| Popis: |
We define a family KV (g,n+1) KV ( g , n + 1 ) of Kashiwara–Vergne problems associated with compact connected oriented 2-manifolds of genus g with n+1 n + 1 boundary components. The problem KV (0,3) KV ( 0 , 3 ) is the classical Kashiwara–Vergne problem from Lie theory. We show the existence of solutions to KV (g,n+1) KV ( g , n + 1 ) for arbitrary g and n . The key point is the solution to KV (1,1) 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) g ( g , n + 1 ) . In more detail, we show that every solution to KV (g,n+1) KV ( g , n + 1 ) induces a Lie bialgebra isomorphism between g (g,n+1) g ( g , n + 1 ) and its associated graded gr g ( g , n + 1 ) . For g=0 g = 0 , a similar result was obtained by G. Massuyeau using the Kontsevich integral. For g≥1 g ≥ 1 , n=0 n = 0 , our results imply that the obstruction to surjectivity of the Johnson homomorphism provided by the Turaev cobracket is equivalent to the Enomoto–Satoh obstruction. |
