The Goldman-Turaev Lie bialgebra and the Kashiwara-Vergne problem in higher genera

Wednesday, 25 April, 2018

Published in: 

arXiv:1804.09566

For a compact oriented surface \Sigmaof genus g with n+1 boundary components, the space g spanned by free homotopy classes of loops in \Sigmacarries the structure of a Lie bialgebra. The Lie bracket was defined by Goldman and it is canonical. The Lie cobracket was defined by Turaev and it depends on the framing of \Sigma. The Lie bialgebra g has a natural decreasing filtration such that both the Goldman bracket and the Turaev cobracket have degree (−2).
In this paper, we address the following Goldman-Turaev formality problem: construct a Lie bialgebra homomorphism θ from g to its associated graded gr\, \mathfrak{g} such that gr\, \theta = {\rm id}. In order to solve it, we define a family of higher genus Kashiwara-Vergne (KV) problems for an element F\in Aut(L), where L is a free Lie algebra. In the case of g=0 and n=2, the problem for F is the classical KV problem from Lie theory. For g>0, these KV problems are new.
Our main results are as follows. On the one hand, every solution of the KV problem induces a GT formality map. On the other hand, higher genus KV problems admit solutions for any g and n. In fact, the solution reduces to two important cases: g=0,n=2 which admits solutions by Alekseev and Torossian and g=1,n=1 for which we construct solutions in terms of certain elliptic associators following Enriquez. By combining these two results, we obtain a proof of the GT formality for any g and n.
We also study the set of solutions of higher genus KV problems and introduce pro-unipotent groups KRV^{(g,n+1)} which act on them freely and transitively. These groups admit graded pro-nilpotent Lie algebras krv^{(g, n+1)}. We show that the elliptic Lie algebra krv^{(1,1)} contains a copy of the Grothendieck-Teichmuller Lie algebra grt_1 as well as symplectic derivations \delta_{2n}.

Author(s): 

Anton Alekseev
Nariya Kawazumi
Yusuke Kuno
Florian Naef