李群、李代数和表示论
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- 作者： Brian C. Hall
- 出版社：世界图书出版公司
- ISBN：9787506282970
- 上架时间：2007-11-6
- 出版日期：2007 年10月
- 开本：16开
- 页码：351
- 版次：1-1
- 所属分类：
数学 > 代数，数论及组合理论 > 综合
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this book provides an introduction to lie groups, lie algebras, and representation theory, aimed at graduate students in mathematics and physics.although there are already several excellent books that cover many of the same topics, this book has two distinctive features that i hope will make it auseful addition to the literature. first, it treats lie groups (not just lie alge bras) in a way that minimizes the amount of manifold theory needed. thus,i neither assume a prior course on differentiable manifolds nor provide a con-densed such course in the beginning chapters. second, this book provides a gentle introduction to the machinery of semisimple groups and lie algebras bytreating the representation theory of su(2) and su(3) in detail before going to the general case. this allows the reader to see roots, weights, and the weyl group "in action" in simple cases before confronting the general theory.
the standard books on lie theory begin immediately with the general case:a smooth manifold that is also a group. the lie algebra is then defined as the space of left-invariant vector fields and the exponential mapping is defined in terms of the flow along such vector fields. this approach is undoubtedly the right one in the long run, but it is rather abstract for a reader encountering such things for the first time. furthermore, with this approach, one must either assume the reader is familiar with the theory of differentiable manifolds (which rules out a substantial part of one's audience) or one must spend considerable time at the beginning of the book explaining this theory (in which case, it takes a long time to get to lie theory proper).
the standard books on lie theory begin immediately with the general case:a smooth manifold that is also a group. the lie algebra is then defined as the space of left-invariant vector fields and the exponential mapping is defined in terms of the flow along such vector fields. this approach is undoubtedly the right one in the long run, but it is rather abstract for a reader encountering such things for the first time. furthermore, with this approach, one must either assume the reader is familiar with the theory of differentiable manifolds (which rules out a substantial part of one's audience) or one must spend considerable time at the beginning of the book explaining this theory (in which case, it takes a long time to get to lie theory proper).
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part i general theory
matrix lie groups
1.1 definition of a matrix lie group
1.1.1 counterexa~ples
1.2 examples of matrix lie groups
1.2.1 the general linear groups gl(n;r) and gl(n;c)
1.2.2 the special linear groups sl(n; r) and sl(n; c)
1.2.3 the orthogonal and special orthogonal groups, o(n) and so(n)
1.2.4 the unitary and special unitary groups, u(n) and su(n)
1.2.5 the complex orthogonal groups, o(n; c) and so(n; c)
1.2.6 the generalized orthogonal and lorentz groups
1.2.7 the symplectic groups sp(n; r), sp(n;c), and $p(n)
1.2.8 the heisenberg group h .
1.2.9 the groups r, c*, s1, and rn
1.2.10 the euclidean and poincaxd groups e(n) and p(n; 1)
1.3 compactness
1.3.1 examples of compact groups
1.3.2 examples of noncompa groups
1.4 connectedness
1.5 simple connectedness
matrix lie groups
1.1 definition of a matrix lie group
1.1.1 counterexa~ples
1.2 examples of matrix lie groups
1.2.1 the general linear groups gl(n;r) and gl(n;c)
1.2.2 the special linear groups sl(n; r) and sl(n; c)
1.2.3 the orthogonal and special orthogonal groups, o(n) and so(n)
1.2.4 the unitary and special unitary groups, u(n) and su(n)
1.2.5 the complex orthogonal groups, o(n; c) and so(n; c)
1.2.6 the generalized orthogonal and lorentz groups
1.2.7 the symplectic groups sp(n; r), sp(n;c), and $p(n)
1.2.8 the heisenberg group h .
1.2.9 the groups r, c*, s1, and rn
1.2.10 the euclidean and poincaxd groups e(n) and p(n; 1)
1.3 compactness
1.3.1 examples of compact groups
1.3.2 examples of noncompa groups
1.4 connectedness
1.5 simple connectedness
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发表于：2010-1-4 13:17:00
本书着重于矩阵李群和李代数以及它们之间的关系和表示。这使得内容似乎有点简单，但是内容却并不单薄。因为，大部分我们感兴趣的李群和李代数都同构于矩阵群和矩阵代数。
本书至少前160页对本人来说是很清晰的，甚至有关于有限群的附录和线性代数的核心内容。在介绍了经典群及其代数以及联系它们的指数映射后，作者开始介绍表示理论，详细讨论了SL(2,C)和SL(3,C)的表示。在仔细讨论了它们的Cartan子代数，权重，根和Weyl群等内容以后，再引入一般性的理论就会让读者感觉十分“舒坦”。
本书一大优点是严谨：很少有牺牲严格性的地方，一些证明留给读者，但不是太多。另一大好处是：印刷错误极少，这对数学书来说是非常重要的。
总之，本书是一本非常好的入门书籍，完全以学生的思维为导向。鲜有作者像Hall这样把书写得通俗易懂又不失严密！
强烈推荐！
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