Holonomy李代数与A_n的子排列的LCS公式
Holonomy Lie algebras and the LCS formula for subarrangements of A_n
摘要：
若X是P^n中超曲面的补，Kohno证明了基本群的幂零完备化同构于X的完整李代数的幂零完备化。当X是超平面排列A的补时，X的基本群的下中心序列子的秩phi_k是已知的，对于两个特殊的类：如果X是超可解的（在这种情况下，上同调环的二次闭包是Koszul），或者如果完整李代数在三度分解为局部分量的直积。本文利用完整性李代数得到了当A是A_n的子排列时φ_k的一个公式。这扩展了Kohno的编织安排的结果，并提供了一个LCS公式的安排，这是不可分解或hypersolvable的一个实例。
英文摘要：
If X is the complement of a hypersurface in P^n, then Kohno showed that the nilpotent completion of the fundamental group is isomorphic to the nilpotent completion of the holonomy Lie algebra of X. When X is the complement of a hyperplane arrangement A, the ranks phi_k of the lower central series quotients of the fundamental group of X are known for isolated examples, and for two special classes: if X is hypersolvable (in which case the quadratic closure of the cohomology ring is Koszul), or if the holonomy Lie algebra decomposes in degree three as a direct product of local components. In this paper, we use the holonomy Lie algebra to obtain a formula for phi_k when A is a subarrangement of A_n. This extends Kohno's result for braid arrangements, and provides an instance of an LCS formula for arrangements which are not decomposable or hypersolvable.
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