Moment Maps and Combinatorial Invariants of Hamiltonian Tn-spaces
Moment Maps and Combinatorial Invariants of Hamiltonian Tn-spaces
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DOI：
10.1007/978-1-4612-0269-1
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发表时间：
1994-12
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影响因子：
--
通讯作者：
V. Guillemin
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文献类型：
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作者：
V. Guillemin
紧李群G在紧辛流形上的作用产生了一些显著的组合不变量。其中最简单和最有趣的是矩多面体，一个凸多面体，它位于G的李代数的对偶之内。这本专著的主要目标之一是描述什么样的几何信息编码在这个多面体。例如，第一章主要致力于德尔赞特定理，该定理说，某些类型的矩多面体和某些类型的辛G-空间之间存在一一对应。(One辛几何中最具挑战性的未解决问题之一是确定Delzant定理在何种程度上对每个紧辛G-空间都是真的。矩多面体还编码了关于G的作用的量子信息。利用几何量子化的方法，人们常常可以把这种作用转化为G在希尔伯特空间上的表示p，在某种意义上，矩多面体是G的不可约表示的图解，这些不可约表示作为p的子表示出现。这一民间传说的精确版本将在第3章和第4章中讨论。此外，在第2章的中途，我们讨论了一个更复杂的对象：杜伊斯特马特-赫克曼测度，作者在第4章中解释了如何从这个测度中读出G在p中的不可约表示的近似重数。这就给了我们一个借口，去触及一些本身就具有很大的现实意义的结果：Duistermaat-Heckman定理，Atiyah-Bott和Berline-Vergne等变上同调的局部化定理，以及最近由维滕、Jeffrey-Kirwan、Lalkman等人对这些结果的极其令人兴奋的推广。这本书的最后两章是一个独立的和有点非正统的治疗理论的环面品种，其中通常的层次关系复杂的辛是扭转。这本书是写给研究人员的
The action of a compact Lie group, G, on a compact sympletic manifold gives rise to some remarkable combinatorial invariants. The simplest and most interesting of these is the moment polytope, a convex polyhedron which sits inside the dual of the Lie algebra of G. One of the main goals of this monograph is to describe what kinds of geometric information are encoded in this polytope. For instance, the first chapter is largely devoted to the Delzant theorem, which says that there is a one-one correspondence between certain types of moment polytopes and certain types of symplectic G-spaces.(One of the most challenging unsolved problems in symplectic geometry is to determine to what extent Delzant’s theorem is true of every compact symplectic G-Space.) The moment polytope also encodes quantum information about the actions of G. Using the methods of geometric quantization, one can frequently convert this action into a representations, p, of G on a Hilbert space, and in some sense the moment polytope is a diagrammatic picture of the irreducible representations of G which occur as subrepresentations of p. Precise versions of this item of folklore are discussed in Chapters 3 and 4. Also, midway through Chapter 2 a more complicated object is discussed: the Duistermaat-Heckman measure, and the author explains in Chapter 4 how one can read off from this measure the approximate multiplicities with which the irreducible representations of G occur in p. This gives an excuse to touch on some results which are in themselves of great current interest: the Duistermaat-Heckman theorem, the localization theorems in equivariant cohomology of Atiyah-Bott and Berline-Vergne and the recent extremely exciting generalizations of these results by Witten, Jeffrey-Kirwan, Lalkman, and others. The last two chapters of this book are a self-contained and somewhat unorthodox treatment of the theory of toric varieties in which the usual hierarchal relation of complex to symplectic is reversed. This book is addressed to researchers