The free Jordan algebra J(m) on m generators is an elusive object. It has been determined when m = 1 (folklore) and m = 2 (Shirshov’s Theorem). Some partial informations are known in the case m = 3, namely the space of Jordan polynomial with three variables which are linear on the last one. We will present two conjectures. Conjecture 1, which determines combinatorially the structure of the homogenous components of J(m) is elementary but mysterious. Then we present Conjecture 2 about Lie algebra cohomology of a class of free Lie algebras in a certain category. Conjecture 2 is natural, but not elementary. Our main result is that Conjecture 2 implies Conjecture 1. The proof, which is quite long, is based on the cyclicity of the Jordan operad. Conjecture 1 has been checked up to degree 15 for m = 2, up to degree 7 for m = 3 and up to degree 6 for m > 3. In the case m = 1, the conjecture is equivalent to Jacobi triple identity. For conjecture 2, the vanishing of the cohomology has been proved up to degree 3 using polynomial functors. In recent work with J. Germoni, we found two new special identities in degree 8 and 4 variables. These identities have been checked by computer, but the interesting point is that they were predicted by our conjecture.
This video is part of the Non-Associative Day in Online, run by the European Non-Associative Algebra Seminar series.
