Cartan subalgebras for quantum symmetric pair coideals
Author:
Gail Letzter
Journal:
Represent. Theory 23 (2019), 88-153
MSC (2010):
Primary 17B37; Secondary 17B10, 17B22
DOI:
https://doi.org/10.1090/ert/523
Published electronically:
January 31, 2019
MathSciNet review:
3904162
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: There is renewed interest in the coideal subalgebras used to form quantum symmetric pairs because of recent discoveries showing that they play a fundamental role in the representation theory of quantized enveloping algebras. However, there is still no general theory of finite-dimensional modules for these coideals. In this paper, we establish an important step in this direction: we show that every quantum symmetric pair coideal subalgebra admits a quantum Cartan subalgebra which is a polynomial ring that specializes to its classical counterpart. The construction builds on Kostant and Sugiura’s classification of Cartan subalgebras for real semisimple Lie algebras via strongly orthogonal systems of positive roots. We show that these quantum Cartan subalgebras act semisimply on finite-dimensional unitary modules and identify particularly nice generators of the quantum Cartan subalgebra for a family of examples.
- Noud Aldenhoven, Erik Koelink, and Pablo Román, Branching rules for finite-dimensional $\mathcal U_q(\mathfrak {su}(3))$-representations with respect to a right coideal subalgebra, Algebr. Represent. Theory 20 (2017), no. 4, 821–842. MR 3669159, DOI https://doi.org/10.1007/s10468-017-9678-z
- Shôrô Araki, On root systems and an infinitesimal classification of irreducible symmetric spaces, J. Math. Osaka City Univ. 13 (1962), 1–34. MR 153782
- Yoshio Agaoka and Eiji Kaneda, Strongly orthogonal subsets in root systems, Hokkaido Math. J. 31 (2002), no. 1, 107–136. MR 1888273, DOI https://doi.org/10.14492/hokmj/1350911773
- M.Balagovic and S. Kolb, Universal K-matrix for quantum symmetric pairs, Journal für die reine und angewandte Mathematik, published online 2016 (arXiv:1507.06276).
- Huanchen Bao and Weiqiang Wang, A new approach to Kazhdan-Lusztig theory of type $B$ via quantum symmetric pairs, Astérisque 402 (2018), vii+134 (English, with English and French summaries). MR 3864017
- Huanchen Bao and Weiqiang Wang, Canonical bases arising from quantum symmetric pairs, Invent. Math. 213 (2018), no. 3, 1099–1177. MR 3842062, DOI https://doi.org/10.1007/s00222-018-0801-5
- Heiko Dietrich, Paolo Faccin, and Willem A. de Graaf, Computing with real Lie algebras: real forms, Cartan decompositions, and Cartan subalgebras, J. Symbolic Comput. 56 (2013), 27–45. MR 3061707, DOI https://doi.org/10.1016/j.jsc.2013.05.007
- 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, DOI https://doi.org/10.1007/BF00116516
- 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, DOI https://doi.org/10.2977/prims/1195143610
- 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, DOI https://doi.org/10.1090/S0002-9947-98-01971-0
- V. G. Drinfel′d, Quantum groups, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986) Amer. Math. Soc., Providence, RI, 1987, pp. 798–820. MR 934283
- Michael Ehrig and Catharina Stroppel, Nazarov-Wenzl algebras, coideal subalgebras and categorified skew Howe duality, Adv. Math. 331 (2018), 58–142. MR 3804673, DOI https://doi.org/10.1016/j.aim.2018.01.013
- A. M. Gavrilik and A. U. Klimyk, $q$-deformed orthogonal and pseudo-orthogonal algebras and their representations, Lett. Math. Phys. 21 (1991), no. 3, 215–220. MR 1102131, DOI https://doi.org/10.1007/BF00420371
- James E. Humphreys, Introduction to Lie algebras and representation theory, Springer-Verlag, New York-Berlin, 1972. Graduate Texts in Mathematics, Vol. 9. MR 0323842
- J. E. Humphreys, Longest element of a finite Coxeter group, unpublished notes (http://people.math.umass.edu/$\sim$jeh/pub/longest.pdf) (2015).
