Abstract
We recall the basic theory of double vector bundles and the canonical pairing of their duals, introduced by the author and by Konieczna and Urbański. We then show that the relationship between a double vector bundle and its two duals can be understood simply in terms of an associated cotangent triple vector bundle structure. In particular, we show that the dihedral group of the triangle acts on this triple via forms of the isomorphisms R, introduced by the author and Ping Xu. We then consider the three duals of a general triple vector bundle and show that the corresponding group is neither the dihedral group of the square nor the symmetry group on four symbols.
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Dedicated to Alan Weinstein on the occasion of his sixtieth birthday.
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© 2005 Birkhäuser Boston
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Mackenzie, K.C.H. (2005). Duality and triple structures. In: Marsden, J.E., Ratiu, T.S. (eds) The Breadth of Symplectic and Poisson Geometry. Progress in Mathematics, vol 232. Birkhäuser Boston. https://doi.org/10.1007/0-8176-4419-9_15
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DOI: https://doi.org/10.1007/0-8176-4419-9_15
Publisher Name: Birkhäuser Boston
Print ISBN: 978-0-8176-3565-7
Online ISBN: 978-0-8176-4419-2
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