Lie algebras

A Lie algebra L, is a vector space over some field together with a bilinear multiplication [,]:LxL-->L, called the bracket, which satisfies two simple properties:

  1. [x,y] = -[y,x] (Anticommutativity)
  2. [x[y,z]] = [[x,y],z] + [x,[y,z]] (Jacobi identity).
The Jacobi identity says that the adjoint action is a derivation.

It turns out that this simple formal definition gives you a vast range of interesting algebras. For example, any associative algebra can be given a Lie structure by defining [x,y] = xy - yx, where we denote the associtive multiplication by juxtaposition. The Lie bracket is then called the commutator and measures how non-commutative your algebra is.

The finite-dimensional simple (i.e., no ideals) Lie algebras over the complex numbers are well-understood. The canonical reference for their structure, classification and representation theory is the book by Humphreys. Over algebraically-closed fields of characteristic p, a huge amount of work has gone into showing that there are no surprises. I don't know of any good expository overviews.

When you start to consider infinite-dimensional (simple) Lie algebras (over C), life becomes much more interesting. Firstly, there are the Cartan algebras, which are Lie algebras of vector fields on finite-dimensional manifolds. These algebras have finite-dimensional analogues in characteristic p, and as simple superalgebras. They also provide prototypical graded algebras and so are canonical examples in graded situations.

The most widely studied class of infinite-dimensional Lie algebras is the Kac-Moody algebras introduced independently by V.G. Kac and R.V. Moody in 1968. A huge literature has grown up on Kac-Moody algebras, partly because they are intrinsically interesting and partly because they have vast numbers of applications within both mathematics and physics. In particular, much of the language of theoretical physics is dominated by Kac-Moody algebras.

Among Kac-Moody algebras, the ones most studied are the affine algebras. These, it turns out, are very closely related to the finite-dimensional simple Lie algebras, and a great deal of the finite theory can be lifted up to the affine setting. There are enough differences to make life interesting, but enough similarities so you know what questions to ask, and what answers you will probably get. The best overviews of Kac-Moody algebras are the books by Kac, Moody and Pianzola and Wan.

Lie algebra is an incredibly exciting and interesting place to be. There are large numbers of important unsolved questions, there is a rich theory in place, and there are hordes of applications. Lie algebras are also very beautiful. Recently, Lie algebras have spawned numerous variants, including Borcherds algebras, color algebras and the enormously popular, if somewhat misnamed, quantum groups.

Humphreys, J.E., Introduction to Lie algebras and representation theory, Springer, 1972.
Kac, V.G., Infinite dimensional Lie algebras, 3d edn, Cambridge University Press, 1990.
Moody, R.V.; Pianzola, A., Lie algebras with triangular decompositions, Wiley, 1995.
Wan, Z.-X., Introduction to Kac-Moody algebra, World Scientific, 1991.

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Last modified: 26 July 1996
Duncan J. Melville
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