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Quantum Groups pp 104-119 | Cite as

Twisted yangians and infinite-dimensional classical Lie algebras

  • G. I. Olshanskii
I. Quantum Groups, Deformation Theory And Representation Theory
Part of the Lecture Notes in Mathematics book series (LNM, volume 1510)

Abstract

The Yangians are quantized enveloping algebras of polynomial current Lie algebras and twisted Yangians should be their analogs for twisted polynomial current Lie algebras. We define and study certain examples of twisted Yangians and describe their relationship to a problem which arises in representation theory of infinite-dimensional classical groups.

Keywords

Hopf Algebra Dynkin Diagram Involutive Automorphism Quantum Determinant Left Coideal 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    I.V. Cherednik, Factorizing particles on a half-line and root systems, Theor. Math. Phys. 16, no. 1 (1984), 35–44.MathSciNetzbMATHGoogle Scholar
  2. [2]
    I.V. Cherednik, Quantum groups as hidden symmetries of classical representation theory, Diff. Geom. Meth. Math. Phys. (Proc. 17-th Intern. Conf.), World Scientific, Singapore, 1989, pp. 47–45.Google Scholar
  3. [3]
    J. Dixmier, Algèbres enveloppantes, Gauthier-Villars, Paris, 1974.zbMATHGoogle Scholar
  4. [4]
    V.G. Drinfeld, Hopf algebras and the quantum Yang-Baxter equation, Dokl. Akad. Nauk SSSR 283 (1985), 1060–1064; Soviet Math. Dokl. 32 no. 1 (1985), 245–258, English transl..MathSciNetGoogle Scholar
  5. [5]
    V.G.Drinfeld, Degenerated affine Hecke algebras and Yangians, Funct. Anal. Appl. 20 no. 1 (1986).Google Scholar
  6. [6]
    V.G. Drinfeld, New realization of the Yangians and the quantized affine algebras, Dokl. Akad. Nauk SSSR 296 (1987), 13–17; Soviet Math. Dokl. 36 (1988), 212–216, English transl..Google Scholar
  7. [7]
    V.G. Drinfeld, Quantum groups, Proc. ICM-86, vol. 1, Berkeley, 1987, pp. 789–820.MathSciNetGoogle Scholar
  8. [8]
    B.L. Feigin and D.B. Fuks, Casimir operators in modules over the Virasoro algebra, Dokl. Akad. Nauk SSSR 269 (1984), 1060–1064; Soviet Math. Dokl. 32 (1985), 1057–1060, English transl..MathSciNetzbMATHGoogle Scholar
  9. [9]
    V.G. Kac, Laplace operators of infinite-dimensional Lie algebras and theta functions, Proc. Nat. Acad. Sci. USA 81 (1984), 645–647.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    A.N. Kirillov and N.Yu. Reshetikhin, The Yangians, Bethe ansatz and combinatorics, Lett. Math. Phys. 12 (1986), 199–208.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    M.L. Nazarov, Quantum Berezinian and the classical Capelli identity, Lett. Math. Phys. 21 (1991), 123–131.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    M.L.Nazarov, Yangians of the “strange” Lie superalgebras, Published in this volume.Google Scholar
  13. [13]
    G.I. Olshanskii, Extension of the algebra U(g) for the infinite-dimensional classical Lie algebras g and the Yanginans Y(g[(m)), Dokl. Akad. Nauk SSSR 297 (1987), 1050–1054; Soviet Math. Dokl. 36 (1988), 569–573, English transl..Google Scholar
  14. [14]
    G.I. Olshanskii, Yanginans and universal enveloping algebras, Diff. Geometry, Lie Groups and Mechanics. IX. Zap. Nauchn. Semin. LOMI 164 (1987), 142–150, Leningrad; J. Soviet Math. 47 no. 2 (1989), 2466–2473, English translation:.Google Scholar
  15. [15]
    G.I.Olshanskii, Representations of infinite-dimensional classical groups, limits of enveloping algebras and Yangians, Preprint (September, 1990), 61, To be published in a collection of the series “Advances in Soviet Mathematics”, Amer. Math. Soc..Google Scholar
  16. [16]
    N.Yu. Reshetikhin and M.A. Semenov-Tian-Shansky, Central extensions of quantum groups, Lett. Math. Phys. 19 (1990), 133–142.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    E.K. Sklyanin, Boundary conditions for integrable quantum systems, J. Phys. A21 (1988), 2375–2389.MathSciNetzbMATHGoogle Scholar
  18. [18]
    D.P.Želobenko, Compact Lie groups and their representations, Transl. Math. Monogr. no. 40 (1973), Amer. Math. Soc., Providence, R.I.Google Scholar

Copyright information

© Springer-Verlag 1992

Authors and Affiliations

  • G. I. Olshanskii
    • 1
  1. 1.Institute of Geography of the USSR Academy of SciencesMoscowRussia

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