本书的内容是关于楼（building）理论及其在几何和拓扑中的应用。楼作为一种组合和几何结构由Jacques Tits引入，作为理解任意域上保距还原线性代数群结构的一种方法，Tits因此项工作获得2008年Abel奖。楼理论是研究代数群及其表示的必要工具，在几个相当不同的领域中具有重要应用。本书的第一部分是作者专为国内学生学习楼理论准备的导读资料，其中特别注重利用例子说明问题，可读性很强；第二部分则综述了楼理论在几何与拓扑方面的应用，不仅总结了近些年楼理论研究的成就，还提出了未来的研究方向。本书是一本观点较高、极具学术价值的数学学习资料，可供我国高等院校代数及相关专业作为教学参考书使用。 Symmetry is an essential concept in mathematics， science and daily life， and an effective mathematical tool to describe symmetry is the notion of groups. For example， the symmetries of the regular solids (or Platonic solids) are described by the finite subgroups of the rotation group SO(3). Therefore， finding the symmetry group of a geometric object or space is a classical and important problem. On the other hand， given a group， how to find a natural geometric space which realizes the group as its symmetries is also interesting and fruitful. One of the most useful or beautiful class of groups consists of algebraic groups， and their corresponding geometric spaces are given by Tits buildings. Originally introduced by Tits to give a geometric description of exceptional simple algebraic groups， buildings have turned out to be extremely useful in a broad range of subjects in contemporary mathematics， including algebra， geometry， topology， number theory， and analysis etc. Since the theory of algebraic groups is complicated， the theory of buildings can be technical and demanding by itself. This book gives an accessible approach by using elementary and concrete examples and by emphasizing many applications in many seemingly unrelated subjects. The reader will learn from this book what buildings are， why they are useful， and how they can be used.
內容試閱：
前辅文
Part 1 Buildings and Groups
1. Combinatorics
1.1 Geometry
1.1.1 Graphs
1.1.2 Trees
1.1.3 Euclidean geometry
1.1.4 Incidence geometry
1.1.5 Projective space
1.2 Coxeter group
1.2.1 Coxeter system
1.2.2 Finite reflection group
1.2.3 Affine reflection group
1.3 Chamber systems
1.3.1 Edge-colored graphs
1.3.2 Buildings
1.4 Chamber complexes
1.4.1 Complexes
1.4.2 Chamber complex
1.4.3 Building
1.5 Conclusion
2. Chevalley Groups
2.1 (B;N) pairs
2.2 Simple Lie algebras
2.2.1 An
2.2.2 Bn
2.2.3 Cn
2.2.4 Dn
2.3 Classical groups
2.3.1 GLn
2.3.2 SLn
2.3.3 Sp2n
2.3.4 SO2n
2.3.5 SO2n+
2.4 Chevalley groups and (B;N) pairs
2.4.1 Chevalley basis
2.4.2 Lie algebra representations
2.4.3 Building of a Chevalley group
2.4.4 SLn
2.5 Chevalley groups over local fields
2.5.1 Affine roots
2.5.2 BN pair
2.6 Examples
2.6.1 Sp
2.6.2 SLn
2.6.3 SL3(Qp)
2.6.4 SL2(Qp)
2.7 Conclusion
3. Reductive Groups over Local Fields
3.1 Root data
3.2 Reductive group
3.2.1 Roots
3.2.2 Root data
3.2.3 Root group data
3.2.4 Pinning
3.3 Apartments
3.3.1 Affine space
3.3.2 Affine apartment
3.3.3 Affine extension
3.3.4 Affine roots
3.4 Building of a reductive group
3.4.1 Quasi-split groups
3.4.2 Filtration on root groups
3.4.3 Construction of the building
3.5 Compactification of buildings
3.5.1 Compactifying apartments
3.5.2 Metric
3.5.3 X
3.6 Congruence subgroup
3.6.1 Models
3.6.2 Smooth models of root subgroups
3.6.3 Filtrations on tori
3.6.4 Smooth
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