On triviality of the Kashiwara-Vergne problem for quadratic Lie algebras

TL;DR

This paper demonstrates that the Kashiwara-Vergne problem for quadratic Lie algebras simplifies to representing the Campbell-Hausdorff series in a specific form, providing insights into explicit solutions and a direct proof of the Duflo theorem.

Contribution

It shows the reduction of the KV problem for quadratic Lie algebras to a specific series representation, explaining existing solutions and offering a direct proof of the Duflo theorem.

Findings

Reduction of the KV problem to series representation in quadratic Lie algebras

Explicit rational solutions explained for quadratic case

Provides a direct proof of the Duflo theorem in this context

Abstract

We show that the Kashiwara-Vergne (KV) problem for quadratic Lie algebras (that is, Lie algebras admitting an invariant scalar product) reduces to the problem of representing the Campbell-Hausdorff series in the form ln(e^xe^y)=x+y+[x,a(x,y)]+[y,b(x,y)], where a(x,y) and b(x,y) are Lie series in x and y. This observation explains the existence of explicit rational solutions of the quadratic KV problem (see M. Vergne, C.R.A.S. 329 (1999), no. 9, 767--772 and A. Alekseev, E. Meinrenken, C.R.A.S. 335 (2002), no. 9, 723--728 arXiv:math/0209346), whereas constructing an explicit rational solution of the full KV problem would probably require the knowledge of a rational Drinfeld associator. It also gives, in the case of quadratic Lie algebras, a direct proof of the Duflo theorem (implied by the KV problem).