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Nondegeneracy of simple Lie algebras of real rank 1

Work in progress with Philippe Monnier

Conn showed that compact simple Lie algebras are smoothly nondegenerate.

Weinstein showed that simple Lie algebras of real rank >= 2 are smoothly degenerate.

What about the case of real rank 1 ?

There are only 3 series of real rank 1: so(n,1), su(n,1) and sp(n,1)

(orthogonal, unitary . . . → Read More: Nondegeneracy of simple Lie algebras of real rank 1

Computation of entropy

This is a small research project that I intend to give to some master students for their memoir:

– Examples of geometric structures of zero entropy. In particular, write down the proof of the fact that linear Poisson structures have zero entropy.

– Example of positive entropy. Explicit computations. Comparison with Lyapunov exponents, Godbillon-Vey . . . → Read More: Computation of entropy

Geometric approach to Levi decomposition

(A work in progress of mine)

Recently there have been very interesting papers by Crainic, Fernandes and company about geometric approaches to normal forms problems in Poisson geometry. Using groupoid techniques (in particular integration & the path method & their integrability iteria) they recover Conn’s linearization theorem (for Poisson structures), and also my linearization . . . → Read More: Geometric approach to Levi decomposition

Levi decomposition for Lie algebras of vector fields

Every geometric structure (e.g. a vector field, a foliation, a metric, etc.) has its associated Lie algebra of vector fields which preserves the structure. These Lie algebra are often infinite-dimensional. Still, they may admit a Levi decomposition similar to the finite-dimensional case.

In our Poisson book with JP Dufour, we wrote down the formal . . . → Read More: Levi decomposition for Lie algebras of vector fields

Linearization of singular foliations

There are still many things to be done about the linearization of singular foliations

First of all, what is a linear singular foliation ?

Actually, the are two different non-equivalent notions of linear foliations:

1) Those given by linear actions of Lie groups

2) Those given by linear integrable multi-vector fields (Nambu structures)

The . . . → Read More: Linearization of singular foliations