Affine Lie algebra

Summary

An affine Lie algebra is a Kac–Moody algebra whose generalized Cartan matrix is of affine type. The untwisted ones are built from a finite-dimensional simple Lie algebra $\mathfrak{g}$. One takes the loop algebra $\mathfrak{g} \otimes \C[t, t^{-1}]$, adds a one-dimensional center and adds a derivation. They are the infinite-dimensional Kac–Moody algebras that are best understood. Their representation theory is tied to modular forms, to identities such as the Macdonald identities, and to two-dimensional conformal field theory.