nLab quantum symmetric pair

Idea

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 $\mathfrak{g}$ and an involution $\theta$ on it and are classified by Satake diagrams, quantum symmetric pairs are pairs of a Drinfeld-Jimbo quantum group and its right coideal subalgebra which in the classical limit gives $U(\mathfrak{g}^\theta)$.

Literature

The notion 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]

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:

• Weiqiang Wang, Quantum symmetric pairs, ICM 2022, EMS (2022) [arXiv:2112.10911, doi:10.4171/icm2022/76]

category: algebra