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Q-Manifolds and Mackenzie Theory

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Abstract

Double Lie algebroids were discovered by Kirill Mackenzie from the study of double Lie groupoids and were defined in terms of rather complicated conditions making use of duality theory for Lie algebroids and double vector bundles. In this paper we establish a simple alternative characterization of double Lie algebroids in a supermanifold language. Namely, we show that a double Lie algebroid in Mackenzie’s sense is equivalent to a double vector bundle endowed with a pair of commuting homological vector fields of appropriate weights. Our approach helps to simplify and elucidate Mackenzie’s original definition; we show how it fits into a bigger picture of equivalent structures on ‘neighbor’ double vector bundles. It also opens ways for extending the theory to multiple Lie algebroids, which we introduce here.

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Authors and Affiliations

  1. School of Mathematics, University of Manchester, Oxford Road, Manchester, M13 9PL, UK

    Theodore Th. Voronov

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  1. Theodore Th. Voronov
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Correspondence to Theodore Th. Voronov.

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Communicated by Y. Kawahigashi

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Voronov, T.T. Q-Manifolds and Mackenzie Theory. Commun. Math. Phys. 315, 279–310 (2012). https://doi.org/10.1007/s00220-012-1568-y

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  • DOI: https://doi.org/10.1007/s00220-012-1568-y

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