$\vee$ -Systems, Holonomy Lie Algebras, and Logarithmic Vector Fields
$\vee$ -Systems, Holonomy Lie Algebras, and Logarithmic Vector Fields
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$vee$ -系统、完整李代数和对数向量场
DOI：
10.1093/imrn/rnw289
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发表时间：
2018
影响因子：
1
通讯作者：
Feigin M
中科院分区：
文献类型：
--
作者：
Feigin M
结果表明，与超平面排列相关的完整李代数的某些表示类别的描述本质上等价于与相关联的系统的分类，相应连接的平坦部分可以解释为向量场，它们都是对数的和梯度的。我们推测任何系统的超平面排列在 Saito 意义上都是自由的，并且对于所有已知系统和称为调和系统的特殊类系统（包括所有 Coxeter 系统）都证明了这一点。在不可约 Coxeter 情况下，相应梯度矢量场的势结果是 Saito 平面坐标，或其单参数变形。我们给出了这些变形以及经典调和系统族的势的公式。
It is shown that the description of certain class of representations of the holonomy Lie algebraassociated with hyperplane arrangementis essentially equivalent to the classification of-systems associated withThe flat sections of the corresponding-connection can be interpreted as vector fields, which are both logarithmic and gradient. We conjecture that the hyperplane arrangement of any-system is free in Saito's sense and show this for all known-systems and for a special class of-systems called harmonic, which includes all Coxeter systems. In the irreducible Coxeter case the potentials of the corresponding gradient vector fields turn out to be Saito flat coordinates, or their one-parameter deformations. We give formulas for these deformations as well as for the potentials of the classical families of harmonic-systems.