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From the Legendre diffeomorphism to double Lie algebroids

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Kirill Mackenzie
Mit, 05/12/2018 - 10:30 - 12:00
MPIM Lecture Hall

For any vector bundle $A$ there is a canonical diffeomorphism $R: T^*(A^*)\to T^*(A)$ introduced by
Ping Xu and the speaker (Duke Math. J. 1994). We describe this in detail and show how it can be extended
to the duals of any double vector bundle. This leads in a natural way to the concept of double Lie algebroid
(Crelle, 2011). We show that the double Lie algebroid structure on the cotangent double of a Lie bialgebroid
is a natural abstraction of the structure of a Poisson Lie group(oid) and we briefly consider other important
instances of double Lie algebroids and alternative approaches. 

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