On certain canonical diffeomorphisms in symplectic and Poisson geometry
نویسندگان
چکیده
Associated with the canonical symplectic structure on a cotangent bundle T M is the diffeomorphism #: T (T M) −→ T (T M). This and the Tulczyjew diffeomorphism T (T M) −→ T (TM) may be derived from the canonical involution T (TM) −→ T (TM) by suitable dualizations. We show that the constructions which yield these maps extend very generally to the double Lie algebroids of double Lie groupoids, where they play a crucial role in the relations between double Lie algebroids and Lie bialgebroids. There have been several talks this meeting about notions of double for Lie bialgebroids. Some of these have derived from the 1997 construction of Liu Zhang– Ju, Alan Weinstein and Xu Ping [4] in which they introduced the notion of Courant algebroid, and some have involved elements of super mathematics. I very much hope that before long there will be a clear account of the relations between these various approaches and even a unification of them. There is another approach to the question of doubles, which was not at first related to Lie bialgebroids, but arose out of broad considerations of what may be called “second–order geometry”. It is not possible to describe this approach from scratch in an hour, but it is appropriate at this conference to indicate the broad features of it. This talk therefore takes a slice through the papers [5], [7], [6], [8], transverse to their chronological sequence and provides an alternative route to approach them. Some aspects of §2 and §4 are new. I am very grateful to Ted Voronov and Mike Prest for the splendid opportunities and good fellowship which the Workshop provided, and to the London Mathematical Society for its support. I also wish to thank Yvette Kosmann–Schwarzbach for her comments on an earlier version. 1. The three canonical diffeomorphisms with which the paper begins are associated with iterated tangent and cotangent bundles: (1) The canonical involution on an iterated tangent bundle J : T M −→ T M, 2000 Mathematics Subject Classification. Primary 58H01. Secondary 17B66, 18D05, 22A22, 58F05. c ©2000 American Mathematical Society
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