- Mini-workshop: Coideal Subalgebras of Quantum Groups, Oberwolfach Rep. 12 (2015), no. 1, 533–569. Abstracts from the mini-workshop held February 15–21, 2015; Organized by István Heckenberger, Stefan Kolb and Jasper V. Stokman. MR 3443118, DOI https://doi.org/10.4171/OWR/2015/10
- N. Z. Iorgov and A. U. Klimyk, Classification theorem on irreducible representations of the $q$-deformed algebra $U_q’({\rm so}_n)$, Int. J. Math. Math. Sci. 2 (2005), 225–262. MR 2143754, DOI https://doi.org/10.1155/IJMMS.2005.225
- Bertram Kostant, On the conjugacy of real Cartan subalgebras. I, Proc. Nat. Acad. Sci. U.S.A. 41 (1955), 967–970. MR 73928, DOI https://doi.org/10.1073/pnas.41.11.967
- Anthony W. Knapp, Lie groups beyond an introduction, Progress in Mathematics, vol. 140, Birkhäuser Boston, Inc., Boston, MA, 1996. MR 1399083
- Stefan Kolb, Quantum symmetric Kac-Moody pairs, Adv. Math. 267 (2014), 395–469. MR 3269184, DOI https://doi.org/10.1016/j.aim.2014.08.010
- Stefan Kolb, Quantum symmetric pairs and the reflection equation, Algebr. Represent. Theory 11 (2008), no. 6, 519–544. MR 2453228, DOI https://doi.org/10.1007/s10468-008-9093-6
- Stefan Kolb and Gail Letzter, The center of quantum symmetric pair coideal subalgebras, Represent. Theory 12 (2008), 294–326. MR 2439008, DOI https://doi.org/10.1090/S1088-4165-08-00332-4
- Stefan Kolb and Jacopo Pellegrini, Braid group actions on coideal subalgebras of quantized enveloping algebras, J. Algebra 336 (2011), 395–416. MR 2802552, DOI https://doi.org/10.1016/j.jalgebra.2011.04.001
- Jens Carsten Jantzen, Lectures on quantum groups, Graduate Studies in Mathematics, vol. 6, American Mathematical Society, Providence, RI, 1996. MR 1359532
- Michio Jimbo, A $q$-difference analogue of $U({\mathfrak g})$ and the Yang-Baxter equation, Lett. Math. Phys. 10 (1985), no. 1, 63–69. MR 797001, DOI https://doi.org/10.1007/BF00704588
- 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
- Anthony Joseph and Gail Letzter, Local finiteness of the adjoint action for quantized enveloping algebras, J. Algebra 153 (1992), no. 2, 289–318. MR 1198203, DOI https://doi.org/10.1016/0021-8693%2892%2990157-H
- Gail Letzter, Harish-Chandra modules for quantum symmetric pairs, Represent. Theory 4 (2000), 64–96. MR 1742961, DOI https://doi.org/10.1090/S1088-4165-00-00087-X
- 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
- Gail Letzter, Quantum symmetric pairs and their zonal spherical functions, Transform. Groups 8 (2003), no. 3, 261–292. MR 1996417, DOI https://doi.org/10.1007/s00031-003-0719-9
- Gail Letzter, Quantum zonal spherical functions and Macdonald polynomials, Adv. Math. 189 (2004), no. 1, 88–147. MR 2093481, DOI https://doi.org/10.1016/j.aim.2003.11.007
- Gail Letzter, Invariant differential operators for quantum symmetric spaces, Mem. Amer. Math. Soc. 193 (2008), no. 903, vi+90. MR 2400554, DOI https://doi.org/10.1090/memo/0903
- George Lusztig, Introduction to quantum groups, Progress in Mathematics, vol. 110, Birkhäuser Boston, Inc., Boston, MA, 1993. MR 1227098
- 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, DOI https://doi.org/10.1006/aima.1996.0066
- 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
- Masatoshi Noumi, Mathijs S. Dijkhuizen, and Tetsuya Sugitani, Multivariable Askey-Wilson polynomials and quantum complex Grassmannians, Special functions, $q$-series and related topics (Toronto, ON, 1995) Fields Inst. Commun., vol. 14, Amer. Math. Soc., Providence, RI, 1997, pp. 167–177. MR 1448686, DOI https://doi.org/10.1090/s0002-9947-98-01971-0
- A. L. Onishchik and E. B. Vinberg, Lie Groups and Lie Algebras, II, Encyclopaedia of Mathematical Sciences, Vol 41, Springer-Verlag, 1991.
- Eric C. Rowell and Hans Wenzl, ${\rm SO}(N)_2$ braid group representations are Gaussian, Quantum Topol. 8 (2017), no. 1, 1–33. MR 3630280, DOI https://doi.org/10.4171/QT/85
- Mitsuo Sugiura, Conjugate classes of Cartan subalgebras in real semi-simple Lie algebras, J. Math. Soc. Japan 11 (1959), 374–434. MR 146305, DOI https://doi.org/10.2969/jmsj/01140374
- C. Stroppel, private communication.
- H. Watanabe, Crystal basis theory for a quantum symmetric pair $(U,U^j)$, arXiv: 1704.01277.
- Hans Wenzl, On centralizer algebras for spin representations, Comm. Math. Phys. 314 (2012), no. 1, 243–263. MR 2954516, DOI https://doi.org/10.1007/s00220-012-1494-z
Retrieve articles in Representation Theory of the American Mathematical Society with MSC (2010): 17B37, 17B10, 17B22
Retrieve articles in all journals with MSC (2010): 17B37, 17B10, 17B22
Additional Information
Gail Letzter
Affiliation:
Mathematics Research Group, National Security Agency, Fort Meade, Maryland 20755-6844
MR Author ID:
228201
Email:
gletzter@verizon.net
Keywords:
Quantized enveloping algebras,
symmetric pairs,
coideal subalgebras,
Cartan subalgebras
Received by editor(s):
June 4, 2017
Received by editor(s) in revised form:
November 23, 2018
Published electronically:
January 31, 2019
Article copyright:
© Copyright 2019
American Mathematical Society