Communications in Mathematical Physics

, Volume 141, Issue 3, pp 599–617 | Cite as

UniversalR-matrix for quantized (super)algebras

  • S. M. Khoroshkin
  • V. N. Tolstoy


For quantum deformations of finite-dimensional contragredient Lie (super)algebras we give an explicit formula for the universalR-matrix. This formula generalizes the analogous formulae for quantized semisimple Lie algebras obtained by M. Rosso, A. N. Kirillov, and N. Reshetikhin, Ya. S. Soibelman, and S. Z. Levendorskii. Our approach is based on careful analysis of quantized rank 1 and 2 (super)algebras, a combinatorial structure of the root systems and algebraic properties ofq-exponential functions. We don't use quantum Weyl group.


Neural Network Statistical Physic Complex System Root System Nonlinear Dynamics 
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  1. 1.
    Kulish, P.P., Reshetikhin, N.Yu.: The quantum linear problem for the sine-Gordon equation and higher representations. Zap. Nauch. Sem. LOMI101, 101–110 (1986)Google Scholar
  2. 2.
    Skylanin, E.K.: Some algebraic structures connected with the Yang-Baxter equation. Func. Analiz i ego pril.16, (4), 27–34 (1982)Google Scholar
  3. 3.
    Drinfeld, V.G.: Hopf algebras and the quantum Yang-Baxter equation. DAN SSSR283, 1060–1064 (1985)Google Scholar
  4. 4.
    Drinfeld, V.G.: Quantum groups. Proc. of Int. Congr. of Mathem. MSRI Berkeley, pp. 798–820 (1986)Google Scholar
  5. 5.
    Jimbo, M.:Q-difference analogue ofU q (G) and the YB equation. Lett. Math. Phys.10, 63–69 (1985)CrossRefGoogle Scholar
  6. 6.
    Reshetikhin, N.Yu., Takhtajan, L.A., Faddeev, L.D.: Quantization of Lie groups and Lie algebras. Algebra i Analiz1, 178–206 (1989)Google Scholar
  7. 7.
    Drinfeld, V.G.: Quasicocomutative Hopf algebras. Algebra Anal.1, 30–45 (1989)Google Scholar
  8. 8.
    Rosso, M.: An analogue of PBW theorem and the universalR-matrix forU h sl(n+1). Commun. Math. Phys.124, 307–318 (1989)CrossRefGoogle Scholar
  9. 9.
    Kirillov, A.N., Reshetikhin, N.:Q-Weyl group and a multiplicative formula for universalR-matrices. Preprint HUTMP 90/B261 (1990)Google Scholar
  10. 10.
    Levendorshii, S.Z., Soibelman, Ya.S.: Some applications of quantum Weyl groups. The multiplicative formula for universalR-matrix for simple Lie algebras. Preprint RGU (1990)Google Scholar
  11. 11.
    Asherova, R.M., Smirnov, Yu.F., Tolstoy, V.N.: Description of some class of projection operators for semisimple complex Lie algebras. Matem. Zametki35, 15–25 (1979)Google Scholar
  12. 12.
    Kac, V.G.: Infinite-dimensional algebras, Dedekind's η-function, classical Mobius function and the very strange formula. Adv. Math.30, 85–136 (1987)CrossRefGoogle Scholar
  13. 13.
    Tolstoy, V.N.: Extremal projectors for quantized Kac-Moody superalgebras and some of their applications. Report at Summer Workshop “Quantum groups,” Clausthal, Germany, July (1989) (to appear in Lecture Notes in Physics)Google Scholar
  14. 14.
    Tolstoy, V.N.: Extremal projectors for reductive classical Lie superalgebras with nondegenerated general Killing form. Usp. Math. Nauk40, 225–226 (1985)Google Scholar
  15. 15.
    Tolstoy, V.N.: Extremal projectors for contragredient Lie algebras and superalgebras of finite growth. Usp. Math. Nauk44, 211–212 (1989)Google Scholar
  16. 16.
    Lusztig, G.: Canonical bases arising from quantized enveloping algebras. Preprint MIT (1990)Google Scholar
  17. 17.
    Lusztig, G.: Quantum groups at roots of 1. Preprint MIT (1990)Google Scholar
  18. 18.
    De Concini, C., Kac, V.: Representations of quantum groups at roots of 1. Preprint MIT (1990)Google Scholar
  19. 19.
    Smirnov, Yu.F., Tolstoy, V.N.: Extremal projectors and their use for solving YB problem. Proceeding of the V International Conference. Selected Topics in QFT and Mathematical Physics. Niederle, J., Fischer, J. (eds.), pp. 347–359 Singapore: World Scientific 1989Google Scholar

Copyright information

© Springer-Verlag 1991

Authors and Affiliations

  • S. M. Khoroshkin
    • 1
  • V. N. Tolstoy
    • 2
  1. 1.Institute of New TechnologiesMoscowUSSR
  2. 2.Institute of Nuclear PhysicsMoscow State UniversityMoscowUSSR

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