Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
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)
Skylanin, E.K.: Some algebraic structures connected with the Yang-Baxter equation. Func. Analiz i ego pril.16, (4), 27–34 (1982)
Drinfeld, V.G.: Hopf algebras and the quantum Yang-Baxter equation. DAN SSSR283, 1060–1064 (1985)
Drinfeld, V.G.: Quantum groups. Proc. of Int. Congr. of Mathem. MSRI Berkeley, pp. 798–820 (1986)
Jimbo, M.:Q-difference analogue ofU q (G) and the YB equation. Lett. Math. Phys.10, 63–69 (1985)
Reshetikhin, N.Yu., Takhtajan, L.A., Faddeev, L.D.: Quantization of Lie groups and Lie algebras. Algebra i Analiz1, 178–206 (1989)
Drinfeld, V.G.: Quasicocomutative Hopf algebras. Algebra Anal.1, 30–45 (1989)
Rosso, M.: An analogue of PBW theorem and the universalR-matrix forU h sl(n+1). Commun. Math. Phys.124, 307–318 (1989)
Kirillov, A.N., Reshetikhin, N.:Q-Weyl group and a multiplicative formula for universalR-matrices. Preprint HUTMP 90/B261 (1990)
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)
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)
Kac, V.G.: Infinite-dimensional algebras, Dedekind's η-function, classical Mobius function and the very strange formula. Adv. Math.30, 85–136 (1987)
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)
Tolstoy, V.N.: Extremal projectors for reductive classical Lie superalgebras with nondegenerated general Killing form. Usp. Math. Nauk40, 225–226 (1985)
Tolstoy, V.N.: Extremal projectors for contragredient Lie algebras and superalgebras of finite growth. Usp. Math. Nauk44, 211–212 (1989)
Lusztig, G.: Canonical bases arising from quantized enveloping algebras. Preprint MIT (1990)
Lusztig, G.: Quantum groups at roots of 1. Preprint MIT (1990)
De Concini, C., Kac, V.: Representations of quantum groups at roots of 1. Preprint MIT (1990)
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 1989
Author information
Authors and Affiliations
Additional information
Communicated by Ya. G. Sinai
Rights and permissions
About this article
Cite this article
Khoroshkin, S.M., Tolstoy, V.N. UniversalR-matrix for quantized (super)algebras. Commun.Math. Phys. 141, 599–617 (1991). https://doi.org/10.1007/BF02102819
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02102819