Kac-Moody Algebra

A Kac-Moody algebra is a complex Lie algebra constructed from a generalized Cartan matrix. These Lie algebras extend the theory of finite-dimensional complex semisimple Lie algebras. A generalized Cartan matrix is an integer matrix with , for , and iff .

Choose a complex vector space of dimension , where is the matrix rank. Choose linearly independent elements , ..., in and linearly independent elements , ..., of its dual vector space satisfying . The auxiliary Lie algebra is generated by the elements of and , ..., , , ..., , subject to the Lie bracket relations

			(1)
			(2)
			(3)
			(4)

where , , and is the Kronecker delta. The Kac-Moody algebra is

(5)

where is the largest Lie ideal of satisfying (Kac 1990). Here a Lie ideal is a subspace with .

For , this construction gives the special linear Lie algebra . The matrix instead gives an infinite-dimensional affine Kac-Moody algebra. Affine types are encoded by affine Dynkin diagrams.