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Double Lie algebroids and the double of a Lie bialgebroid

K. C. H. Mackenzie
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Abstract

We define a general notion of abstract double Lie algebroid. We show (1) that the double Lie algebroid of a double Lie groupoid is a double Lie algebroid in this sense; (2) that the double cotangent constructed from Lie algebroid structures on a vector bundle A and its dual A* is a double Lie algebroid if and only if (A, A*) is a Lie bialgebroid; (3) that a vacant double Lie algebroid structure is equivalent to a matched pair structure on the side Lie algebroids.

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K. C. H. Mackenzie. 1998-08-17. Double Lie algebroids and the double of a Lie bialgebroid. https://arxiv.org/abs/math/9808081

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