Kac-Moody algebras

One way of classifying the finite-dimensional simple Lie algebras is in terms of their Cartan matrices. If you slightly relax the defining conditions on the Cartan matrix, you get a wider class of algebras called the Kac-Moody algebras. These were introduced independently in 1968 by Victor G. Kac and Robert V. Moody. Kac-Moody algebras are mostly infinite-dimensional and provide a very natural generalization of the finite-dimensional simples.

The easiest class of Kac-Moody algebras to study is the affine algebras, since these have a particularly nice realization as (roughly) a finite-dimensional simple tensored with Laurent polynomials. This tensoring effectively gives you an infinite number of copies of the underlying finite-dimensional simple with a well-defined multiplication of any two elements, whether from the same or different copies. Because of this nice realization, a great deal of the finite-dimensional theory lifts up to the affine case and gains only enough complexity to make life interesting. It also turns out that affine Kac-Moody algebras have many applications in theoretical physics and many diverse areas of mathematics.

Beyond the affine algebras lie the indefinite ones. Indefinite Kac-Moody algebras are very poorly understood, in large part because we do not know any good concrete realization for them. Probably the most interesting and tractable class of indefinte Kac-Moody algebras is the type called Lorentzian. These have recently been studied in a group of papers by Nikulin and others. In general it seems that there are just too many Kac-Moody algebras to get a good realization that will apply to all of them. However, Kac-Moody theory is an extremely active area of mathematical research and we may be surprised.

The basic reference in Kac-Moody algebras is the book by Kac. For an easier introduction, try Wan; for a different approach, see the book by Moody and Pianzola.

Cartan matrices

Finite-dimensional simple Lie algebras are in one-to-one correspondence with Cartan matrices defined as follows:
A Cartan matrix is an nxn matrix A=(a_{ij}) with integer entries such that

all terms on the leading diagonal are 2 (a_{ii} = 2 for all i);
all off-diagonal are non-positive (a_{ij} \le 0, i \neq j);
off-diagonal zeros are matched by their mirro images (if a_{ij} = 0 then a_{ji}=0);
all principal minors of A are positive.

For Kac-Moody algebras, you drop the last condition.

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
Comments to dmel@music.stlawu.edu