Degenerations of Lie algebras and geometry of Lie groups
Degenerations of Lie algebras and geometry of Lie groups
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李代数的简并和李群几何
DOI：
10.1016/s0926-2245(02)00146-8
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发表时间：
2003
影响因子：
0.5
通讯作者：
J. Lauret
中科院分区：
文献类型：
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作者：
J. Lauret
真实的李代数簇的每一点都自然地与李群上的一个左不变黎曼度量相一致。我们研究不变量理论和黎曼方面的这种多样性之间的相互作用。特别地，利用簇上某些自然泛函的特殊临界点行为，确定了所有在等距和标度下只能被赋予一个左不变度量的李群，证明了它们对应于唯一退化为交换的李代数.利用已知的左不变度量满足某些pinching曲率条件的描述，我们还找到了所有退化为双曲空间的李代数，以及所有可能退化为三维真实的李代数的李代数.最后，作为另一种相互作用，对簇上的SL（n）-闭轨道进行了分类，并利用表示Λ2SL（m）<$SL（n）的闭轨道曲线给出了Einstein解流形的显式曲线.
Each point of the variety of real Lie algebras is naturally identified with a left invariant Riemannian metric on a Lie group. We study the interplay between invariant-theoretic and Riemannian aspects of this variety. In particular, using the special critical point behavior of certain natural functional on the variety, we determine all the Lie groups which can be endowed with only one left invariant metric up to isometry and scaling, proving first that they correspond to Lie algebras whose only degeneration is to the abelian one. We also find all the Lie algebras which degenerate to the Lie algebra of the hyperbolic space, and all the possible degenerations for 3-dimensional real Lie algebras, by using well known descriptions of left invariant metrics satisfying some pinching curvature conditions. Finally, as another interaction, the closed SL(n)-orbits on the variety are classified, and explicit curves of Einstein solvmanifolds are provided by using curves of closed orbits of the representation Λ2SL(m)⊗SL(n).