The pro-unipotent automorphism group of the associated graded Goldman-Turaev Lie bialgebra on a closed surface is explicitly described in terms of a divergence map and the kernel of a reduced coproduct.
Modular vector fields in non-commutative geometry
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We construct a non-commutative analogue of the modular vector field on a Poisson manifold for a given pair of a double bracket and a connection on a space of 1-forms. The key ingredient, the triple divergence map, is directly constructed from a connection on a linear category to deal with multiple base points. As an application, we give an algebraic description of the framed, groupoid version of Turaev's loop operation $\mu$ similar to the one obtained by Alekseev-Kawazumi-Kuno-Naef and the author.
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The formality of the Goldman-Turaev Lie bialgebra on a closed surface
The pro-unipotent automorphism group of the associated graded Goldman-Turaev Lie bialgebra on a closed surface is explicitly described in terms of a divergence map and the kernel of a reduced coproduct.