环面流形的根系与对称性
Root systems and symmetries of torus manifolds
摘要：
将根系与自由阿尔贝群中的有限集联系起来，证明了它的不可约子系统是A、B或D型的。我们将这个一般结果应用到环面流形上，其中环面流形是一个$2n$维连通闭光滑流形，其光滑有效作用是一个具有不动点的$n$维紧环面，并证明了如果环面作用扩展到一个连通紧李群$G$的光滑作用，则$G$的李代数的一个单因子是A，B或D型的.这给出了Wiemeler定理的另一种证明。对于具有稳定不变复结构的环面流形，我们也讨论了类似的问题。在这种情况下，只有A型出现。
英文摘要：
We associate a root system to a finite set in a free abelian group and prove that its irreducible subsystem is of type A, B or D. We apply this general result to a torus manifold, where a torus manifold is a $2n$-dimensional connected closed smooth manifold with a smooth effective action of an $n$-dimensional compact torus having a fixed point, and show that if the torus action extends to a smooth action of a connected compact Lie group $G$, then a simple factor of the Lie algebra of $G$ is of type A, B or D. This gives an alternative proof to Wiemeler's theorem. We also discuss a similar problem for a torus manifold with an invariant stably complex structure. In this case only type A appears.
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