Duke Mathematical Journal

Lie bialgebroids and Poisson groupoids

Kirill C. H. Mackenzie and Ping Xu
Source: Duke Math. J. Volume 73, Number 2 (1994), 415-452.
First Page: Show Hide
Primary Subjects: 58H05
Secondary Subjects: 17B99, 22E99, 58F05, 58F07
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Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.dmj/1077288817
Mathematical Reviews number (MathSciNet): MR1262213
Zentralblatt MATH identifier: 0844.22005
Digital Object Identifier: doi:10.1215/S0012-7094-94-07318-3

References

[1] R. Abraham and J. Marsden, Foundations of Mechanics, 2nd ed., Addison-Wesley, New York, 1985.
[2] C. Albert and P. Dazord, Théorie des groupoïdes symplectiques. Chapitre II. Groupoïdes symplectiques, Publications du Département de Mathématiques. Nouvelle série, Publ. Dép. Math. Nouvelle Sér., vol. 1990, Univ. Claude-Bernard, Lyon, 1990, pp. 27–99.
Mathematical Reviews (MathSciNet): MR95m:58134
Zentralblatt MATH: 0849.58075
[3] A. L. Besse, Manifolds All of Whose Geodesics Are Closed, Ergeb. Math. Grenzgeb. (3), vol. 93, Springer-Verlag, Berlin, 1978.
Mathematical Reviews (MathSciNet): MR80c:53044
Zentralblatt MATH: 0387.53010
[4] A. Coste, P. Dazord, and A. Weinstein, Groupoïdes symplectiques, Publications du Département de Mathématiques. Nouvelle Série. A, Vol. 2, Publ. Dép. Math. Nouvelle Sér. A, vol. 87, Univ. Claude-Bernard, Lyon, 1987, i–ii, 1–62.
Mathematical Reviews (MathSciNet): MR90g:58033
Zentralblatt MATH: 0668.58017
[5] T. J. Courant, Dirac manifolds, Trans. Amer. Math. Soc. 319 (1990), no. 2, 631–661.
Mathematical Reviews (MathSciNet): MR90m:58065
Zentralblatt MATH: 0850.70212
Digital Object Identifier: doi:10.2307/2001258
[6] T. J. Courant, Tangent Dirac structures, J. Phys. A 23 (1990), no. 22, 5153–5168.
Mathematical Reviews (MathSciNet): MR92d:58064
Zentralblatt MATH: 0715.58013
Digital Object Identifier: doi:10.1088/0305-4470/23/22/010
[7] V. G. Drinfel'd, Hamiltonian structures on Lie groups, Lie bialgebras and the geometric meaning of classical Yang-Baxter equation, Soviet Math. Dokl. 27 (1983), 68–71.
Zentralblatt MATH: 0526.58017
Mathematical Reviews (MathSciNet): MR688240
[8] V. G. Drinfel'd, Quantum groups, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), Amer. Math. Soc., Providence, RI, 1987, pp. 798–820.
Mathematical Reviews (MathSciNet): MR89f:17017
Zentralblatt MATH: 0667.16003
[9] P. J. Higgins and K. C. H. Mackenzie, Algebraic constructions in the category of Lie algebroids, J. Algebra 129 (1990), no. 1, 194–230.
Mathematical Reviews (MathSciNet): MR92e:58241
Zentralblatt MATH: 0696.22007
Digital Object Identifier: doi:10.1016/0021-8693(90)90246-K
[10] P. J. Higgins and K. C. H. Mackenzie, Duality for base-changing morphisms of vector bundles, modules, Lie algebroids and Poisson bundles, to appear in Math. Proc. Cambridge Philos. Soc.
Mathematical Reviews (MathSciNet): MR1235995
Zentralblatt MATH: 0812.55007
Digital Object Identifier: doi:10.1017/S0305004100071760
[11] M. V. Karasev, Analogues of the objects of Lie group theory for nonlinear Poisson brackets, Math. USSR-Izv. 28 (1987), 497–527.
Zentralblatt MATH: 0624.58007
[12] Y. Kosmann-Schwarzbach and F. Magri, Poisson-Nijenhuis structures, Ann. Inst. H. Poincaré Phys. Théor. 53 (1990), no. 1, 35–81.
Mathematical Reviews (MathSciNet): MR92b:17026
