有限辛向量空间中的根系
Root Systems In Finite Symplectic Vector Spaces
摘要：
研究了有限域上可能退化辛向量空间中的子集，这些子集在给定的Coxeter/Weyl反射群下是稳定的.这些辛根系为复数[Len 13 a]上的幂零群G的有限维Nichols代数的分类提供了重要的组合数据，其中辛形式由群的换位子映射给出.例如，辛根系的退化度决定了G的中心的大小。本文对有限域F_2上的辛根系进行了分类，这里辛的意思是迷向的。我们证明了每一个Dynkin图都有一个唯一的极小辛根系，从而要求辛向量空间具有特定的退化度。任何一个非极小辛根系都是极小辛根系的商，且具有泛性质。作为例子和进一步的应用，我们显式地构造了Cartan矩阵的所有辛根系。ADE型Dynkin图。
英文摘要：
We study subsets in possibly degenerate symplectic vector spaces over finite fields, which are stable under a given Coxeter/Weyl reflection group. These symplectic root systems provide crucial combinatorical data to classify finite-dimensional Nichols algebras for nilpotent groups G over the complex numbers [Len13a], where the symplectic form is given by the group's commutator map. For example, the degree of degeneracy of the symplectic root system determines the size of the center of G. In this article we classify symplectic root systems over the finite field F_2, where symplectic just means isotropic. We prove that every Dynkin diagram admits, up to symplectic isomorphisms, a unique minimal symplectic root system over F_2 and thus requires a specific degree of degeneracy of the symplectic vector space. Any non-minimal symplectic root system turns out to be a quotient of a minimal one by a universal property. As examples and for further applications we explicitly construct all symplectic root systems for Cartan matrices resp. Dynkin diagrams of type ADE.
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