2007/09/26 by Theodore Voronov, Voronov, Theodore
Mathematics · Physics and Astronomy · #17B66 #18D05 #58A50 #58C50 #58H05 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Homotopy and Cohomology in Algebraic Topology #Mathematical Physics (math-ph) #Primary 53D17. Secondary 17B62 #Symplectic Geometry (math.SG) #math-ph #math.DG #math.MP #math.SG #msc:17B62 #msc:17B66 #msc:18D05 #msc:53D17. #msc:58A50 #msc:58C50 #msc:58H05
paper · pdf · doi:10.48550/arxiv.0709.4232
LaTeX, 17 pages; based on a talk at ESI, August/September 2007
arxiv created 2007/09/26 · openalex publication_date 2007/09/26 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/31
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 (super)manifolds 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 Z2-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.