The Duflo Isomorphism Theorem states that the center of the universal enveloping algebra of a finite dimensional Lie algebra is isomorphic to the ring of invariant polynomials. The isomorphism is given by an explicit formula which involves Bernoulli numbers. While the original proof of Duflo (of 1977) was based on the structure theory, the search for a universal (structure theory independent) proof started as early as 1978 when Kashiwara and Vergne put forward a conjecture on the properties of the Campbell-Hausdorff series which implies the Duflo Isomorphism Theorem.
In the talk, we will present a solution of the Kashiwara-Vergne conjecture based on the theory of Drinfeld associators. In particular, the Bernoulli numbers in the explicit form of the Duflo isomorphism are traced back to certain coefficients in the associator. We also address a question of whether the Duflo Isomorphism Theorem and the Kashiwara-Vergne conjecture are hard (need associators for their proof) or soft statement. This leads us to an interesting conjecture which involves the Grothendieck-Teichmüller Lie algebra grt.
The talk is based on joint works with B. Enriquez and C. Torossian.