Tanaka prolongation of free Lie algebras

Abstract

With the exception of the three step real free Lie algebra on two generators, all real free Lie algebras of step at least three are shown to have trivial Tanaka prolongation. This result, together with the known results concerning the step two real free Lie algebras and the step three real free Lie algebra on two generators, gives a complete list of Tanaka prolongations for real free Lie algebras.

On the maximum conjecture

“…Thus, a basis for D −2 is the second Hall basis of f (see [17, §4.1] for definitions): Proof. The proof of this assertion is actually an adjustment of Warhurst's main proof in [21] about the triviality of Tanaka prolongation of free algebras. We present here a sketch of it.…”

“…For the sake of brevity, we denote simply by c • x and d • Jx some two certain combinations r =i,j c •r x r and r ′ =i,j d •r ′ Jx r ′ . Applying Lemma 1 of [21], for r = ρ, X = x i and Y = x j gives:…”

“…Performing (somehow) long computations like those at the lower half of the page 65 of [21], one obtains:…”

A Liouville Type Theorem for Carnot Groups: A Case Study

“…In the noncommutative case, as is the situation we are interested in, the procedure was generalized by Tanaka [15] and it was used to generalize the study of infinitesimal automorphisms of G-structures by different authors. For the contact structures, the Tanaka prolongation theory was used in [18] and more recently by the authors of this article in different collaborations [4,9,10,17].…”

Rigid Two-Step Nilpotent Lie Groups Relative to Multicontact Structures

“…These questions are formulated by Tanaka and his school in terms of prolongation Lie algebras and rather deep results are presented in [19,20,21]. Other very interesting results along these lines have been obtained by Reimann on H-type groups [18], by Warhurst on filiform groups [22], jet spaces [23] and free algebras [24], and by Ottazzi on Hessenberg manifolds [16].…”