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The incipient infinite cluster in high-dimensional percolation
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## Drinfel'd doubles and Ehresmann doubles for
Lie algebroids and Lie bialgebroids

### K. C. H. Mackenzie

**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 \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`

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