$Q$-Manifolds and Mackenzie Theory

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.

This is a substantial re-work of our earlier paper arXiv:math.DG/0608111. In particular, we included various details as well as some new statements that may have independent interest

References in corpus (1)

• Duality functors for triple vector bundles

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• Categorification of VB-Lie algebroids and VB-Courant algebroids