Higher genus Kashiwara–Vergne problems and the Goldman–Turaev Lie bialgebra

Abstract

Abstract. We define a family KV (g,n+1) of Kashiwara-Vergne problems associated with compact connected oriented 2-manifolds of genus g with n + 1 boundary components. The problem KV (0,3) is the classical Kashiwara-Vergne problem from Lie theory. We show the existence of solutions of KV (g,n+1) for arbitrary g and n. The key point is the solution of KV(1,1) based on the results by B. Enriquez on elliptic associators. Our construction is motivated by applications to the formality problem for the Goldman-Turaev Lie bialgebra g (g,n+1) . In more detail, we show that every solution of KV (g,n+1) induces a Lie bialgebra isomorphism between g (g,n+1) and its associated graded gr g (g,n+1) . For g = 0, a similar result was obtained by G. Massuyeau using the Kontsevich integral.This paper is a summary of our results. Details and proofs will appear elsewhere.

Topology and Geometry

“…Turaev's cobracket played a major role in a series of papers by him and Massuyeau published between 2013 and 2018 [45,46,47,48], and in other papers published during the same period by N. Kawazumi, Y. Kuno, A. Alekseev and F. Naef [2,35,3]. We shall mention some of these developments in §5.10 below.…”

Vladimir Turaev, friend and colleague

“…In the papers [2] and [3] by Alekseev, Kawazumi, Kuno and Naef, a highergenus Kashiwara-Vergne problem is introduced using the Turaev cobracket and an isomorphism is established between the Goldman-Turaev Lie bialgebra and another Lie bialgebra structure arising from this Kashiwara-Vergne problem. In the case of surfaces of genus zero, a similar result was obtained by Massuyeau in [44], using the Kontsevich integral.…”

Stable cohomology of graph complexes

“…Enriquez defines the elliptic Grothendieck-Teichmüller Lie algebra as a semi-direct product grt ell 1 = grt 1 ⋉ r ell where r ell is an explicitly defined Lie subalgebra of the Lie algebra of derivations of the free Lie algebra on two generators [9]. By [12,Corollary 3] r ell is moreover isomorphic to H 0 (GC (1) ). On the other hand, an elliptic variant of krv was introduced in [1,2] motivated by string topology of surfaces, and in [25] in mould theory.…”

“…By [12,Corollary 3] r ell is moreover isomorphic to H 0 (GC (1) ). On the other hand, an elliptic variant of krv was introduced in [1,2] motivated by string topology of surfaces, and in [25] in mould theory. It is also naturally described as a Lie subalgebra of derivations of the free Lie algebra on two generators and is denoted krv 1,1 .…”