Lie-Poisson structure on some Poisson Lie groups. (English) Zbl 0766.58018

Ginzburg, Viktor L.; Weinstein, Alan

J. Am. Math. Soc. 5, No. 2, 445-453 (1992).

Let \(K\) be a compact semisimple group endowed with the standard Poisson structure and let \(K^*\) be its Poisson dual (also endowed with its standard Poisson structure). Lu and Ratiu have recently used this framework to obtain a new proof of the nonlinear convexity theorem of Kostant. Let \({\mathcal G}\) be a complex semisimple Lie algebra and \(K\) its compact real form. Let \({\mathcal L}\) be the Lie algebra of \(K\) viewed as the compact part in the Iwasawa decomposition of \(G\) considered as a real group.
The main result of this paper is that the standard Poisson structure on the Poisson dual \(K^*\) is actually globally diffeomorphic to the linear one on the dual of \({\mathcal L}\).

Reviewer: J.Lacroix (Paris)

MSC:

53D17	Poisson manifolds; Poisson groupoids and algebroids
53D10	Contact manifolds (general theory)
53C15	General geometric structures on manifolds (almost complex, almost product structures, etc.)
17B56	Cohomology of Lie (super)algebras
58H05	Pseudogroups and differentiable groupoids

Keywords:

Poisson structure; compact Lie algebras; cohomology

References:

[1]	M. F. Atiyah, Convexity and commuting Hamiltonians, Bull. London Math. Soc. 14 (1982), no. 1, 1 – 15. · Zbl 0482.58013
[2]	M. F. Atiyah and R. Bott, The moment map and equivariant cohomology, Topology 23 (1984), no. 1, 1 – 28. · Zbl 0521.58025
[3]	Glen E. Bredon, Introduction to compact transformation groups, Academic Press, New York-London, 1972. Pure and Applied Mathematics, Vol. 46. · Zbl 0246.57017
[4]	Jack F. Conn, Normal forms for smooth Poisson structures, Ann. of Math. (2) 121 (1985), no. 3, 565 – 593. · Zbl 0592.58025
[5]	Thomas Delzant, Hamiltoniens périodiques et images convexes de l’application moment, Bull. Soc. Math. France 116 (1988), no. 3, 315 – 339 (French, with English summary). · Zbl 0676.58029
[6]	V. G. Drinfel\(^{\prime}\)d, Hamiltonian structures on Lie groups, Lie bialgebras and the geometric meaning of classical Yang-Baxter equations, Dokl. Akad. Nauk SSSR 268 (1983), no. 2, 285 – 287 (Russian).
[7]	-, Quantum groups, Proc. ICM, Berkeley 1 (1986), 789-820.
[8]	J. J. Duistermaat, On the similarity between the Iwasawa projection and the diagonal part, Mém. Soc. Math. France (N.S.) 15 (1984), 129 – 138. Harmonic analysis on Lie groups and symmetric spaces (Kleebach, 1983). · Zbl 0564.22007
[9]	W. T. van Est, Une application d’une méthode de Cartan-Leray, Nederl. Akad. Wetensch. Proc. Ser. A. 58 = Indag. Math. 17 (1955), 542 – 544 (French). · Zbl 0067.26203
[10]	D. B. Fuks, Cohomology of infinite-dimensional Lie algebras, Contemporary Soviet Mathematics, Consultants Bureau, New York, 1986. Translated from the Russian by A. B. Sosinskiĭ. · Zbl 0667.17005
[11]	V. L. Ginzburg and J. -H. Lu, Poisson calculus: Vector fields and differential forms (in preparation).
[12]	V. Guillemin and S. Sternberg, Convexity properties of the moment mapping, Invent. Math. 67 (1982), no. 3, 491 – 513. · Zbl 0503.58017
[13]	Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, vol. 80, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. · Zbl 0451.53038
[14]	Frances Clare Kirwan, Cohomology of quotients in symplectic and algebraic geometry, Mathematical Notes, vol. 31, Princeton University Press, Princeton, NJ, 1984. · Zbl 0553.14020
[15]	Bertram Kostant, On convexity, the Weyl group and the Iwasawa decomposition, Ann. Sci. École Norm. Sup. (4) 6 (1973), 413 – 455 (1974). · Zbl 0293.22019
[16]	Jean-Louis Koszul, Crochet de Schouten-Nijenhuis et cohomologie, Astérisque Numéro Hors Série (1985), 257 – 271 (French). The mathematical heritage of Élie Cartan (Lyon, 1984).
[17]	J.-H. Lu, Multiplicative and affine Poisson structures on Lie groups, Berkeley Thesis, 1990.
[18]	Jiang-Hua Lu and Tudor Ratiu, On the nonlinear convexity theorem of Kostant, J. Amer. Math. Soc. 4 (1991), no. 2, 349 – 363. · Zbl 0785.22019
[19]	Jiang-Hua Lu and Alan Weinstein, Poisson Lie groups, dressing transformations, and Bruhat decompositions, J. Differential Geom. 31 (1990), no. 2, 501 – 526. · Zbl 0673.58018
[20]	Jürgen Moser, On the volume elements on a manifold, Trans. Amer. Math. Soc. 120 (1965), 286 – 294. · Zbl 0141.19407
[21]	Alan Weinstein, Some remarks on dressing transformations, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 35 (1988), no. 1, 163 – 167. · Zbl 0653.58012
[22]	Alan Weinstein and Ping Xu, Extensions of symplectic groupoids and quantization, J. Reine Angew. Math. 417 (1991), 159 – 189. · Zbl 0722.58021