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

Mackenzie, K.

doi:10.1090/S1079-6762-98-00050-X

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
DOI: https://doi.org/10.1090/S1079-6762-98-00050-X
Published electronically: October 22, 1998
MathSciNet review: 1650045
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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.

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