The aim of this book

Why another introduction to Lie algebras? The subject of this book is one of the areas of algebra that has been most written about. The basic theory was unearthed more than a century ago and has been polished in a long chain of textbooks to a sheen of classical perfection. Experts' shelves are graced by the three volumes of Bourbaki [1]; for students with the right background and motivation to learn from them, the expositions in the books by Humphreys [10], Fulton and Harris [6], and Carter [2] could hardly be bettered; and there is a recent undergraduate-level introduction by Erdmann and Wildon [4]. So where is the need for this book?

The answer comes from my own experience in teaching courses on Lie algebras to Australian honours-level undergraduates (see the Acknowledgements section). Such courses typically consist of 24 one-hour lectures. At my own university the algebraic background knowledge of the students would be: linear algebra up to the Jordan canonical form, the basic theory of groups and rings, the rudiments of group representation theory, and a little multilinear algebra in the context of differential forms. From that starting point, I have found it difficult to reach any peak of the theory by following the conventional route. My definition of a peak includes the classification of simple Lie algebras, the highest-weight classification of their modules, and the combinatorics of characters, tensor products, and crystal bases; by ‘the conventional route’ I mean the path signposted by the theorems of Engel and Lie (about solvability), Cartan (about the Killing form), Weyl (about complete reducibility), and Serre, as in the book by Humphreys [10].