Quasi-Lie bialgebras of loops in quasi-surfaces

Closed string operators in topology leading to Lie bialgebras and higher string algebra

This paper explains the conjectured algebraic duality between genus zero Gromov-Witten theory and genus zero "Closed String topology". This duality in another perspective is discussed on page 87 of the book "Frobenius manifold, quantum cohomology, and moduli spaces" (by Yuri Manin). This paper also discusses Fulton MacPherson strata.

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。

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.

The Goldman-Turaev Lie bialgebra and the Johnson homomorphisms

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

闭曲面上Goldman-Turaev李双代数的形式性

The formality of the Goldman-Turaev Lie bialgebra on a closed surface

该研究利用非交换联络工具重述高亏格Kashiwara-Vergne群与配极子，解决了闭曲面情形下Goldman-Turaev李双代数相伴分次的幂幺自同构群的确定问题。

On Lie bialgebras of loops on orientable surface

Goldman (Invent. Math. 85(2) (1986) 263) and Turaev (Ann. Sci. Ecole Norm. Sup. (4) 24 (6)(1991) 635) found a Lie bialgebra structure on the vector space generated by non-trivial free homotopy classes of loops on an orientable surface. Chas (Combinatorial Lie bialgebras of curves on surfaces, Topology 43 (2004) 543), by the aid of the computer, found a negative answer to Turaev's question about the characterization of multiples of simple classes in terms of the cobracket, in every surface of negative Euler characteristic and positive genus. However, she left open Turaev's conjecture, namely if…