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Licensed Unlicensed Requires Authentication Published by De Gruyter August 19, 2011

Ehresmann doubles and Drinfel'd doubles for Lie algebroids and Lie bialgebroids

Kirill C. H. Mackenzie
From the journal

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.

Received: 2008-08-26
Revised: 2010-08-12
Published Online: 2011-08-19
Published in Print: 2011-September

© Walter de Gruyter Berlin · New York 2011

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