Leibniz algebras, Courant algebroids, and multiplications on reductive homogeneous spaces
Leibniz algebras, Courant algebroids, and multiplications on reductive homogeneous spaces
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DOI：
10.1353/ajm.2001.0017
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发表时间：
2000-06
影响因子：
1.7
通讯作者：
M. Kinyon;A. Weinstein
中科院分区：
文献类型：
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作者：
M. Kinyon;A. Weinstein
我们证明了每个莱布尼茨代数 S 上的斜对称积可以通过李代数中的子代数的约简补来实现。因此，我们在 S 上构造了一个非关联乘法，当 S 是李代数时，它是从积分伴随表示导出的。我们应用这种构造来实现对库朗代数体的截面和第二作者最近介绍的“全李代数”的括号运算。 1. 简介。满足雅可比恒等式弱化版本的斜对称双线性运算源自代数和微分几何中的许多构造。本文的目的是展示这些运算的某些操作，特别是李双代数胚上的双数上的库朗括号，可以在约简齐次空间的切空间上以自然的方式实现。我们使用我们的构造来采取步骤寻找“整合”这些不完全李代数的类群对象。我们的构建背后的主要思想来自 K. Nomizu、K. Yamaguti 和 M. Kikkawa 的工作。 Nomizu (18) 表明，具有平行挠率和曲率的仿射连接局部等价于还原齐次空间上的不变连接，并且每个这样的空间都有一个规范连接，其中沿测地线的平行平移与群的自然作用一致。 Yamaguti (22) 将 Nomizu 规范联系的挠率和曲率张量描述为一对代数运算，一个是双线性的，另一个是三线性的，满足定义他所谓的“一般李三重系统”的公理，以及 Kikkawa 后来称为“李三重代数”的公理。在本文中，我们将这些对象称为 Lie-Yamaguti 代数。当三线性运算为零时，双线性运算是李代数运算，齐次空间是局部李群，其上的连接是使左不变向量场平行的连接。最后，Kikkawa (7) 展示了如何将 Lie-Yamaguti 代数“整合”到还原齐次空间上的非结合乘法，并以公理方式描述了这些乘法。不幸的是，Kikkawa 的构造在我们的设置中应用时并不能完全再现曲率为零时李群上的乘法；而是它
We show that the skew-symmetrized product on every Leibniz algebra S can be realized on a reductive complement to a subalgebra in a Lie algebra. As a consequence, we construct a nonassociative multiplication on S which, when S is a Lie algebra, is derived from the integrated adjoint representation. We apply this construction to realize the bracket operations on the sections of Courant algebroids and on the "omni-Lie algebras" recently introduced by the second author. 1. Introduction. Skew-symmetric bilinear operations which satisfy weak ened versions of the Jacobi identity arise from a number of constructions in algebra and differential geometry. The purpose of this paper is to show how cer tain of these operations, in particular the Courant brackets on the doubles of Lie bialgebroids, can be realized in a natural way on the tangent spaces of reduc tive homogeneous spaces. We use our construction to take steps toward finding group-like objects which "integrate" these not-quite-Lie algebras. The main ideas behind our construction come from work of K. Nomizu, K. Yamaguti, and M. Kikkawa. Nomizu (18) showed that affine connections with parallel torsion and curvature are locally equivalent to invariant connec tions on reductive homogeneous spaces, and that each such space has a canonical connection for which parallel translation along geodesies agrees with the natu ral action of the group. Yamaguti (22) characterized the torsion and curvature tensors of Nomizu's canonical connection as pairs of algebraic operations, one bilinear and the other trilinear, satisfying axioms defining what he called a "gen eral Lie triple system," and what Kikkawa later called a "Lie triple algebra." In this paper, we will call these objects Lie-Yamaguti algebras. When the trilinear operation is zero, the bilinear operation is a Lie algebra operation, and the homo geneous space is locally a Lie group on which the connection is the one which makes left-invariant vector fields parallel. Finally, Kikkawa (7) showed how to "integrate" Lie-Yamaguti algebras to nonassociative multiplications on reductive homogeneous spaces, and he characterized these multiplications axiomatically. Unfortunately, Kikkawa's construction when applied in our setting does not quite reproduce the multiplication on a Lie group when the curvature is zero; rather it