The Goldman-Turaev Lie bialgebra in genus zero and the Kashiwara-Vergne problem
The Goldman-Turaev Lie bialgebra in genus zero and the Kashiwara-Vergne problem
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DOI：
10.1016/j.aim.2017.12.005
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发表时间：
2017-03
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影响因子：
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通讯作者：
A. Alekseev;Nariya Kawazumi;Y. Kuno;Florian Naef
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文献类型：
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作者：
A. Alekseev;Nariya Kawazumi;Y. Kuno;Florian Naef
本文描述了零属曲面上的Goldman-Turaev李双代数理论与李论中的Kashiwara-Vergne （KV）问题之间的惊人联系。设Σ为具有非空边界的二维定向流形，K为特征为零的域。Goldman - Turaev Lie双代数由Σ上自由循环的同伦类的k -张成空间上的Goldman括号{−，−}和Turaev协括号δ定义。应用展开式θ: K π→K< x 1，…，x n>得到了用非交换变量x 1，…，x n表示的运算{−，−}和δ的代数描述。如果Σ是一个g= 0的曲面，则最低次部分{−，−}- 1和δ−1是标准定义的（并且与θ无关）。他们定义了一个循环词空间上的李双代数结构，该结构由Schedler[31]引入并研究。第二个和第三个作者推测可以定义一个展开式θ使得{−，−}={−，−}−1和δ= δ−1。本文的主要结果表明，对于零属曲面，构造这样的展开式本质上等价于KV问题。在1986年，Massuyeau用Kontsevich积分构造了这样的展开式。为了证明这一结果，我们证明了Turaev协环δ可以用双括号（对Goldman括号的升级）和在KV理论中起核心作用的非交换散度协环来构造。除此之外，这一观察给出了KV问题的一种新的拓扑解释，并允许将其扩展到具有任意数目的边界分量（和任意属，见[2]）的曲面。
In this paper, we describe a surprising link between the theory of the Goldman–Turaev Lie bialgebra on surfaces of genus zero and the Kashiwara–Vergne (KV) problem in Lie theory. Let Σ be an oriented 2-dimensional manifold with non-empty boundary and K a field of characteristic zero. The Goldman–Turaev Lie bialgebra is defined by the Goldman bracket {−,−} and Turaev cobracket δ on the K-span of homotopy classes of free loops on Σ. Applying an expansion θ: K π→ K< x 1,…, x n> yields an algebraic description of the operations {−,−} and δ in terms of non-commutative variables x 1,…, x n. If Σ is a surface of genus g= 0 the lowest degree parts {−,−}− 1 and δ− 1 are canonically defined (and independent of θ). They define a Lie bialgebra structure on the space of cyclic words which was introduced and studied by Schedler [31]. It was conjectured by the second and the third authors that one can define an expansion θ such that {−,−}={−,−}− 1 and δ= δ− 1. The main result of this paper states that for surfaces of genus zero constructing such an expansion is essentially equivalent to the KV problem. In [24], Massuyeau constructed such expansions using the Kontsevich integral. In order to prove this result, we show that the Turaev cobracket δ can be constructed in terms of the double bracket (upgrading the Goldman bracket) and the non-commutative divergence cocycle which plays the central role in the KV theory. Among other things, this observation gives a new topological interpretation of the KV problem and allows to extend it to surfaces with arbitrary number of boundary components (and of arbitrary genus, see [2]).