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摘要翻译：
对于任意有限图Gamma和任意特征不等于2的域K，我们构造了一个K上的代数簇X，它的K-点参数化由对应于图顶点的极值元生成的K-李代数，并给出了对应于非边的交换关系。在此基础上，我们研究了Gamma是有限型或仿射型的连通、简单加边Dynkin图的情形。我们证明了X是仿射空间，并且X的开稠密子集中的所有点都是同构于单个固定李代数的参数化李代数。如果Gamma是仿射型，那么这个固定李代数就是对应于关联有限型Dynkin图的分裂有限维单李代数。这给出了这些李代数的一个新的构造，其中它们与有趣的退化结合在一起，对应于开稠密子集外的点。我们的结果对于识别这些李代数是有用的。
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英文标题：
《Constructing simply laced Lie algebras from extremal elements》
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作者：
Jan Draisma and Jos in 't panhuis
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最新提交年份：
2008
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分类信息：
一级分类：Mathematics 数学
二级分类：Rings and Algebras 环与代数
分类描述：Non-commutative rings and algebras, non-associative algebras, universal algebra and lattice theory, linear algebra, semigroups
非交换环与代数，非结合代数，泛代数与格论，线性代数，半群
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一级分类：Mathematics 数学
二级分类：Algebraic Geometry 代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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英文摘要：
For any finite graph Gamma and any field K of characteristic unequal to 2 we construct an algebraic variety X over K whose K-points parameterise K-Lie algebras generated by extremal elements, corresponding to the vertices of the graph, with prescribed commutation relations, corresponding to the non-edges. After that, we study the case where Gamma is a connected, simply laced Dynkin diagram of finite or affine type. We prove that X is then an affine space, and that all points in an open dense subset of X parameterise Lie algebras isomorphic to a single fixed Lie algebra. If Gamma is of affine type, then this fixed Lie algebra is the split finite-dimensional simple Lie algebra corresponding to the associated finite-type Dynkin diagram. This gives a new construction of these Lie algebras, in which they come together with interesting degenerations, corresponding to points outside the open dense subset. Our results may prove useful for recognising these Lie algebras.
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PDF链接：
https://arxiv.org/pdf/0707.2927
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