Florian Schätz

Equivalences of higher derived brackets

This note elaborates on Th. Voronov’s construction [Th. Voronov, Higher derived brackets and homotopy algebras, J. Pure Appl. Algebra 202 (1–3) (2005) 133–153; Th. Voronov, Higher derived brackets for arbitrary derivations, Travaux Math. XVI (2005) 163–186] of L∞-structures via higher derived brackets with a Maurer–Cartan element. It is shown that gauge equivalent Maurer–Cartan elements induce L∞-isomorphic structures. Applications in symplectic, Poisson and Dirac geometry are discussed.

Deformations of Coisotropic Submanifolds for Fibrewise Entire Poisson Structures

We show that deformations of a coisotropic submanifold inside a fibrewise entire Poisson manifold are controlled by the L∞-algebra introduced by Oh–Park (for symplectic manifolds) and Cattaneo–Felder. In the symplectic case, we recover results previously obtained by Oh–Park. Moreover we consider the extended deformation problem and prove its obstructedness.

Holohonies for connections with values in

Given a flat connection on a manifold with values in a filtered L-infinity-algebra, we construct a morphism of coalgebras that generalizes the holonomies of flat connections with values in Lie algebras. The construction is based on Gugenheims A-infinity version of de Rhams theorem, which in turn is based on Chens iterated integrals. Finally, we discuss examples related to the geometry of configuration spaces of points in Euclidean space, and to generalizations of the holonomy representations of braid groups.

L_\infty

We define the notion of a formal connection for a smooth family of star products with fixed underlying symplectic structure. Such a formal connection allows one to relate star products at different points in the family. This generalizes the formal Hitchin connection, which was introduced by the first author. We establish a necessary and sufficient condition that guarantees the existence of a formal connection, and we describe the space of formal connections for a family as an affine space modelled on the formal symplectic vector fields. Moreover, we showthat if the parameter space has trivial first cohomology group, any two flat formal connections are related by an automorphism of the family of star products.