Skip to main content
SpringerLink
Log in

Affine and degenerate affine BMW algebras: actions on tensor space

  • Published:
Selecta Mathematica Aims and scope Submit manuscript

Abstract

The affine and degenerate affine Birman–Murakami–Wenzl (BMW) algebras arise naturally in the context of Schur–Weyl duality for orthogonal and symplectic quantum groups and Lie algebras, respectively. Cyclotomic BMW algebras, affine and cyclotomic Hecke algebras, and their degenerate versions are quotients of the affine and degenerate affine BMW algebras. In this paper, we explain how the affine and degenerate affine BMW algebras are tantalizers (tensor power centralizer algebras) by defining actions of the affine braid group and the degenerate affine braid algebra on tensor space and showing that, in important cases, these actions induce actions of the affine and degenerate affine BMW algebras. We then exploit the connection to quantum groups and Lie algebras to determine universal parameters for the affine and degenerate affine BMW algebras. Finally, we show that the universal parameters are central elements—the higher Casimir elements for orthogonal and symplectic enveloping algebras and quantum groups.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ariki, S., Mathas, A., Rui, H.: Cyclotomic Nazarov-Wenzl algebras. Nagoya Math. J. 182, 47–134 (2006). MR2235339, arXiv:math.QA/0506467

    Google Scholar 

  2. Baumann, P.: On the center of quantized enveloping algebras. J. Algebra 203(1), 244–260 (1998). MR1620662

    Article  MathSciNet  MATH  Google Scholar 

  3. Beliakova, A., Blanchet, C.: Skein construction of idempotents in Birman–Murakami–Wenzl algebras. Math. Ann. 321(2), 347–373 (2001). MR 1866492

    Article  MathSciNet  MATH  Google Scholar 

  4. Bourbaki, N.: Groupes et algèbres de Lie. Masson, Paris (1990)

    Google Scholar 

  5. Chari, V., Pressley, A.: A guide to quantum groups. Cambridge University Press, Cambridge (1995). MR1358358

    MATH  Google Scholar 

  6. Daugherty, Z., Ram, A., Virk, R.: Affine and graded BMW algebras: the center. arXiv:1105.4207

  7. Drinfel’d, V.G.: On almost cocommutative Hopf algebras. Leningr. Math. J. 1, 321–342 (1990). MR1025154

    MathSciNet  MATH  Google Scholar 

  8. Gelfand, I.M.: Center of the infinitesimal group ring. Mat. Sb. Nov. Ser. 26(68), 103–112 (1950). Zbl. 35, 300

    Google Scholar 

  9. Goodman, F.: Comparison of admissibility conditions for cyclotomic Birman–Wenzl–Murakami algebras. J. Pure Appl. Algebra 214, 2009–2016 (2010). MR2645333 arXiv:0905.4258

    Google Scholar 

  10. Goodman, F.: Admissibility conditions for degenerate cyclotomic BMW algebras. Comm. Algebra 39(2), 452–461 (2011). MR2773313 arXiv:0905.4253

    Google Scholar 

  11. Goodman, F., de la Harpe, P., Jones, V.F.R.: Coxeter Graphs and Towers of Algebras, Mathematical Sciences Research Institute Publications 14. Springer, Berlin (1980). MR0999799

    Google Scholar 

  12. Goodman, F., Hauschild, H.: Affine Birman–Wenzl–Murakami algebras and tangles in the solid torus. Fundam. Math. 190, 77–137 (2006). MR2232856 arXiv:math.QA/0411155

    Google Scholar 

  13. Goodman, F., Mosley, H.: Cyclotomic Birman–Wenzl–Murakami algebras I: freeness and realization as tangle algebras. J. Knot Theory Its Ramif. 18(8), 1089–1127 (2009). MR2554337 arXiv:math/0612064

    Google Scholar 

  14. Goodman, F., Mosley, H.: Cyclotomic Birman-Wenzl-Murakami algebras II: admissibility relations and freeness. Algebras Represent. Theory 14(1), 1–39 (2011). MR2763289 arXiv:math.QA/0612065

    Google Scholar 

  15. Häring-Oldenburg, R.: The reduced Birman-Wenzl algebra of Coxeter type B. J. Algebra 213, 437–466 (1999). MR1673464

    Article  MathSciNet  MATH  Google Scholar 

  16. Halverson, T., Ram, A.: Characters of algebras containing a Jones basic construction: the Temperley-Lieb, Brauer, Okada and Birman-Wenzl algebras. Adv. Math. 116, 263–321 (1995). MR1363766

    Article  MathSciNet  MATH  Google Scholar 

  17. Koike, K., Terada, I.: Young diagrammatic methods for the representation theory of the classical groups of type \(B_n,\, C_n,\, D_n\). J. Algebra 107, 466–511 (1987). MR0885807

