Two results on a class of Poisson structures on Lie groups. (English) Zbl 0760.58016
We investigate the Hamiltonian character of the monodromy map with respect to a class of Poisson structures on Lie groups. Under suitable assumptions on the \(r\)-matrix, we also show that the Lie group \(G\) equipped with an associated Sklyanin bracket is globally Poisson diffeomorphic to a Poisson submanifold of \(G^ 2\) equipped with a so- called twisted structure.
Reviewer: L.-C.Li (University Park)
MSC:
| 37J99 | Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems |
| 17B66 | Lie algebras of vector fields and related (super) algebras |
References:
| [1] | [A] Adler, M.: On a trace functional for formal pseudo-differential operators and the symplectic strucure of the Korteweg-de Vries type equations. Invent. Math.50, 219–248 (1979) · Zbl 0393.35058 · doi:10.1007/BF01410079 |
| [2] | [AvM] Adler, M., van Moerbeke, P.: Completely integrable systems, Euclidean Lie algebras, and curves. Adv. Math.36, 267–317 (1980) · Zbl 0455.58017 · doi:10.1016/0001-8708(80)90007-9 |
| [3] | [DL] Deift, P., Li, L.C.: Poisson geometry of the analog of the Miura maps and Bäcklund-Darboux transformations for equations of Toda type and periodic Toda flows. Commun. Math. Phys. 143, 201–214 (1991) · Zbl 0755.58050 · doi:10.1007/BF02100291 |
| [4] | [Dr] Drinfeld, V.: Hamiltonian structures on Lie groups, Lie bialgebras and the geometric meaning of the classical Yang-Baxter equations. Sov. Math., Dokl.27, 68–71 (1983) · Zbl 0526.58017 |
| [5] | [FT] Faddeev, L.D., Takhtadzhyan, L.A.: Hamiltonian, methods in the theory of solitons. Berlin Heidelberg New York: Springer 1987 · Zbl 0632.58004 |
| [6] | [H] Helgason, S.: Differential geometry, Lie groups, and symmetric spaces. New York: Academic Press 1978 · Zbl 0451.53038 |
| [7] | [K] Kostant, B.: Quantization and represetation theory. In: Proc. Research Symposium on Representations of Lie Groups. Oxford 1977. (Lond. Math. Soc. Lect. Notes Ser., vol. 34) Cambridge: Cambridge University Press 1979 · Zbl 0474.58010 |
| [8] | [L] Li, L.C.: The SVD flow on generic symplectic leaves are completely integrable (preprint) · Zbl 0899.58024 |
| [9] | [LP1] Li, L. C., Parmentier, S.: A new class of quadratic Poisson structures and the Yang-Baxter equation. C.R. Acad. Sci., Paris Sér. I307, 279–281 (1988); Nonlinear Poisson structures andr-matrices. Commun. Math. Phys.125, 545–563 (1989) · Zbl 0649.58012 |
| [10] | [LP2] Li, L.C., Parmentier, S.: Modified structures associated with Poisson Lie groups. In: Ratiu, T. (ed.): The Geometry of Hamiltonian Systems (Publ., Math. Sci. Res. Inst., vol. 22). Berlin Heidelberg New York: Springer 1991 · Zbl 0736.17023 |
| [11] | [LW] Lu, J.H., Weinstein, A.: Poisson Lie groups, dressing transformation and Bruhat decompositions. J. Differ. Geom.31, 502–526 (1990) · Zbl 0673.58018 |
| [12] | [STS1] Semenov-Tian-Shansky, M.A.: What is a classicalr-matrix? Funct. Anal. Appl.17, 259–272 (1983) · Zbl 0535.58031 · doi:10.1007/BF01076717 |
| [13] | [STS2] Semenov-Tian-Shansky, M.A.: Dressing transformations and Poisson Lie group actions. Publ. Res. Inst. Math. Sci.21, 1237–1260 (1985) · Zbl 0674.58038 · doi:10.2977/prims/1195178514 |
| [14] | [S] Symes, W. W.: Systems of Toda type, inverse spectral problems and representation theory. Invent. Math.59, 13–51 (1980) · Zbl 0474.58009 · doi:10.1007/BF01390312 |
| [15] | [V] Varadarajan, V.S.: Lie groups, Lie algebras, and their representations. Berlin Heidelberg New York: Springer 1984 · Zbl 0955.22500 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
