XXV Workshop on Geometric Methods in Physics | 2-8.07.2006 | |||||

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## Kirill Mackenzie## DOUBLE LIE ALGEBROIDS IN POISSON GEOMETRYFor any Poisson structure on a smooth manifold M the bracket of functions induces
a bracket of 1--forms which makes the cotangent bundle T^*M a Lie algebroid. On
the other
hand, for any Lie group G there is a Lie groupoid structure on T^*G with
base manifold the dual of the Lie algebra. It is a basic fact that a Poisson structure on
a Lie group G makes G a Poisson Lie group if and only if the two structures
on T^*G commute in the
categorical sense, that is, if and only if T^*G is a
Lie groupoid object in the category of Lie algebroids. This observation shows that the Drinfel'd double of the Lie bialgebra is a double in the categorical sense, as well as in the sense of Drinfel'd. We will describe this and further applications of the method to Poisson actions. |