Contact and Frobeniusian forms on Lie groups

Abstract

We are interested in the construction of contact forms and Frobeniusian forms on a Lie group. Since the notions of contact and symplectic forms on a manifold can be given in terms of Cartan class of differential forms, we investigate the general behavior of the Cartan class of left invariant forms on a Lie group G, especially when G is nilpotent or semi-simple. Since any left invariant form on a Lie group is given by a linear form on the Lie algebra g of G, we study Lie algebras provided with a linear contact form or a Frobeniusian form. We construct the class of contact Lie algebras, for a given dimension, in terms of linear or quadratic deformations of the Heisenberg algebra. We also classify up to contraction the class of Frobenius Lie algebras and characterize in this case the principal element. Since semi-simple Lie algebras of rank greater than 2 are never contact Lie algebras, we study on a Lie group G the Pfaffian forms invariant by a proper subgroup J. We classify, for example, all the J-invariant Pfaffian forms on the Heisenberg group. We describe also a contact form on the simple Lie group SL(2p) invariant by the maximal compact subgroup.