李群.李代数和表示论 本书特色

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).

李群.李代数和表示论 内容简介

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).

李群.李代数和表示论 目录

part i general theory
matrix lie groups
1.1definition of a matrix lie group
1.1.1counterexa~ples
1.2examples of matrix lie groups
1.2.1the general linear groups gl(n;r) and gl(n;c)
1.2.2 the special linear groups sl(n; r) and sl(n; c)
1.2.3the orthogonal and special orthogonal groups, o(n) and so(n)
1.2.4the unitary and special unitary groups, u(n) and su(n)
1.2.5 the complex orthogonal groups, o(n; c) and so(n; c)
1.2.6the generalized orthogonal and lorentz groups
1.2.7 the symplectic groups sp(n; r), sp(n;c), and $p(n)
1.2.8the heisenberg group h.
1.2.9the groups r, c*, s1,and rn
1.2.10 the euclidean and poincaxd groups e(n) and p(n; 1)
1.3compactness
1.3.1examples of compact groups
1.3.2examples of noncompa groups
1.4connectedness
1.5simple connectedness
1.6homomorpliisms and isomorphisms
1.6.1 example: su(2) and s0(3)
1.7 the polar decomposition for s[(n; r) and sl(n; c)
1.8lie groups
1.9exercises
2 lie algebras and the exponential mapping
2.1the matrix exponential
2.2computing the exponential of a matrix
2.2.1case 1: x is diagonalizable
2.2.2case 2: x is nilpotent
2.2.3case 3: x arbitrary
2.3the matrix logarithm
2.4further properties of the matrix exponential
2.5the lie algebra of a matrix lie group
2.5.1physicists' convention
2.5.2the general linear groups
2.5.3the special linear groups
2.5.4the unitary groups
2.5.5the orthogonal groups
2.5.6the generalized orthogonal groups
2.5.7the symplectic groups
2.5.8the heisenberg group
2.5.9the euclidean and poincar6 groups
2.6properties of the lie algebra
2.7the exponential mapping
2.8lie algebras
2.8.1structure constants
2.8.2direct sums
2.9the complexification of a real lie algebra
2.10 exercises
3the baker-campbell-hausdorff formula
3.1the baker-campbell-hausdorff formula for the heisenberg group
3.2the general baker-campbell-hausdorff formula
3.3the derivative of the exponential mapping
3.4proof of the baker-campbell-hausdorff formula
3.5the series form of the baker-campbell-hausdorff formula
3.6group versus lie algebra homomorphisms
3.7covering groups
3.8subgroups and subalgebras
3.9exercises
4basic representation theory
4.1representations
4.2why study representations?
4.3examples of representations
4.3.1the standard representation
4.3.2the trivial representation
4.3.3the adjoint representation
4.3.4some representations of s(,1(2)
4.3.5two unitary representations of s0(3)
4.3.6a unitary representation of the reals
……
part ii semistmple theory
references
index

李群.李代数和表示论 节选

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).