Zentralblatt MATH: 0707.58048
[13] B. Kostant and S. Sternberg, Anti-Poisson algebras and current algebras, preprint.
[14] A. Lichnerowicz, Les variétés de Poisson et leurs algèbres de Lie associées, J. Differential Geometry 12 (1977), no. 2, 253–300.
Mathematical Reviews (MathSciNet): MR58:18565
Zentralblatt MATH: 0405.53024
Project Euclid: euclid.jdg/1214433987
[15] Jiang-Hua Lu and A. Weinstein, Groupoïdes symplectiques doubles des groupes de Lie-Poisson, C. R. Acad. Sci. Paris Sér. I Math. 309 (1989), no. 18, 951–954.
Mathematical Reviews (MathSciNet): MR91i:58045
Zentralblatt MATH: 0701.58025
[16] Jiang-Hua Lu and A. Weinstein, Poisson Lie groups, dressing transformations, and Bruhat decompositions, J. Differential Geom. 31 (1990), no. 2, 501–526.
Mathematical Reviews (MathSciNet): MR91c:22012
Zentralblatt MATH: 0673.58018
Project Euclid: euclid.jdg/1214444324
[17] K. C. H. Mackenzie, Lie Groupoids and Lie Algebroids in Differential Geometry, London Math. Soc. Lecture Note Ser., vol. 124, Cambridge University Press, Cambridge, 1987.
Mathematical Reviews (MathSciNet): MR89g:58225
Zentralblatt MATH: 0683.53029
[18] K. C. H. Mackenzie, Double Lie algebroids and second-order geometry. I, Adv. Math. 94 (1992), no. 2, 180–239.
Mathematical Reviews (MathSciNet): MR93f:58255
Zentralblatt MATH: 0765.57025
Digital Object Identifier: doi:10.1016/0001-8708(92)90036-K
[19] S. Majid, Matched pairs of Lie groups associated to solutions of the Yang-Baxter equations, Pacific J. Math. 141 (1990), no. 2, 311–332.
Mathematical Reviews (MathSciNet): MR91a:17009
Zentralblatt MATH: 0735.17017
Project Euclid: euclid.pjm/1102646609
[20] J. Pradines, Fibrés vectoriels doubles et calculs des jets non holonomes, Amiens, 1974, notes polycopiées.
Mathematical Reviews (MathSciNet): MR629107
Zentralblatt MATH: 0396.53016
[21] J. Pradines, Remarque sur le groupoïde cotangent de Weinstein-Dazord, C. R. Acad. Sci. Paris Sér. I Math. 306 (1988), no. 13, 557–560.
Mathematical Reviews (MathSciNet): MR89h:58222
Zentralblatt MATH: 0659.18009
[22] N. Reshetikhin and M. A. Semenov-Tian-Shansky, Quantum $R$-matrices and factorization problems, J. Geom. Phys. 5 (1988), no. 4, 533–550 (1989).
Mathematical Reviews (MathSciNet): MR92g:17019
Zentralblatt MATH: 0711.17008
Digital Object Identifier: doi:10.1016/0393-0440(88)90018-6
[23] W. M. Tulczyjew, A symplectic formulation of particle dynamics, Differential Geometric Methods in Mathematical Physics (Proc. Sympos., Univ. Bonn, Bonn, 1975) eds. K. Bleuler and A. Reetz, Lecture Notes in Math., vol. 570, Springer-Verlag, Berlin, 1977, pp. 457–463.
Mathematical Reviews (MathSciNet): MR56:6736a
Zentralblatt MATH: 0353.53021
Digital Object Identifier: doi:10.1007/BFb0087795
[24] A. Weinstein, Coisotropic calculus and Poisson groupoids, J. Math. Soc. Japan 40 (1988), no. 4, 705–727.
Mathematical Reviews (MathSciNet): MR90b:58091
Zentralblatt MATH: 0642.58025
Digital Object Identifier: doi:10.2969/jmsj/04040705
Project Euclid: euclid.jmsj/1230129807
[25] A. Weinstein and Ping Xu, Classical solutions of the quantum Yang-Baxter equation, Comm. Math. Phys. 148 (1992), no. 2, 309–343.
Mathematical Reviews (MathSciNet): MR93k:58102
Zentralblatt MATH: 0849.17015
Digital Object Identifier: doi:10.1007/BF02100863
Project Euclid: euclid.cmp/1104250952

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