Advertisement

Israel Journal of Mathematics

, Volume 221, Issue 2, pp 705–729 | Cite as

Diagonal reduction algebra and the reflection equation

  • S. Khoroshkin
  • O. Ogievetsky
Article

Abstract

We describe the diagonal reduction algebra D(gl n ) of the Lie algebra gl n in the R-matrix formalism. As a byproduct we present two families of central elements and the braided bialgebra structure of D(gl n ).

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [ABRR98]
    D. Arnaudon, E. Buffenoir, E. Ragoucy and P. Roche, Universal solutions of quantum dynamical Yang-Baxter equations, Lett. Math. Phys. 44 (1998), 201–214.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [AST73]
    R. M. Ašerova, J. F. Smirnov and V. N. Tolstoi, Projection operators for simple Lie groups. II. General scheme for the construction of lowering operators. The case of the groups SU(n), Teoret. Mat. Fiz. 15 (1973), 107–119.MathSciNetzbMATHGoogle Scholar
  3. [AST79]
    R. M. Ašerova, J. F. Smirnov and V. N. Tolstoi, Description of a certain class of projection operators for complex semisimple Lie algebras, Mat. Zametki 26 (1979), 15–25, 156.MathSciNetzbMATHGoogle Scholar
  4. [ES01]
    P. Etingof and O. Schiffmann, Lectures on the dynamical Yang-Baxter equations, in Quantum groups and Lie theory (Durham, 1999), London Math. Soc. Lecture Note Ser., Vol. 290, Cambridge Univ. Press, Cambridge, 2001, pp. 89–129.Google Scholar
  5. [HIO+99]
    L. K. Hadjiivanov, A. P. Isaev, O. V. Ogievetsky, P. N. Pyatov and I. T. Todorov, Hecke algebraic properties of dynamical R-matrices. Application to related quantum matrix algebras, J. Math. Phys. 40 (1999), 427–448.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [HO]
    B. Herlemont and O. Ogievetsky, Rings of h-deformed differential operators, Theoret. and Math. Phys., to appear.Google Scholar
  7. [Isa96]
    A. P. Isaev, Twisted Yang-Baxter equations for linear quantum (super)groups, J. Phys. A 29 (1996), 6903–6910.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [Jos14]
    A. Joseph, Modules for relative Yangians (family algebras) and Kazhdan-Lusztig polynomials, Transform. Groups 19 (2014), 105–129.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [Kho04]
    S. M. Khoroshkin, An extremal projector and a dynamical twist, Teoret. Mat. Fiz. 139 (2004), 158–176.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [KN12]
    S. Khoroshkin and M. Nazarov, Mickelsson algebras and representations of Yangians, Trans. Amer. Math. Soc. 364 (2012), 1293–1367.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [KNV11]
    S. Khoroshkin, M. Nazarov and E. Vinberg, A generalized Harish-Chandra isomorphism, Adv. Math. 226 (2011), 1168–1180.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [KO08]
    S. Khoroshkin and O. Ogievetsky, Mickelsson algebras and Zhelobenko operators, J. Algebra 319 (2008), 2113–2165.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [KO10]
    S. Khoroshkin and O. Ogievetsky, Diagonal reduction algebras of gl type, Funktsional. Anal. i Prilozhen. 44 (2010), 27–49.zbMATHGoogle Scholar
  14. [KO11]
    S. Khoroshkin and O. Ogievetsky, Structure constants of diagonal reduction algebras of gl type, SIGMA Symmetry Integrability Geom. Methods Appl. 7 (2011), Paper 064, 34.Google Scholar
  15. [Kir01]
    A. A. Kirillov, Introduction to family algebras, Mosc. Math. J. 1 (2001), 49–63.MathSciNetzbMATHGoogle Scholar
  16. [Mic73]
    J. Mickelsson, Step algebras of semi-simple subalgebras of Lie algebras, Rep.Mathematical Phys. 4 (1973), 307–318.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [Ogi02]
    O. Ogievetsky, Uses of quantum spaces, in Quantum symmetries in theoretical physics and mathematics (Bariloche, 2000), Contemp. Math., Vol. 294, Amer. Math. Soc., Providence, RI, 2002, pp. 161–232.CrossRefGoogle Scholar
  18. [Zhe94]
    D. P. Zhelobenko, Representations of reductive lie algebras, VO “Nauka”, Moscow, 1994.zbMATHGoogle Scholar

Copyright information

© Hebrew University of Jerusalem 2017

Authors and Affiliations

  1. 1.ITEPMoscowRussia
  2. 2.National Research University Higher School of EconomicsMoscowRussia
  3. 3.Aix Marseille Univ, Université de Toulon, CNRS, CPTMarseilleFrance
  4. 4.Kazan Federal UniversityKazanRussia

Personalised recommendations