Kac-Moody algebra

In mathematics, a Kac-Moody algebra is a Lie algebra, usually infinite-dimensional, that can be defined through a generalized root system. Kac-Moody algebras have applications throughout mathematics and mathematical physics.

Definition

A Kac-Moody algebra is specified by the following pieces of data:

1. An n by n generalized Cartan matrix <math>C = (c_{ij})<math> of rank r.
2. A vector space <math>\mathfrak{h}<math> over the complex numbers of dimension 2n − r.
3. A set of n linearly independent elements <math>\alpha_i\ <math> of <math>\mathfrak{h}<math> and a set of n linear independent elements <math>\alpha_i^*<math> of the dual space, such that <math>\alpha_i^*(\alpha_j) = c_{ij}<math>. The <math>\alpha_i\ <math> are known as coroots, while the <math>\alpha_i^*<math> are known as roots.

The Kac-Moody algebra is defined by generators <math>e_i<math> and <math>f_i<math> and the elements of <math>\mathfrak{h}<math> and relations

• <math>[e_i,f_i] = \alpha_i\ <math>.
• <math>[e_i,f_j] = 0\ <math> for <math>i \neq j<math>.
• <math>[e_i,x]=\alpha_i^*(x)e_i<math>, for <math>x \in \mathfrak{h}<math>.
• <math>[f_i,x]=-\alpha_i^*(x)f_i<math>, for <math>x \in \mathfrak{h}<math>.
• <math>[x,x'] = 0<math> for <math>x,x' \in \mathfrak{h}<math>.
• <math>[e_i,[e_i,\ldots,[e_i,e_j]]] = 0<math> for <math>1-c_{ij}\ <math> applications of <math>e_i\ <math>.
• <math>[f_i,[f_i,\ldots,[f_i,f_j]]] = 0<math> for <math>1-c_{ij}\ <math> applications of <math>f_i\ <math>.

A real (possible infinite-dimensional) Lie algebra is also considered a Kac-Moody algebra if its complexification is a Kac-Moody algebra.

Interpretation

<math>\mathfrak{h}<math> is a Cartan subalgebra of the Kac-Moody algebra.

If g is an element of the Kac-Moody algebra such that

<math>\forall g\in \mathfrak{h}\,[g,x]=\omega(x)g<math>

where ω is an element of <math>\mathfrak{h}^*<math>, then g is said to have weight ω. The Kac-Moody algebra can be diagonalized into weight eigenvectors. The Cartan subalgebra h has weight zero, e_i has weight α*_i and f_i has weight −α*_i. If the Lie bracket of two weight eigenvectors is nonzero, then its weight is the sum of their weights. The condition <math>[e_i,f_j] = 0\ <math> for <math>i \neq j<math> simply means the α*_i are simple roots.

Types of Kac-Moody algebras

C can be decomposed as DS where D is a positive diagonal matrix and S is a symmetric matrix.

• finite-dimensional simple Lie algebras (S is positive definite)
• affine (S is positive semidefinite)
• hyperbolic (S is indefinite)

S can never be negative definite or negative semidefinite because its trace is positive.