi:10.1090/ert/469]
Martina Balagović, Stefan Kolb, Universal K-matrix for quantum symmetric pairs, J. Reine Angew. Math. 747 (2019) 299–353 [arXiv:1507.06276, doi:10.1515/crelle-2016-0012]
A modern survey:
Last revised on October 3, 2024 at 08:35:42. See the history of this page for a list of all contributions to it.