Lectures on Lie Groups and Lie Algebras

by Roger W. Carter , Ian G. MacDonald , Graeme B. Segal , Foreword by M. Taylor

• ISBN13: 9780521499224
• ISBN10: 0521499224
• Format: Paperback
• Copyright: 1995-09-29
• Publisher: Cambridge University Press

Summary

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.

Table of Contents

Foreword	vii
M. J. Taylor
Lie Algebras and Root Systems	1	(44)
R. W. Carter
Preface	3	(2)
Introduction to Lie algebras	5	(7)
Basic concepts	5	(2)
Representations and modules	7	(1)
Special kinds of Lie algebra	8	(2)
The Lie algebras sln (C)	10	(2)
Simple Lie algebras over C	12	(13)
Cartan subalgebras	12	(1)
The Cartan decomposition	13	(2)
The Killing form	15	(1)
The Weyl group	16	(2)
The Dynkin diagram	18	(7)
Representations of simple Lie algebras	25	(11)
The universal enveloping algebra	25	(1)
Verma modules	26	(1)
Finite dimensional irreducible modules	27	(2)
Weyl's character and dimension formulae	29	(3)
Fundamental representations	32	(4)
Simple groups of Lie type	36	(9)
A Chevalley basis of g	36	(2)
Chevalley groups over an arbitrary field	38	(1)
Finite Chevalley groups	39	(2)
Twisted groups	41	(2)
Suzuki and Ree groups	43	(1)
Classification of finite simple groups	44	(1)
Lie Groups	45	(88)
Graeme Segal
Introduction	47	(2)
Examples	49	(4)
SU2, SO3, and SL2R	53	(6)
Homogeneous spaces	59	(4)
Some theorems about matrices	63	(6)
Lie theory	69	(13)
Representation theory	82	(3)
Compact groups and integration	85	(4)
Maximal compact subgroups	89	(2)
The Peter-Weyl theorem	91	(9)
Functions on Rn'' and Sn-1	100	(4)
Induced representations	104	(4)
The complexification of a compact group	108	(2)
The unitary and symmetric groups	110	(5)
The Borel-Weil theorem	115	(5)
Representations of non-compact groups	120	(4)
Representations of SL2R	124	(4)
The Heisenberg group	128	(5)
Linear Algebraic Groups	133	(53)
I.G. Macdonald
Preface	135	(2)
Introduction	137	(2)
Affine algebraic varieties	139	(7)
Linear algebraic groups: definition and elementary properties	146	(11)
Interlude	154	(3)
Projective algebraic varieties	157	(5)
Tangent spaces. Separability	162	(4)
The Lie algebra of a linear algebraic group	166	(6)
Homogeneous spaces and quotients	172	(5)
Borel subgroups and maximal tori	177	(5)
The root structure of a linear algebraic group	182	(4)
Notes and references	186	(1)
Bibliography	187	(2)
Index	189