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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 \lq\lq double\rq\rq\ is now used in the Ehresmann sense) which unifies many iterated constructions in differential geometry.

Copyright 1998 American Mathematical Society

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Article Info

• ERA Amer. Math. Soc. 04 (1998), pp. 74-87
• Publisher Identifier: S 1079-6762(98)00050-X
• 1991 Mathematics Subject Classification. Primary 58F05
• Key words and phrases.
• Received by the editors July 12, 1998
• Posted on October 22, 1998
• Communicated by Frances Kirwan
• Comments (When Available)

K. C. H. Mackenzie

School of Mathematics and Statistics, University of Sheffield, Sheffield, S3 7RH, England

E-mail address: K.Mackenzie@sheffield.ac.uk