Book Review: Potential theory and geometry on Lie groups
Book Review: Potential theory and geometry on Lie groups
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书评：李群的势理论和几何
DOI：
10.1090/bull/1785
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发表时间：
2023
影响因子：
1.3
通讯作者：
Saloff-Coste, Laurent
中科院分区：
文献类型：
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作者：
Saloff-Coste, Laurent
这本书关注的是（好的）概率测度μ的迭代卷积幂μ（n）的大n行为。其目的是了解这种卷积幂的行为如何与基础群G的代数结构以及其几何和拓扑结构相关。假设群G是一个真实的连通李群，但没有进一步的假设。假设测度μ是对称的（即，对于任何Borel集A，μ（A）= μ（A− 1）），并且具有连续紧支撑密度，在G的单位元e处为正。连通李群是代数对象，其代数结构在很大程度上由它们的李代数捕获。它们也是几何对象时，配备了左不变的黎曼结构（即，选择一个线性基础的李代数）。重要的是，左不变黎曼结构的不同可能选择都导致相同的大尺度几何，我们可以认为它本质上与G相连（一个适当的定义需要准等距的概念）。
This book is concerned with the large n behavior of the iterated convolution powers μ (n) of (nice) probability measures μ. The aim is to understand how the behavior of such convolution powers relates to the algebraic structure of underlying group G, as well as its geometry and topology. The group G is assumed to be a real connected Lie group but no further assumptions are made. The measure μ is assumed to be symmetric (ie, μ (A)= μ (A− 1) for any Borel set A) and to have a continuous compactly supported density, which is positive at the identity element e of G. Connected Lie groups are algebraic objects whose algebraic structure is largely captured by their Lie algebra. They are also geometric objects when equipped with a left-invariant Riemannian structure (ie, the choice of a linear basis for their Lie algebra). Importantly, the different possible choices of a left-invariant Riemannian structure all lead to the same large scale geometry, which we can think of as intrinsically attached to G (a proper definition entails the notion of quasiisometry).