Full Description
Lie groups, Lie algebras, and representation theory are the main focus of this text. In order to keep the prerequisites to a minimum, the author restricts attention to matrix Lie groups and Lie algebras. This approach keeps the discussion concrete, allows the reader to get to the heart of the subject quickly, and covers all of the most interesting examples. The book also introduces the often-intimidating machinery of roots and the Weyl group in a gradual way, using examples and representation theory as motivation. The text is divided into two parts. The first covers Lie groups and Lie algebras and the relationship between them, along with basic representation theory. The second part covers the theory of semisimple Lie groups and Lie algebras, beginning with a detailed analysis of the representations of SU(3). The author illustrates the general theory with numerous images pertaining to Lie algebras of rank two and rank three, including images of root systems, lattices of dominant integral weights, and weight diagrams. This book is sure to become a standard textbook for graduate students in mathematics and physics with little or no prior exposure to Lie theory.
Brian Hall is an Associate Professor of Mathematics at the University of Notre Dame.
Contents
Preface Part I General Theory 1 Matrix Lie Groups 1.1 Definition of a Matrix Lie Group 1.2 Examples of Matrix Lie Groups 1.3 Compactness 1.4 Connectedness 1.5 Simple Connectedness 1.6 Homomorphisms and Isomorphisms 1.7 (Optional) The Polar Decomposition for $ {SL}(n; {R})$ and $ {SL}(n; {C})$ 1.8 Lie Groups 1.9 Exercises 2 Lie Algebras and the Exponential Mapping 2.1 The Matrix Exponential 2.2 Computing the Exponential of a Matrix 2.3 The Matrix Logarithm 2.4 Further Properties of the Matrix Exponential 2.5 The Lie Algebra of a Matrix Lie Group 2.6 Properties of the Lie Algebra 2.7 The Exponential Mapping 2.8 Lie Algebras 2.9 The Complexification of a Real Lie Algebra 2.10 Exercises 3 The Baker--Campbell--Hausdorff Formula 3.1 The Baker--Campbell--Hausdorff Formula for the Heisenberg Group 3.2 The General Baker--Campbell--Hausdorff Formula 3.3 The Derivative of the Exponential Mapping 3.4 Proof of the Baker--Campbell--Hausdorff Formula 3.5 The Series Form of the Baker--Campbell--Hausdorff Formula 3.6 Lie Algebra Versus Lie Group Homomorphisms 3.7 Covering Groups 3.8 Subgroups and Subalgebras 3.9 Exercises 4 Basic Representation Theory 4.1 Representations 4.2 Why Study Representations? 4.3 Examples of Representations 4.4 The Irreducible Representations of $ {su}(2)$ 4.5 Direct Sums of Representations 4.6 Tensor Products of Representations 4.7 Dual Representations 4.8 Schur's Lemma 4.9 Group Versus Lie Algebra Representations 4.10 Complete Reducibility 4.11 Exercises Part II Semisimple Theory 5 The Representations of $ {SU}(3)$ 5.1 Introduction 5.2 Weights and Roots 5.3 The Theorem of the Highest Weight 5.4 Proof of the Theorem 5.5 An Example: Highest Weight $( 1,1) $ 5.6 The Weyl Group 5.7 Weight Diagrams 5.8 Exercises 6 Semisimple Lie Algebras 6.1 Complete Reducibility and Semisimple Lie Algebras 6.2 Examples of Reductive and
