Tangent Lie Algebra of a Diffeomorphism Group and Application to Holonomy Theory
Tangent Lie Algebra of a Diffeomorphism Group and Application to Holonomy Theory
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一微分同胚群的切李代数及其在完整理论中的应用
DOI：
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发表时间：
2018
影响因子：
1.1
通讯作者：
Z. Muzsnay
中科院分区：
文献类型：
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作者：
B. Hubicska;Z. Muzsnay
在本文中，我们介绍了切空间的概念 $${\mathcal {T}}_{o} {\mathcal {G}}$$ T o G 的（不一定光滑）子群 $${\mathcal {G}}$$ G 微分同胚群 $${\mathcal {D}}i\!f\!f^{\infty }(M)$$ D i f f ∞ ( M ) 紧流形 M 。我们证明 $${\mathcal {T}}_{o} {\mathcal {G}}$$ T o G 是 M 上平滑向量场李代数的李子代数。该构造可以推广到任何（有限或无限维）李群的子群。这样引入的正切李代数 $${\mathcal {T}}_{o} {\mathcal {G}}$$ T o G 是光滑情况下经典李代数的推广。作为一个工作示例，我们详细讨论芬斯勒流形的完整群和纤维完整群的切线结构。
In this paper we introduce the notion of tangent space $${\mathcal {T}}_{o} {\mathcal {G}}$$ T o G of a (not necessary smooth) subgroup $${\mathcal {G}}$$ G of the diffeomorphism group $${\mathcal {D}}i\!f\!f^{\infty }(M)$$ D i f f ∞ ( M ) of a compact manifold M . We prove that $${\mathcal {T}}_{o} {\mathcal {G}}$$ T o G is a Lie subalgebra of the Lie algebra of smooth vector fields on M . The construction can be generalized to subgroups of any (finite- or infinite-dimensional) Lie groups. The tangent Lie algebra $${\mathcal {T}}_{o} {\mathcal {G}}$$ T o G introduced this way is a generalization of the classical Lie algebra in the smooth cases. As a working example we discuss in detail the tangent structure of the holonomy group and fibered holonomy group of Finsler manifolds.