Theory of connections on graded principal bundles
Theory of connections on graded principal bundles
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分级主丛上的联系理论
DOI：
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发表时间：
1996
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通讯作者：
T. Stavracou
中科院分区：
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作者：
T. Stavracou
在Kostant-Berezin-Leites分次流形理论的框架下讨论了分次主丛的几何性质。我们首先回顾了这一理论的基本内容，同时建立了分次李群及其作用的补充性质。特别强调在分级背景下引入和研究自由动作。其次，我们研究分次主丛的几何，我们证明了它们有几个性质类似于普通主丛。特别地，我们证明了分次主丛上的垂直导子层与由结构分次李群的作用引起的分次分布相一致。这一结果导致一个自然的定义的分次联络的分次分布，它的关系与李超代数值分次微分形式也展示。最后，我们定义了分次联络的曲率，并证明了曲率控制着与分次联络对应的水平分次分布的对合性。
The geometry of graded principal bundles is discussed in the framework of graded manifold theory of Kostant–Berezin–Leites. We first review the basic elements of this theory establishing at the same time supplementary properties of graded Lie groups and their actions. Particular emphasis is given in introducing and studying free actions in the graded context. Next, we investigate the geometry of graded principal bundles; we prove that they have several properties analogous to those of ordinary principal bundles. In particular, we show that the sheaf of vertical derivations on a graded principal bundle coincides with the graded distribution induced by the action of the structure graded Lie group. This result leads to a natural definition of the graded connection in terms of graded distributions; its relation with Lie superalgebra-valued graded differential forms is also exhibited. Finally, we define the curvature for the graded connection and we prove that the curvature controls the involutivity of the horizontal graded distribution corresponding to the graded connection.