Ehresmann doubles and Drinfel'd doubles for Lie algebroids and Lie bialgebroids
By K. Mackenzie. 2006
81 citations recorded by the source corpus.
Abstract The word ‘double’ was used by Ehresmann to mean ‘an object X in the category of all X’. Double categories, double groupoids and double vector bundles are instances, but the notion of Lie algebroid cannot readily be doubled in the Ehresmann sense, since a Lie algebroid bracket cannot be defined diagrammatically. In this paper we use the duality of double vector bundles to define a notion of double Lie algebroid, and we show that this abstracts the infinitesimal structure (at second order) of a double Lie groupoid. We further show that the cotangent of either Lie algebroid in a Lie bialgebroid has a double Lie algebroid structure, that reduces, in the case of a Lie bialgebra, to the classical Drinfel'd double. Thus one may say that the Drinfel'd double of a Lie bialgebroid is an Ehresmann double, and it follows that double Lie groupoids provide global models for the cotangent double of a Lie bialgebroid. A double vector bundle is called vacant if it is constructed from vector bundles A, B on a common base M as the simultaneous pullback A × M B. Given a matched pair (A, B) of Lie algebroids over base M, we show that A × M B has a double Lie algebroid structure, and that any double Lie algebroid structure on a vacant double vector bundle A × M B arises in this way. In particular, double Lie algebras in the sense of Lu and Weinstein, Kosmann-Schwarzbach and Magri, and Majid, have the structure of a vacant double Lie algebroid. Lastly we extend the construction of Lu (Duke Math. J. 86: 261–304, 1997) which associates a matched pair of Lie algebroids to any Poisson group action, to actions of Lie bialgebroids; this yields a double Lie algebroid which in general does not correspond to a matched pair. The methods of the paper are entirely ‘classical’ rather than utilizing super techniques.
Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/8aa7cde71c6770f045c9528caa6133ff3e8dfa36
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Citations in the corpus
- Lie algebroid structures on double vector bundles and representation theory of Lie algebroids (2008, background): “We note that this result was already established in [15].”
- VB-groupoids and representation theory of Lie groupoids (2010, methodology): “VB-groupoids were introduced by Pradines [15] in relation to the theory of symplectic groupoids [3] and have played an important role in the study of double structures by Kirill Mackenzie and his collaborators [8, 10, 11, 12, 14].”