Duflo isomorphism, the Kashiwara-Vergne conjecture and Drinfeld associators

摘要

The Duflo Isomorphism Theorem establishes an isomorphism between the center of the universal enveloping algebra Z(Ug) and the ring of invariant polynomials (Sg)~g for a finite dimensional Lie algebra g over a field K of characteristic zero. One way to give a universal (independent of the structure theory) proof of the Duflo isomorphism is via the Kashiwara-Vergne conjecture on properties of the Campbell-Hausdorff series ch(x,y) = ln(e~xe~y). In this note, we review the proof of the Kashiwara-Vergne conjecture which uses the theory of Drinfeld associators following [6] and [2]. We show that in some cases, notably for g solvable and g quadratic, the Kashiwara-Vergne conjecture and the Duflo Isomorphism Theorem can be established by elementary means (we refer to it as Soft Duflo Theorems). Following [6], we address the uniqueness issue for the Kashiwara-Vergne problem. This naturally leads to a definition of the pro-unipotent group KV acting freely and transitively on the set of solutions of the Kashiwara-Vergne problem. It turns out that the group KV contains a subgroup isomorphic to the Grothendieck-Teichmüller group GRT. Conjecturally, KV ≈ Kt × GRT, where Kt is a central line spanned by the infinitesimal pure braid t. In this conjecture holds true, one can reconstruct a Drinfeld associator starting from a solution of the Kashiwara-Vergne problem.