Cartan class of Invariant forms on Lie groups

Abstract

We are interested in the class, in the Elie Cartan sense, of left invariant forms on a Lie group. We construct the class of Lie algebras provided with a contact form and classify the frobeniusian Lie algebras up to contraction. We also study forms which are invariant by a subgroup. We show that the simple group SL(2n, R) which doesn't admit left invariant contact form, yet admits a contact form which is invariant by a maximal compact subgroup.

Sasakian Nilmanifolds

“…Therefore In other words, α2m+1 ∈ g * is an algebraic contact structure on g. Since (4.15) dα i = 0, for 1 ≤ i ≤ 2m and dα 2m+1 is a linear combination of αi ∧ αj with i, j ≤ 2m the Lie algebra g is 2-step nilpotent. By Proposition 19 in [19], we conclude that g is the Heisenberg algebra h(1, m).…”