Simple Lie algebras, algebraic prolongations and contact structures

The story starts with the result of Mukai that every complex simple finite dimensional Lie algebra has a faithful realization as a subalgebra of an algebra of polynomials with the Legendre bracket. Every such realization is determined by a unique polynomial of degree 4. This generalizes the result of Cartan that found a 14- dimensional vector space of polynomials in 5 variables which is a Lie algebra of type G₂ with respect to the Legendre bracket. To prove his result Mukai uses the notion of algebraic prolongation of a negatively graded Lie algebra. He observes that the algebraic prolongation of a graded Heisenberg Lie algebra of dimension 2d+1 is the algebra of polynomials in 2d+1 variables with the Legendre bracket. We present a different approach to Mukai's result by using the geometry of generalized flag varieties and coadjoint orbits. We made the connection between the embedding of the simple Lie algebras into the Legendre algebra and the contact structure on the minimal nilpotent orbit on of the simple algebra. The existence is due to the fact that the big cell of the minimal orbit is a Heisenberg group and the contact structure has a canonical representative 1-form over it. We show that the polynomial of degree 4 determining the embedding is up to a precisely determined factor the generator of the algebra of invariants of a particular representation considered by Wallach and Gross [GW96] in connection with quaternionic real forms of complex Lie algebras (the so called invariant of degree 4). This invariant of degree 4 is also used by Wallach [Wal03] in the study of quaternionic discrete series, where an element is generic precisely when the invariant does not vanish at it. We provide a complete proof of the result of Tanaka that the algebraic prolongation of the negative graded Heisenberg is the Legendre algebra using results on cohomology of Lie algebras by Wallach. We interpreted the Legendre algebra as a dual Verma module and provided an intrinsic reason for the existence of the multiplicative structure. Moreover, we give a uniform description of the algebraic prolongation of a class of negatively graded algebras. Related to the invariant of degree 4, we show, using results of Kac-Popov and Sato-Kimura that the situation arising from the minimal nilpotent orbit is the only one of a symplectic representation with a free algebra of invariants. Boothby proved the converse that every complex compact simply connected homogenous contact manifold is the projectivized minimal nilpotent orbit of a unique complex simple Lie algebra. We give a new simplified proof of Boothby's result. Moreover we give a description of homogenous contact manifolds, relating them to nilpotent orbits. We mention that there is an overlap of some of our results with results of Landsberg, see [Lan08], [LR07]