Drinfel'd doubles and ehresmann doubles for lie algebroids and lie bialgebroids

Electronic Research Announcements of the American Mathematical Society

Volumn 4, Issue 11, 1998, Pages 74-87

  Mackenzie, K C H ^a  

^a UNIVERSITY OF SHEFFIELD   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 85009827370     PISSN: None     EISSN: 10796762     Source Type: Journal    
DOI: 10.1090/S1079-6762-98-00050-X     Document Type: Article

References (31)

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 Math- ematik 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 Interna- tional Congress of Mathematicians, Berkeley, 1986, pages 798-820. American Mathematical Society, Providence, RI, 1987. MR 89f:17017
9. A. C. Ehresmann, editor. Charles Ehresmann: Evres complétes 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. Urbański. 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. Diérential 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. Diérential Geom., 31:501-526, 1990. MR 91c:22012
18. K. Mackenzie. Lie groupoids and Lie algebroids in diérential 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 élds. 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