[English]
17:00 -- 18:30 Ȋwȓ(wLpX)
Tea: 16:30 -- 17:00 R[

Last updated March 23, 2017
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411 -- 056, 17:00 -- 18:30

Alexander Voronov (University of Minnesota)

Homotopy Lie algebroids

Abstract: A well-known result of A. Vaintrob [Vai97] characterizes Lie algebroids and their morphisms in terms of homological vector fields on supermanifolds. We give an interpretation of Lie bialgebroids and their morphisms in terms of odd symplectic dg-manifolds, building on the approach of D. Roytenberg[Roy99]. This extends naturally to the homotopy Lie case and leads to the notion of L-bialgebroids and L-morphisms between them.


425 -- 056, 17:00 -- 18:30

v Y (Ócmw)

Formality of the Goldman-Turaev Lie bialgebra and the Kashiwara-Vergne problem in positive genus

Abstract: This talk is based on a joint work with A. Alekseev, N. Kawazumi and F. Naef. Given a compact oriented surface with non-empty boundary and a framing of the surface, one can define the Lie bracket (Goldman bracket) and the Lie cobracket (Turaev bracket) on the vector space spanned by free homotopy classes of loops on the surface. These maps are of degree minus two with respect to a certain filtration. Then one can ask the formality of this Lie bialgebra: is the Goldman-Turaev Lie bialgebra isomorphic to its associated graded?
For surfaces of genus zero, we showed that this question is closely related to the Kashiwara-Vergne (KV) problem in Lie theory (arXiv:1703.05813). A similar result was obtained by G. Massuyeau by using the Kontsevich integral.
Our new topological interpretation of the classical KV problem leads us to introduce a generalization of the KV problem in connection with the formality of the Goldman-Turaev Lie bialgebra for surfaces of positive genus. We will discuss the existence and uniqueness of solutions to the generalized KV problem.


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