210521 Charlotte Kirchhoff-Lukat

The concept of modular class is best known for Poisson structures, but is naturally defined for any Lie algebroid: It is a class in the first Lie algebroid cohomology. Poisson structures as Lie algebroids have the special feature that their dual is isomorphic to the tangent bundle and thus representatives are vector fields, which allows for the definition of the so-called modular foliation, locally spanned by Hamiltonian vector fields and the modular vector field. This modular foliation can in turn be viewed as the foliation of a Poisson structure on the total space of the real line bundle det(T^*M) (Gualtieri-Pym).

In this talk, I will show how to extend these concepts to general real or complex Dirac structures in exact Courant algebroids and discuss the information contained in the modular class of a Dirac structure in some non-Poisson examples.

This is joint work in progress with Ralph Klaasse.