    Article  MathSciNet  MATH  Google Scholar 

  18. Leduc, R., Ram, A.: A ribbon Hopf algebra approach to the irreducible representations of centralizer algebras: The Brauer, Birman-Wenzl, and Type A Iwahori-Hecke algebras. Adv. Math. 125, 1–94 (1997). MR1427801

    Article  MathSciNet  MATH  Google Scholar 

  19. Molev, A.: Yangians and classical Lie algebras, Mathematical Surveys and Monographs 143. American Mathematical Society, Providence, RI (2007). ISBN: 978-0-8218-4374-1 MR2355506

  20. Molev, A., Rozhkovskaya, N.: Characteristic maps for the Brauer algebra. arXiv:1112.0620v1

  21. Nazarov, M.: Young’s orthogonal form for Brauer’s centralizer algebra. J. Algebra 182, 664–693 (1996). MR1398116

    Article  MathSciNet  MATH  Google Scholar 

  22. Orellana R., Ram, A.: Affine braids, Markov traces and the category \(\cal O\). In: Mehta V.B. (ed.) Proceedings of the International Colloquium on Algebraic Groups and Homogeneous Spaces Mumbai 2004, Tata Institute of Fundamental Research, Narosa Publishing House, Amer. Math. Soc., pp. 423–473 (2007). MR2348913 arXiv:math/0401317

  23. Perelomov, A.M., Popov, V.S.: A generating function for Casimir operators. Dokl. Akad. Nauk SSSR 174, 1021–1023 (1967). (Russian) MR0214703

    MathSciNet  Google Scholar 

  24. Perelomov, A.M., Popov, V.S.: Casimir operators for the orthogonal and symplectic groups. Jadernaja Fiz. 3, 1127–1134 (Russian); translated as Soviet J. Nuclear Phys. 3, 819–824 (1996). MR0205621

    Google Scholar 

  25. Reshetikhin, NYu., Takhtadzhyan, L.A., Faddeev, L.D.: Quantization of Lie groups and Lie algebras. Algebra i Analiz 1, 178–206 (1989) (Russian); translation in Leningrad Math. J. 1, 193–225 (1990)

  26. Turaev, V., Wenzl, H.: Quantum invariants of 3-manifolds associated with classical simple Lie algebras. Int. J. Math. 4(2), 323–358 (1993). MR1217386

    Article  MathSciNet  MATH  Google Scholar 

  27. Weyl, H.: Classical groups, their invariants and representations. Princeton University Press, Princeton, NJ (1946). MR0000255

    MATH  Google Scholar 

  28. Wilcox, S., Yu, S.: The cyclotomic BMW algebra associated with the two string type B braid group. Commun. Algebra (to appear). arXiv:math.RT/0611518

  29. Wilcox, S., Yu, S.: On the freeness of the cyclotomic BMW algebras: admissibility and an isomorphism with the cyclotomic Kauffman tangle algebras. arXiv:0911.5284

  30. Yu, S.: The cyclotomic Birman-Murakami-Wenzl algebras. Ph.D. Thesis, University of Sydney (2007). arXiv:0810.0069

  31. Zhelobenko, D.P.: Compact Lie groups and their representations, Translated from the Russian by Israel Program for Scientific Translations. Translations of Mathematical Monographs, vol. 40. American Mathematical Society, Providence, RI (1973). MR0473097

Download references

Acknowledgments

Significant work on this paper was done while the authors were in residence at the Mathematical Sciences Research Institute (MSRI) in 2008. We thank MSRI for hospitality, support, and a wonderful working environment. We thank F. Goodman and A. Molev for their helpful comments and references. This research has been partially supported by the National Science Foundation DMS-0353038 and the Australian Research Council DP0986774 and DP120101942.

Author information

Authors and Affiliations

  1. Department of Mathematics, Dartmouth College, Hanover, NH, 03755, USA

    Zajj Daugherty

  2. Department of Mathematics and Statistics, University of Melbourne, Parkville, VIC, 3010, Australia

    Arun Ram

  3. Department of Mathematics, Campus Box 395, Boulder, CO, 80309, USA

    Rahbar Virk

Authors
  1. Zajj Daugherty
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Arun Ram
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Rahbar Virk
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zajj Daugherty.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Daugherty, Z., Ram, A. & Virk, R. Affine and degenerate affine BMW algebras: actions on tensor space. Sel. Math. New Ser. 19, 611–653 (2013). https://doi.org/10.1007/s00029-012-0105-3

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00029-012-0105-3

Keywords

Mathematics Subject Classification (2010)

Search

Navigation