This bold and refreshing approach to Lie algebras assumes only modest prerequisites (linear algebra up to the Jordan canonical form and a basic familiarity with groups and rings), yet it reaches a major result in representation theory: the highest-weight classification of irreducible modules of the general linear Lie algebra. The author's exposition is focused on this goal rather than aiming at the widest generality and emphasis is placed on explicit calculations with bases and matrices. The book begins with a motivating chapter explaining the context and relevance of Lie algebras and their representations and concludes with a guide to further reading. Numerous examples and exercises with full solutions are included. Based on the author's own introductory course on Lie algebras, this book has been thoroughly road-tested by advanced undergraduate and beginning graduate students and it is also suited to individual readers wanting an introduction to this important area of mathematics.
Anthony Henderson is an Associate Professor in the School of Mathematics and Statistics at the University of Sydney, where he has worked since graduating in 2001 from the Massachusetts Institute of Technology. He won the 2011 Christopher Heyde Medal of the Australian Academy of Science for his research in geometric and combinatorial aspects of representation theory and currently holds an Australian Research Council Future Fellowship. He has taught a wide range of undergraduate courses and in 2009 was awarded a Faculty of Science Citation for Excellence in Teaching.
1. Motivation: representations of Lie groups; 2. Definition of a Lie algebra; 3. Basic structure of a Lie algebra; 4. Modules over a Lie algebra; 5. The theory of sl2-modules; 6. General theory of modules; 7. Integral gln-modules; 8. Guide to further reading; Appendix: solutions to the exercises; References; Index.