论文标题
光滑的循环和循环捆
Smooth loops and loop bundles
论文作者
论文摘要
循环是具有标识元素和分裂的相当通用的代数结构，但不一定是关联的。平滑环是谎言组的直接概括。非Lie平滑环路的一个关键示例是单位八圈的循环。在本文中，我们研究了平滑环及其相关切线代数的特性，包括毛尔 - 卡丹方程的循环类似物。然后，给定一个歧管，我们将循环束作为相关的捆绑包给特定的主捆绑包。鉴于主束上的连接，我们定义了循环束结构的扭转，并显示了它与曲率的关系，还考虑了某些相关功能的关键点。在整个过程中，我们看到了$ g_ {2} $的一些已知属性 - 从这个更一般的设置可以看到结构。
A loop is a rather general algebraic structure that has an identity element and division, but is not necessarily associative. Smooth loops are a direct generalization of Lie groups. A key example of a non-Lie smooth loop is the loop of unit octonions. In this paper, we study properties of smooth loops and their associated tangent algebras, including a loop analog of the Mauer-Cartan equation. Then, given a manifold, we introduce a loop bundle as an associated bundle to a particular principal bundle. Given a connection on the principal bundle, we define the torsion of a loop bundle structure and show how it relates to the curvature, and also consider the critical points of some related functionals. Throughout, we see how some of the known properties of $G_{2}$-structures can be seen from this more general setting.
