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 Q-manifolds and Mackenzie theory: an overview

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This text is meant to be a brief overview of the topics announced in the title and is based on my talk in Vienna August-September 2007. It does not contain new results except probably for a remark concerning Q-manifold homology, which I wish to elaborate elsewhere. -Mackenzie theory- stands for the rich circle of notions that have been put forward by Kirill Mackenzie solo or in collaboration: double structures such as double Lie groupoids and double Lie algebroids, Lie bialgebroids and their doubles, nontrivial dualities for double and multiple vector bundles, etc. -Q-manifolds- are supermanifolds with a homological vector field, i.e., a self-commuting odd vector field. They may have an extra Z-grading called weight not necessarily linked with the Z 2-grading parity. I discuss double Lie algebroids discovered by Mackenzie and explain how this quite complicated fundamental notion is equivalent to a very simple one if the language of Q-manifolds is used. In particular, it shows how the two seemingly different notions of a -Drinfeld double- of a Lie bialgebroid due to Mackenzie and Roytenberg respectively, turn out to be the same thing if properly understood.

Author: Theodore Voronov

Source: https://archive.org/


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