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Voronov, Theodore Th.

\(Q\)-manifolds and Mackenzie theory. (English) Zbl 1261.53080

Commun. Math. Phys. 315, No. 2, 279-310 (2012).
The main result is a characterization of double Lie algebroids, defined by K. C. H. Mackenzie in [Adv. Math. 94, No. 2, 180–239 (1992; Zbl 0765.57025); Adv. Math. 154, No. 1, 46–75 (2000; Zbl 0971.58015)], as a double vector bundle endowed with a pair of commuting homological vector fields of appropriate weights. This is achieved by the fiberwise parity reversion of the original double vector bundle in both directions, which should be contrasted with Mackenzie’s original approach based on dualizations. Though Voronov’s statement looks very natural, the proof is really hard because it requires a supergeometric rephrasing of Mackenzie’s conditions from his original definition that heavily rely on the non-trivial duality theory for double vector bundles. In fact, the paper contains two proofs of the main result, which are more or less independent: one by a brute calculation, and a more conceptual one. This theory is extended to multiple Lie algebroids which are introduced here. Since the appearance of preliminary preprint versions of this paper, it has already influenced some further work, for example, [K. C. H. Mackenzie, J. Reine Angew. Math. 658, 193–245 (2011; Zbl 1246.53112)].
Reviewer: Cornelia-Livia Bejan (Iaşi)

Cited in 2 Reviews
Cited in 39 Documents

MSC:

53D17 Poisson manifolds; Poisson groupoids and algebroids
58H05 Pseudogroups and differentiable groupoids

Keywords:

\(Q\)-manifolds; Mackenzie theory; double Lie algebroids; supermanifold; double vector bundles

Citations:

Zbl 0765.57025; Zbl 0971.58015; Zbl 1246.53112
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Full Text: DOI arXiv

References:

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