Abstract
It was proved by Montaner and Zelmanov that up to classical twisting Lie bialgebra structures on \({\mathfrak{g}[u]}\) fall into four classes. Here \({\mathfrak{g}}\) is a simple complex finite-dimensional Lie algebra. It turns out that classical twists within one of these four classes are in a one-to-one correspondence with the so-called quasi-trigonometric solutions of the classical Yang–Baxter equation. In this paper we give a complete list of the quasi-trigonometric solutions in terms of sub-diagrams of the certain Dynkin diagrams related to \({\mathfrak{g}}\) . We also explain how to quantize the corresponding Lie bialgebra structures.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Belavin A., Drinfeld V.: Triangle equations and simple Lie algebras. Math. Phys. Rev. 4, 93–165 (1984) Harwood Academic
Delorme P.: Classification des triples de Manin pour les algèbres de Lie réductives complexes. J. Algebra 246, 97–174 (2001)
Drinfeld, V.: Quantum groups. In: Proceedings ICM (Berkeley 1986), vol. 1, pp. 798–820. American Mathematical Society (1987)
Halbout G.: Formality theorem for Lie bialgebras and quantization of twists and coboundary r-matrices. Adv. Math. 207, 617–633 (2006)
Jimbo M.: Quantum R-matrix for the generalized Toda system. Commun. Math. Phys. 102(1), 537–547 (1986)
Khoroshkin, S., Pop, I., Stolin, A., Tolstoy, V.: On some Lie bialgebra structures on polynomial algebras and their quantization. Preprint No. 21, Mittag-Leffler Institute, Sweden (2003/2004)
Khoroshkin S., Pop I., Samsonov M., Stolin A., Tolstoy V.: On some Lie bialgebra structures on polynomial algebras and their quantization. Commun. Math. Phys. 282, 625–662 (2008)
Khoroshkin S., Tolstoy V.: Universal R-matrix for quantized (super) algebras. Commun. Math. Phys. 141, 599–617 (1991)
Montaner F., Zelmanov E.: Bialgebra structures on current Lie algebras. Preprint, University of Wisconsin, Madison (1993)
Stolin A.: On rational solutions of Yang–Baxter equations. Maximal orders in loop algebra. Commun. Math. Phys. 141, 533–548 (1991)
Stolin A.: Some remarks on Lie bialgebra structures for simple complex Lie algebras. Commun. Algebra 27(9), 4289–4302 (1999)
Stolin A., Yermolova-Magnusson J.: The 4th structure. Czech. J. Phys. 56(10/11), 1293–1297 (2006)
Tolstoy, V.: From quantum affine Kac–Moody algebras to Drinfeldians and Yangians. In: Kac–Moody Lie Algebras and Related Topics. Contemp. Math., vol. 343, pp. 349–370. American Mathematical Society (2004)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Pop, I., Stolin, A. Lagrangian Subalgebras and Quasi-trigonometric r-Matrices. Lett Math Phys 85, 249–262 (2008). https://doi.org/10.1007/s11005-008-0264-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11005-008-0264-5