The center of quantum symmetric pair coideal subalgebras

Kolb, Stefan; Letzter, Gail

doi:10.1090/S1088-4165-08-00332-4

The center of quantum symmetric pair coideal subalgebras

Authors: Stefan Kolb and Gail Letzter
Journal: Represent. Theory 12 (2008), 294-326
MSC (2000): Primary 17B37
DOI: https://doi.org/10.1090/S1088-4165-08-00332-4
Published electronically: August 27, 2008
MathSciNet review: 2439008
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The theory of quantum symmetric pairs as developed by the second author is based on coideal subalgebras of the quantized universal enveloping algebra for a semisimple Lie algebra. This paper investigates the center of these coideal subalgebras, proving that the center is a polynomial ring. A basis of the center is given in terms of a submonoid of the dominant integral weights.

[Ara62] Shôrô Araki, On root systems and an infinitesimal classification of irreducible symmetric spaces, J. Math. Osaka City Univ. 13 (1962), 1–34. MR 0153782
[CK90] Corrado De Concini and Victor G. Kac, Representations of quantum groups at roots of 1, Operator algebras, unitary representations, enveloping algebras, and invariant theory (Paris, 1989) Progr. Math., vol. 92, Birkhäuser Boston, Boston, MA, 1990, pp. 471–506. MR 1103601
[Dij96] Mathijs S. Dijkhuizen, Some remarks on the construction of quantum symmetric spaces, Acta Appl. Math. 44 (1996), no. 1-2, 59–80. Representations of Lie groups, Lie algebras and their quantum analogues. MR 1407040, https://doi.org/10.1007/BF00116516
[DN98] Mathijs S. Dijkhuizen and Masatoshi Noumi, A family of quantum projective spaces and related 𝑞-hypergeometric orthogonal polynomials, Trans. Amer. Math. Soc. 350 (1998), no. 8, 3269–3296. MR 1432197, https://doi.org/10.1090/S0002-9947-98-01971-0
[DS99] Mathijs S. Dijkhuizen and Jasper V. Stokman, Some limit transitions between 𝐵𝐶 type orthogonal polynomials interpreted on quantum complex Grassmannians, Publ. Res. Inst. Math. Sci. 35 (1999), no. 3, 451–500. MR 1710751, https://doi.org/10.2977/prims/1195143610
[FM98] Florence Fauquant-Millet, Sur une algèbre parabolique 𝑃 de 𝑈_{𝑞}(𝑠𝑙_{𝑛+1}) et ses semi-invariants par l’action adjointe de 𝑃, Bull. Sci. Math. 122 (1998), no. 7, 495–519 (French, with English and French summaries). MR 1653466, https://doi.org/10.1016/S0007-4497(99)80002-5
[FMJ01] F. Fauquant-Millet and A. Joseph, Sur les semi-invariants d’une sous-algèbre parabolique d’une algèbre enveloppante quantifiée, Transform. Groups 6 (2001), no. 2, 125–142 (French, with English summary). MR 1835668, https://doi.org/10.1007/BF01597132
[Hel78] Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, vol. 80, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 514561
[IK05] N. Z. Iorgov and A. U. Klimyk, Classification theorem on irreducible representations of the 𝑞-deformed algebra 𝑈_{𝑞}’(𝑠𝑜_{𝑛}), Int. J. Math. Math. Sci. 2 (2005), 225–262. MR 2143754, https://doi.org/10.1155/IJMMS.2005.225
[JL94] Anthony Joseph and Gail Letzter, Separation of variables for quantized enveloping algebras, Amer. J. Math. 116 (1994), no. 1, 127–177. MR 1262429, https://doi.org/10.2307/2374984
[Jos95] Anthony Joseph, Quantum groups and their primitive ideals, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 29, Springer-Verlag, Berlin, 1995. MR 1315966
[Kéb99] Mâlek Stefan Kébé, Sur la classification des 𝒪-algèbres quantiques, J. Algebra 212 (1999), no. 2, 626–659 (French). MR 1676857, https://doi.org/10.1006/jabr.1998.7585
[Kol] S. Kolb, Quantum symmetric pairs and the reflection equation, Algebr. Represent. Theory, in press, DOI 10.1007/s10468-008-9093-6.
[Let99a] Gail Letzter, Harish-Chandra modules for quantum symmetric pairs, Represent. Theory 4 (2000), 64–96. MR 1742961, https://doi.org/10.1090/S1088-4165-00-00087-X
[Let99b] Gail Letzter, Symmetric pairs for quantized enveloping algebras, J. Algebra 220 (1999), no. 2, 729–767. MR 1717368, https://doi.org/10.1006/jabr.1999.8015
[Let02] Gail Letzter, Coideal subalgebras and quantum symmetric pairs, New directions in Hopf algebras, Math. Sci. Res. Inst. Publ., vol. 43, Cambridge Univ. Press, Cambridge, 2002, pp. 117–165. MR 1913438
[Let03] Gail Letzter, Quantum symmetric pairs and their zonal spherical functions, Transform. Groups 8 (2003), no. 3, 261–292. MR 1996417, https://doi.org/10.1007/s00031-003-0719-9
[Let04] Gail Letzter, Quantum zonal spherical functions and Macdonald polynomials, Adv. Math. 189 (2004), no. 1, 88–147. MR 2093481, https://doi.org/10.1016/j.aim.2003.11.007
[Let08] Gail Letzter, Invariant differential operators for quantum symmetric spaces, Mem. Amer. Math. Soc. 193 (2008), no. 903, vi+90. MR 2400554, https://doi.org/10.1090/memo/0903
[Mac00] I. G. Macdonald, Orthogonal polynomials associated with root systems, Sém. Lothar. Combin. 45 (2000/01), Art. B45a, 40. MR 1817334
[Nou96] Masatoshi Noumi, Macdonald’s symmetric polynomials as zonal spherical functions on some quantum homogeneous spaces, Adv. Math. 123 (1996), no. 1, 16–77. MR 1413836, https://doi.org/10.1006/aima.1996.0066
[NS95] Masatoshi Noumi and Tetsuya Sugitani, Quantum symmetric spaces and related 𝑞-orthogonal polynomials, Group theoretical methods in physics (Toyonaka, 1994) World Sci. Publ., River Edge, NJ, 1995, pp. 28–40. MR 1413733
[OS05] Alexei A. Oblomkov and Jasper V. Stokman, Vector valued spherical functions and Macdonald-Koornwinder polynomials, Compos. Math. 141 (2005), no. 5, 1310–1350. MR 2157139, https://doi.org/10.1112/S0010437X05001636
[OV94] È. B. Vinberg (ed.), Lie groups and Lie algebras, III, Encyclopaedia of Mathematical Sciences, vol. 41, Springer-Verlag, Berlin, 1994. Structure of Lie groups and Lie algebras; A translation of Current problems in mathematics. Fundamental directions. Vol. 41 (Russian), Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1990 [ MR1056485 (91b:22001)]; Translation by V. Minachin [V. V. Minakhin]; Translation edited by A. L. Onishchik and È. B. Vinberg. MR 1349140
[Ros90] Marc Rosso, Analogues de la forme de Killing et du théorème d’Harish-Chandra pour les groupes quantiques, Ann. Sci. École Norm. Sup. (4) 23 (1990), no. 3, 445–467 (French). MR 1055444