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 (or left) coideal subalgebra which in the classical limit gives $U(\mathfrak{g}^\theta)$, where $\mathfrak{g}^\theta$ is a $\theta$-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 $q$-analogues of $U(\mathfrak{g}^\theta)$ depending on solutions of the reflection equation; the two approaches are shown compatible.

Literature

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]

• Gail Letzter, Coideal subalgebras and quantum symmetric pairs, In: S. Montgomery, H.-J. Schneider(eds.) New directions in Hopf algebras (Cambridge), vol. 43, pp. 117–166, MSRI Publications, Cambridge University Press 2002

• Gail Letzter, Quantum zonal spherical functions and Macdonald polynomials, Adv. Math. 189 (2004) 88–147 doi

The earlier approach via reflection equation is from

• Masatoshi Noumi, Macdonald’s symmetric polynomials as zonal spherical functions on some quantum homogeneous spaces, Adv. Math. 123 (1996) 16–77 arXiv:math/9503224
• Masatoshi Noumi, T. Sugitani, Quantum symmetric spaces and related $q$-orthogonal polynomials, Group theoretical methods in physics (Singapore) (A. Arima et. al., ed.), World Scientific 1995, pp. 28–40
• Mathijs S. Dijkhuizen, Some remarks on the construction of quantum symmetric spaces, Acta Appl. Math. 44 (1996), no. 1-2, 59–80 doi
• Mathijs S. Dijkhuizen, Masatoshi Noumi, A family of quantum projective spaces and related $q$-hypergeometric orthogonal polynomials, Trans. Amer. Math. Soc. 350 (1998) 3269–3296 doi

The comparison between the two approaches is made in

• Stefan Kolb, Quantum symmetric pairs and the reflection equation Algebras and Representation Theory 11, 519–544 (2008) doi

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]

• H. Bao, W. Wang, Canonical bases arising from quantum symmetric pairs, Invent. math. 213 (2018) 1099–1177 doi

A modern survey:

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

Classical symmetric spaces have R-matrix Poisson structure which could be quantized as another approach to quantum symmetric spaces

• J. Donin, Steven Shnider, Quantum symmetric spaces, J. Pure Appl. Algebra 100 (1995) 103–117 doi

category: algebra