Lectures on Lie Groups and Lie Algebras (London Mathematical Society Student Texts, #32)

In This Excellent Introduction To The Theory Of Lie Groups And Lie Algebras, Three Of The Leading Figures In This Area Have Written Up Their Lectures From An Lms/serc Sponsored Short Course In 1993. Together These Lectures Provide An Elementary Account Of The Theory That Is Unsurpassed. In The First Part Roger Carter Concentrates On Lie Algebras And Root Systems. In The Second Graeme Segal Discusses Lie Groups. And In The Final Part, Ian Macdonald Gives An Introduction To Special Linear Groups. Anybody Requiring An Introduction To The Theory Of Lie Groups And Their Applications Should Look No Further Than This Book. Roger Carter, Graeme Segal, Ian Macdonald. Based On The Three Introductory Lecture Courses Given At The Lms-serc Instructional Conference On Lie Theory And Algebraic Groups Held At Lancaster University In September 1993--fwd. Includes Bibliographical References (p.187 - 188) And Index. Cover; Series Page; Title; Copyright; Contents; Foreword; Lie Algebras and Root Systems R.W. Carter; Preface; 1 Introduction to Lie algebras; 1.1 Basic concepts; 1.2 Representations and modules; 1.3 Special kinds of Lie algebra; 1.4 The Lie algebras sln(C); 2 Simple Lie algebras over C; 2.1 Cartan subalgebras; 2.2 The Cartan decomposition; 2.3 The Killing form; 2.4 The Weyl group; 2.5 The Dynkin diagram; 3 Representations of simple Lie algebras; 3.1 The universal enveloping algebra; 3.2 Verma modules; 3.3 Finite dimensional irreducible modules; 3.4 Weyl's character and dimension formulae 3.5 Fundamental representations4 Simple groups of Lie type; 4.1 A Chevalley basis of g; 4.2 Chevalley groups over an arbitrary field; 4.3 Finite Chevalley groups; 4.4 Twisted groups; 4.5 Suzuki and Ree groups; 4.6 Classification of finite simple groups; Lie Groups Graeme Segal; Introduction; 1 Examples; Matrix groups; Low dimensional examples; Local isomorphism; 2 SU2, S03, and SL2R; 3 Homogeneous spaces; Symmetric spaces; Complex structures on R2n; 4 Some theorems about matrices; A The polar decomposition; B The Gram-Schmidt process; C Reduced echelon form: the Bruhat decomposition 14 The Borel-Weil theorem15 Representations of non-compact groups; 16 Representations of S L2R; 17 The Heisenberg group the metaplectic representation, and the spin representation; The spin representation; Linear Algebraic Groups I.G. Macdonald; Preface; Introduction; 1 Affine algebraic varieties; Morphisms; Products; The image of a morphism; Dimension; 2 Linear algebraic groups: definition and elementary properties; Jordan decomposition; Interlude; 3 Projective algebraic varieties; Prevarieties and varieties; Projective Varieties; Complete varieties; 4 Tangent spaces. Separability D Diagonalization and maximal tori5 Lie theory; Smooth manifolds; Tangent spaces; One-parameter subgroups and the exponential map; Lie's theorems; 6 Fourier series and representation theory; General remarks about representations; 7 Compact groups and integration; A formula for integration on Un; 8 Maximal compact subgroups; 9 The Peter-Weyl theorem; The structure of Calg(G); 10 Functions on Rn and sn-l; The Radon transform; 11 Induced representations; 12 The complexification of a compact group; 13 The unitary groups and the symmetric groups; Weyl's correspondence; Quantum groups Three of the leading figures in the field have composed this excellent introduction to the theory of Lie groups and Lie algebras. Together these lectures provide an elementary account of the theory that is unsurpassed. In the first part, Roger Carter concentrates on Lie algebras and root systems. In the second Graeme Segal discusses Lie groups. And in the final part, Ian Macdonald gives an introduction to special linear groups. Graduate students requiring an introduction to the theory of Lie groups and their applications should look no further than this book. Separability5 The Lie algebra of a linear algebraic group; The adjoint representation; 6 Homogeneous spaces and quotients; 7 Borel subgroups and maximal tori; Borel subgroups; Maximal tori; 8 The root structure of a linear algebraic group; Characters and one-parameter subgroups of tori; The root system R(G, T); The root datum R(G, T); Notes and references; Bibliography; Index