On Lie bialgebras of loops on orientable surface

A regular homotopy version of the Goldman-Turaev Lie bialgebra, the Enomoto-Satoh traces and the divergence cocycle in the Kashiwara-Vergne problem

By introducing a refinement of the Goldman-Turaev Lie bialgebra, we interpret the divergence cocycle in the Kashiwara-Vergne problem and the Enomoto-Satoh obstructions for the surjectivity of the Johnson homomorphisms as some part of a regular homotopy version of the Turaev cobracket.

Partitions of cyclic words and Goldman-Turaev Lie bialgebra

The free $\mathbb{Z}$-module generated from the set of non-trivial homotopy classes of closed curves on an oriented surface has the structure of Lie bialgebra by two operations, the Goldman bracket and Turaev cobracket. M. Chas gave a combinatorial redefinition of these two operations through a natural identification of the homotopy classes of closed curves on the surface with the cyclic words in the generators and their inverses of the fundamental group of the surface. We present a new approach to give a combinatorial definition of the bracket and cobracket, focusing on the information given …

Combinatorial Lie bialgebras of curves on surfaces

Goldman and Turaev found a Lie bialgebra structure on the vector space generated by non-trivial free homotopy classes of curves on a surface. When the surface has non-empty boundary, this vector space has a basis of cyclic reduced words in the generators of the fundamental group and their inverses. We give a combinatorial algorithm to compute this Lie bialgebra on this vector space of cyclic words. Using this presentation, we prove a generalization of Goldman's result relating the bracket to disjointness of curve representatives when one of the classes is simple. We exhibit some examples we fo…

Quasi-Lie bialgebras of loops in quasi-surfaces

We discuss natural operations on loops in a quasi-surface and show that these operations define a structure of a quasi-Lie bialgebra in the module generated by the set of free homotopy classes of non-contractible loops.

The Goldman-Turaev Lie bialgebra and the Johnson homomorphisms

We survey a geometric approach to the Johnson homomorphisms using the Goldman-Turaev Lie bialgebra.

沿着简单分离曲线拆分戈德曼 - 图拉耶夫李双代数

Splitting the Goldman-Turaev Lie bialgebra along a simple separating curve

研究沿着简单分离曲线拆分戈德曼 - 图拉耶夫李双代数，引入双李双模范畴概念及代数构造，通过几何应用将相关运算组装成双李双模范畴，证明了该李双代数的分解定理。