Defining relations of quantum symmetric pair coideal subalgebras

Abstract We explicitly determine the defining relations of all quantum symmetric pair coideal subalgebras of quantised enveloping algebras of Kac–Moody type. Our methods are based on star products on noncommutative ${\mathbb N}$-graded algebras. The resulting defining relations are expressed in terms of continuous q-Hermite polynomials and a new family of deformed Chebyshev polynomials.

[1]  C. Stroppel,et al.  Nazarov–Wenzl algebras, coideal subalgebras and categorified skew Howe duality , 2013, Advances in Mathematics.

[2]  S. Kolb,et al.  Universal K-matrix for quantum symmetric pairs , 2015, Journal für die reine und angewandte Mathematik (Crelles Journal).

[3]  Eugene P. Wigner,et al.  Formulas and Theorems for the Special Functions of Mathematical Physics , 1966 .

[4]  S. Kolb.  Quantum symmetric Kac–Moody pairs , 2012, 1207.6036.

[5]  G. Letzter.  Symmetric Pairs for Quantized Enveloping Algebras , 1999 .

[6]  Weiqiang Wang,et al.  A New Approach to Kazhdan-lusztig Theory of Type $b$ Via Quantum Symmetric Pairs , 2013, 1310.0103.

[7]  G. Letzter.  Cartan subalgebras for quantum symmetric pair coideals , 2017, Representation Theory of the American Mathematical Society.

[8]  Rene F. Swarttouw,et al.  Hypergeometric Orthogonal Polynomials , 2010 .

[9]  P. Podles,et al.  Introduction to Quantum Groups , 1998 .

[10]  Coideal Subalgebras,et al.  NAZAROV-WENZL ALGEBRAS, COIDEAL SUBALGEBRAS AND CATEGORIFIED SKEW HOWE DUALITY , 2013 .

[11]  Stefan Kolb,et al.  The bar involution for quantum symmetric pairs -- hidden in plain sight , 2021, 2104.06120.

[12]  Quantum Symmetric Pairs and Their Zonal Spherical Functions , 2002, math/0204103.

[13]  Weiqiang Wang,et al.  Canonical bases arising from quantum symmetric pairs of Kac–Moody type , 2018, Compositio Mathematica.

[14]  Weiqiang Wang,et al.  Serre–Lusztig relations for ı quantum groups II , 2021 .

[15]  Shahn Majid,et al.  Cross Products by Braided Groups and Bosonization , 1994 .

[16]  Coideal Subalgebras and Quantum Symmetric Pairs , 2001, math/0103228.

[17]  V. Regelskis,et al.  Quasitriangular coideal subalgebras of Uq(g) in terms of generalized Satake diagrams , 2018, Bulletin of the London Mathematical Society.

[18]  S. Kolb,et al.  The bar involution for quantum symmetric pairs , 2014, 1409.5074.

[19]  M. Yakimov,et al.  Symmetric pairs for Nichols algebras of diagonal type via star products , 2019, Advances in Mathematics.

[20]  D. Radford.  The structure of Hopf algebras with a projection , 1985 .