[en] representations up to homotopy ; Lie algebroids ; higher holonomies

[en] We use Chen's iterated integrals to integrate representations up to homotopy. That is,
we construct an A-infinity functor from the representations up to homotopy of a Lie algebroid A to those of its infinity groupoid. This construction extends the usual integration of representations in Lie theory. We discuss several examples including Lie algebras and Poisson manifolds. The construction is based on an A-infinity version of de Rham's theorem due to Gugenheim. The integration procedure we explain here amounts to extending the construction of parallel transport for superconnections, introduced by Igusa and Block-Smith, to the case of certain differential graded manifolds.

Center for Mathematical Analysis, Geometry and Dynamical Systems, IST Lisbon (Lisbon, Portugal)

Swiss National Science Foundation (SNF-grant 200020-121640/1) ; Erwin Schrödinger Institute (Vienna, Austria) ; Center for Mathematical Analysis, Geometry and Dynamical Systems, IST Lisbon (Lisbon, Portugal) ; FCT/POCTI/FEDER through project PTDC/MAT/098936/2008 ; FCT postdoc grant SFRH/BPD/69197/2010