Abstract
Poisson-Lie structures on the Lorentz group are completely classified. A method applicable to an arbitrary semisimple complex Lie group (treated as real) is developed.
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References
Drinfeld, V. G., Hamiltonian structures on Lie groups, Lie bialgebras and the meaning of the classical Yang-Baxter equations,Soviet Math. Dokl. 27, 68–71 (1983).
Drinfeld, V. G., Quantum groups,Proc. ICM, Berkeley, 1986, vol. 1, pp. 789–820.
Semenov-Tian-Shansky, M. A., Dressing transformations and Poisson Lie group actions,Publ. RIMS, Kyoto Univ. 21, 1237–1260 (1985).
Lu, J.-H. and Weinstein, A., Poisson Lie groups, dressing transformations and Bruhat decompositions,J. Differential Geom. 31, 501–526 (1990).
Lu, J.-H., Multiplicative and affine Poisson structures on Lie groups, PhD thesis, University of California, Berkeley (1990).
Belavin, A. A. and Drinfeld, V. G., Triangle equations and simple Lie algebras,Soviet Sci. Rev. C 4, 93–165 (1984). Harwood Acad. Publ., Chur (Switzerland), New York.
Woronowicz, S. L. and Zakrzewski, S., Quantum deformations of the Lorentz group. The Hopf *-algebra level,Compositio Math. 90, 211–243 (1994).
Zakrzewski, S., Realifications of complex quantum groups, in R. Gieleraket al. (eds.),Groups and Related Topics, Kluwer Acad. Pubs., Dordrecht, 1992, pp. 83–100.
Jacobson, N.,Lie Algebras, Dover, New York, 1979.
Greub, W., Halperin, S., and Van Stone, R.,Curvature, Connection and Cohomology, Pure Appl. Math. 47, III, Academic Press, New York, 1976.
Cahen, M., Gutt, S., Ohn, C., and Parker, M., Lie-Poisson groups: Remarks and examples,Lett. Math. Phys. 19, 343–353 (1990).