Remote Access Electronic Research Announcements

Electronic Research Announcements

ISSN 1079-6762



Drinfel'd doubles and Ehresmann doubles for
Lie algebroids and Lie bialgebroids

Author: K. C. H. Mackenzie
Journal: Electron. Res. Announc. Amer. Math. Soc. 4 (1998), 74-87
MSC (1991): Primary 58F05; Secondary 17B66, 18D05, 22A22, 58H05
Published electronically: October 22, 1998
MathSciNet review: 1650045
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We show that the Manin triple characterization of Lie bialgebras in terms of the Drinfel'd double may be extended to arbitrary Poisson manifolds and indeed Lie bialgebroids by using double cotangent bundles, rather than the direct sum structures (Courant algebroids) utilized for similar purposes by Liu, Weinstein and Xu. This is achieved in terms of an abstract notion of double Lie algebroid (where double is now used in the Ehresmann sense) which unifies many iterated constructions in differential geometry.

References [Enhancements On Off] (What's this?)

  • 1. C. Albert and P. Dazord. Théorie des groupoïdes symplectiques: Chapitre II, Groupoïdes symplectiques. In Publications du Département de Mathématiques de l'Université Claude Bernard, Lyon I, nouvelle série, pages 27-99, 1990. MR 95m:58134
  • 2. A. L. Besse. Manifolds all of whose geodesics are closed, volume 93 of Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer-Verlag, 1978. MR 80c:53044
  • 3. R. Brown. From groups to groupoids: a brief survey. Bull. London Math. Soc., 19:113-134, 1987. MR 87m:18009
  • 4. R. Brown and K. C. H. Mackenzie. Determination of a double Lie groupoid by its core diagram. J. Pure Appl. Algebra, 80(3):237-272, 1992. MR 93g:55022
  • 5. A. Coste, P. Dazord, and A. Weinstein. Groupoïdes symplectiques. In Publications du Département de Mathématiques de l'Université de Lyon, I, number 2/A-1987, pages 1-65, 1987. MR 90g:58033
  • 6. T. J. Courant. Dirac manifolds. Trans. Amer. Math. Soc., 319:631-661, 1990. MR 90m:58065
  • 7. V. G. Drinfel'd. Hamiltonian structures on Lie groups, Lie bialgebras and the geometric meaning of the classical Yang-Baxter equation. Soviet. Math. Dokl., 27:68-71, 1983. MR 84i:58044
  • 8. V. G. Drinfel'd. Quantum groups. In A. M. Gleason, editor, Proceedings of the International Congress of Mathematicians, Berkeley, 1986, pages 798-820. American Mathematical Society, Providence, RI, 1987. MR 89f:17017
  • 9. A. C. Ehresmann, editor. Charles Ehresmann: OEuvres complètes et commentées. Seven volumes. Imprimerie Evrard, Amiens, 1984.
  • 10. P. J. Higgins and K. C. H. Mackenzie. Algebraic constructions in the category of Lie algebroids. J. Algebra, 129:194-230, 1990. MR 92e:58241
  • 11. J. Huebschmann. Poisson cohomology and quantization. J. Reine Angew. Math., 408:57-113, 1990. MR 92e:17027
  • 12. K. Konieczna and P. Urbanski. Double vector bundles and duality. Preprint. dg-ga/9710014.
  • 13. Y. Kosmann-Schwarzbach. Exact Gerstenhaber algebras and Lie bialgebroids. Acta Appl. Math., 41:153-165, 1995. MR 97i:17021
  • 14. Zhang-Ju Liu, Alan Weinstein, and Ping Xu. Manin triples for Lie bialgebroids. J. Differential Geom., 45:547-574, 1997. MR 98f:58203
  • 15. Jiang-Hua Lu. Poisson homogeneous spaces and Lie algebroids associated to Poisson actions. Duke Math. J., 86:261-304, 1997. MR 98d:58204
  • 16. Jiang-Hua Lu and A. Weinstein. Groupoïdes symplectiques doubles des groupes de Lie-Poisson. C. R. Acad. Sci. Paris Sér. I Math., 309:951-954, 1989. MR 91i:58045
  • 17. Jiang-Hua Lu and A. Weinstein. Poisson Lie groups, dressing transformations, and Bruhat decompositions. J. Differential Geom., 31:501-526, 1990. MR 91c:22012
  • 18. K. Mackenzie. Lie groupoids and Lie algebroids in differential geometry. London Mathematical Society Lecture Note Series, no. 124. Cambridge University Press, 1987. MR 89g:58225
  • 19. K. C. H. Mackenzie. Double Lie algebroids and second-order geometry, I. Adv. Math., 94(2):180-239, 1992. MR 93f:58255
  • 20. K. C. H. Mackenzie. Double Lie algebroids and iterated tangent bundles. Submitted, 1998. 27pp.
  • 21. K. C. H. Mackenzie. On symplectic double groupoids and duality for Poisson groupoids. Submitted, 1998. 21pp.
  • 22. K. C. H. Mackenzie. Double Lie algebroids and the double of a Lie bialgebroid. Preprint, 1998. 25pp.
  • 23. K. C. H. Mackenzie and Ping Xu. Lie bialgebroids and Poisson groupoids. Duke Math. J., 73(2):415-452, 1994. MR 95b:58171
  • 24. K. C. H. Mackenzie and Ping Xu. Classical lifting processes and multiplicative vector fields. Quarterly J. Math. Oxford (2), 49:59-85, 1998. CMP 98:11
  • 25. K. C. H. Mackenzie and Ping Xu. Integrability of Lie bialgebroids. Submitted, 1997.
  • 26. S. Majid. Matched pairs of Lie groups associated to solutions of the Yang-Baxter equations. Pacific J. Math., 141:311-332, 1990. MR 91a:17009
  • 27. T. Mokri. Matched pairs of Lie algebroids. Glasgow Math. J., 39:167-181, 1997. CMP 97:15
  • 28. J. Pradines. Fibrés vectoriels doubles et calcul des jets non holonomes. Notes polycopiées, Amiens, 1974. MR 83b:58010
  • 29. J. Pradines. Remarque sur le groupoïde cotangent de Weinstein-Dazord. C. R. Acad. Sci. Paris Sér. I Math., 306:557-560, 1988. MR 89h:58222
  • 30. W. M. Tulczyjew. Geometric formulation of physical theories, volume 11 of Monographs and Textbooks in Physical Science. Bibliopolis, Naples, 1989. MR 91d:58084
  • 31. A. Weinstein. Coisotropic calculus and Poisson groupoids. J. Math. Soc. Japan, 40:705-727, 1988. MR 90b:58091

Similar Articles

Retrieve articles in Electronic Research Announcements of the American Mathematical Society with MSC (1991): 58F05, 17B66, 18D05, 22A22, 58H05

Retrieve articles in all journals with MSC (1991): 58F05, 17B66, 18D05, 22A22, 58H05

Additional Information

K. C. H. Mackenzie
Affiliation: School of Mathematics and Statistics, University of Sheffield, Sheffield, S3 7RH, England

Received by editor(s): July 12, 1998
Published electronically: October 22, 1998
Communicated by: Frances Kirwan
Article copyright: © Copyright 1998 American Mathematical Society

American Mathematical Society