Alekseev–Torossian conjecture on the origin of KV solutions from Drinfel’d associators

Determine whether every solution to the Kashiwara–Vergne (KV) equations can be constructed from a Drinfel’d associator via the Alekseev–Torossian procedure; equivalently, prove that all KV solutions arise from associators.

Background

The Kashiwara–Vergne (KV) equations have deep connections to convolution on Lie groups and Lie algebras, and admit an algebraic reformulation in terms of automorphisms of free Lie algebras. It is known that Drinfel’d associators yield explicit solutions to the KV equations (e.g., via constructions by Alekseev–Enriquez–Torossian).

The outstanding question asks whether this construction is exhaustive—namely, whether every KV solution is induced by some associator. This is widely referred to as the Alekseev–Torossian conjecture and has significant implications for the interplay between topological homomorphic expansions (e.g., for Goldman–Turaev structures and welded foams) and algebraic structures governed by associators.