This book is intended for a one year graduate course on Lie groups and Lie algebras. The author proceeds beyond the representation theory of compact Lie groups (which is the basis of many texts) and provides a carefully chosen range of material to give the student the bigger picture. For compact Lie groups, the Peter-Weyl theorem, conjugacy of maximal tori (two proofs), Weyl character formula and more are covered. The book continues with the study of complex analytic groups, then general noncompact Lie groups, including the Coxeter presentation of the Weyl group, the Iwasawa and Bruhat decompositions, Cartan decomposition, symmetric spaces, Cayley transforms, relative root systems, Satake diagrams, extended Dynkin diagrams and a survey of the ways Lie groups may be embedded in one another. The book culminates in a ``topics'' section giving depth to the student's understanding of representation theory, taking the Frobenius-Schur duality between the representation theory of the symmetric group and the unitary groups as a unifying theme, with many applications in diverse areas such as random matrix theory, minors of Toeplitz matrices, symmetric algebra decompositions, Gelfand pairs, Hecke algebras, representations of finite general linear groups and the cohomology of Grassmannians and flag varieties.

1 Haar Measure.- 2 Schur Orthogonality.- 3 Compact Operators.- 4 The Peter-Weyl Theorem.- 5 Lie Subgroups of GL(n, ?).- 6 Vector Fields.- 7 Left-Invariant Vector Fields.- 8 The Exponential Map.- 9 Tensors and Universal Properties.- 10 The Universal Enveloping Algebra.- 11 Extension of Scalars.- 12 Representations of sl(2, ?).- 13 The Universal Cover.- 14 The Local Frobenius Theorem.- 15 Tori.- 16 Geodesics and Maximal Tori.- 17 Topological Proof of Cartan's Theorem.- 18 The Weyl Integration Formula.- 19 The Root System.- 20 Examples of Root Systems.- 21 Abstract Weyl Groups.- 22 The Fundamental Group.- 23 Semisimple Compact Groups.- 24 Highest-Weight Vectors.- 25 The Weyl Character Formula.- 26 Spin.- 27 Complexification.- 28 Coxeter Groups.- 29 The Iwasawa Decomposition.- 30 The Bruhat Decomposition.- 31 Symmetric Spaces.- 32 Relative Root Systems.- 33 Embeddings of Lie Groups.- 34 Mackey Theory.- 35 Characters of GL(n, ?).- 36 Duality between Sk and GL(n, ?).- 37 The Jacobi-Trudi Identity.- 38 Schur Polynomials and GL(n, ?).- 39 Schur Polynomials and Sk.- 40 Random Matrix Theory.- 41 Minors of Toeplitz Matrices.- 42 Branching Formulae and Tableaux.- 43 The Cauchy Identity.- 44 Unitary Branching Rules.- 45 The Involution Model for Sk.- 46 Some Symmetric Algebras.- 47 Gelfand Pairs.- 48 Hecke Algebras.- 49 The Philosophy of Cusp Forms.- 50 Cohomology of Grassmannians.- References.