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Existence of R-matrix for a quantized Kac–Moody algebra

Published online by Cambridge University Press:  24 October 2008

Volodimir Lyubashenko
Volodimir Lyubashenko
Affiliation:
Department of Mathematics, University of York, Heslington, York, YO1 5DD
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Abstract

There is a pairing between two Borel subalgebras of a quantized Kac–Moody algebra, which plays the rôle of R-matrix. Over the field ℚ(q) this pairing is non-degenerate. We show the existence of a braiding in some categories of representations of a quantized Kac-Moody algebra.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 116 , Issue 2 , September 1994 , pp. 193 - 207
Copyright
Copyright © Cambridge Philosophical Society 1994

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References

REFERENCES

[1]Drinfeld, V. G., Hopf algebras and the quantum Yang–Baxter equation. Soviet Math. Dokl. 32 (1985), 254–258.Google Scholar
[2]Drinfeld, V. G.. Unpublished, 1986.Google Scholar
[3]Drinfeld, V. G.. Quantum groups. Proceedings of the ICM, AMS, Providence, R.I. 1 (1987), 798–820.Google Scholar
[4]Jimbo, M.. A q-difference analog of U(g) and the Yang–Baxter equation. Lett. Math. Phys. 10 (1985), 63–69.CrossRefGoogle Scholar
[5]Joyal, A. and Street, R.. Tortile Yang–Baxter operators in tensor categories. J. Pure Appl. Alg. 71 (1991), 43–51.CrossRefGoogle Scholar
[6]Kashiwara, M.. On Crystal Bases of the q-Analog of Universal Enveloping Algebras. Duke Math. J. 63 (1991), 465–516.CrossRefGoogle Scholar
[7]Kirillov, A. N. and Reshetikhin, N.. q-Weyl group and a multiplicative formula for universal R-matrices. Comm. Math. Phys. 134 (1990), no. 2, 421–431.CrossRefGoogle Scholar
[8]Khoroshkin, S. M. and Tolstoy, V. N.. Universal R-matrix for quantized (super)algebras. Comm. Math. Phys. 141 (1991), no. 3, 599–617.CrossRefGoogle Scholar
[9]Levendorskii, S. Z. and Soibelman, Ya. S.. Quantum Weyl group and multiplicative formula for the R-matrix of a simple Lie algebra. Funct. Analysis and its Appl. 25 (1991), no. 2, 143–145.CrossRefGoogle Scholar
[10]Lyubashenko, V. V.. Superanalysis and solutions to the triangles equation. Candidate's Dissertation Phys.-Mat. Sciences, Kiev, 1987.Google Scholar
[11]Majid, S.. More examples of bicrossproduct and double cross product Hopf algebras. Israel J. Math. 72 (1990), no. 1–2, 133–148.CrossRefGoogle Scholar
[12]Majid, S.. Doubles of quasitriangular Hopf algebras. Commun. Alg. 19 (1991), no. 11, 3061–3073.CrossRefGoogle Scholar
[13]Okado, M. and Yamane, H.. R-matrices with gauge parameters and multi-parameter quantized enveloping algebras (preprint), 1991.CrossRefGoogle Scholar
[14]Reshetikhin, N.. Multiparameter Quantum Groups and Twisted Quasitriangular Hopf Algebras. Lett. Math. Phys. 20 (1990), 331–335.CrossRefGoogle Scholar
[15]Tanisaki, T.. Killing Forms, Harish–Chandra Isomorphisms, and Universal R-Matrices for Quantum Algebras. Int. J. Modern Phys. A 7 (1992), 941–961.CrossRefGoogle Scholar
[16]Yetter, D. N.. Quantum groups and representations of monoidal categories. Math. Proc. Camb. Phil. Soc. 108 (1990), 261–290.CrossRefGoogle Scholar