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AuthorHall, Brian C. author
TitleLie Groups, Lie Algebras, and Representations [electronic resource] : An Elementary Introduction / by Brian C. Hall
ImprintNew York, NY : Springer New York : Imprint: Springer, 2003
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Descript XIV, 354 p. online resource


This book provides an introduction to Lie groups, Lie algebras, and repreยญ sentation 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 a useful addition to the literature. First, it treats Lie groups (not just Lie algeยญ 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 conยญ densed such course in the beginning chapters. Second, this book provides a gentle introduction to the machinery of semi simple groups and Lie algebras by treating 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


I General Theory -- 1 Matrix Lie Groups -- 2 Lie Algebras and the Exponential Mapping -- 3 The Baker-Campbell-Hausdorff Formula -- 4 Basic Representation Theory -- II Semisimple Theory -- 5 The Representations of SU(3) -- 6 Semisimple Lie Algebras -- 7 Representations of Complex Semisimple Lie Algebras -- 8 More on Roots and Weights -- A A Quick Introduction to Groups -- A.1 Definition of a Group and Basic Properties -- A.2 Examples of Groups -- A.2.1 The trivial group -- A.2.2 The integers -- A.2.4 Nonzero real numbers under multiplication -- A.2.5 Nonzero complex numbers under multiplication -- A.2.6 Complex numbers of absolute value 1 under multiplication -- A.2.7 The general linear groups -- A.2.8 Permutation group (symmetric group) -- A.3 Subgroups, the Center, and Direct Products -- A.4 Homomorphisms and Isomorphisms -- A.5 Quotient Groups -- A.6 Exercises -- B Linear Algebra Review -- B.1 Eigenvectors, Eigenvalues, and the Characteristic Polynomial -- B.2 Diagonalization -- B.3 Generalized Eigenvectors and the SN Decomposition -- B.4 The Jordan Canonical Form -- B.5 The Trace -- B.6 Inner Products -- B.7 Dual Spaces -- B.8 Simultaneous Diagonalization -- C More on Lie Groups -- C.1 Manifolds -- C.1.1 Definition -- C.1.2 Tangent space -- C.1.3 Differentials of smooth mappings -- C.1.4 Vector fields -- C.1.5 The flow along a vector field -- C.1.6 Submanifolds of vector spaces -- C.1.7 Complex manifolds -- C.2 Lie Groups -- C.2.1 Definition -- C.2.2 The Lie algebra -- C.2.3 The exponential mapping -- C.2.4 Homomorphisms -- C.2.5 Quotient groups and covering groups -- C.2.6 Matrix Lie groups as Lie groups -- C.2.7 Complex Lie groups -- C.3 Examples of Nonmatrix Lie Groups -- C.4 Differential Forms and Haar Measure -- D Clebsch-Gordan Theory for SU(2) and the Wigner-Eckart Theorem -- D.1 Tensor Products of sl(2; ?) Representations -- D.2 The Wigner-Eckart Theorem -- D.3 More on Vector Operators -- E Computing Fundamental Groups of Matrix Lie Groups -- E.1 The Fundamental Group -- E.2 The Universal Cover -- E.3 Fundamental Groups of Compact Lie Groups I -- E.4 Fundamental Groups of Compact Lie Groups II -- E.5 Fundamental Groups of Noncompact Lie Groups -- References

Mathematics Algebra Group theory Topological groups Lie groups Physics Mathematics Algebra Group Theory and Generalizations Topological Groups Lie Groups Mathematical Methods in Physics


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