KacMoody

Kac–Moody algebras are a broad class of Lie algebras that generalize finite-dimensional semisimple Lie algebras to infinite dimensions. They are defined from a generalized Cartan matrix and were developed independently by Victor Kac and Robert Moody in the late 1960s and early 1970s. The construction provides a unifying framework for many infinite-dimensional symmetry algebras that appear in mathematics and theoretical physics.

A Kac–Moody algebra is built from generators e_i, f_i, h_i for i = 1,…,n, subject to standard Cartan relations: [h_i,h_j]=0, [h_i,e_j]=a_ij e_j, [h_i,f_j]=-a_ij f_j, and [e_i,f_j]=δ_ij h_i, together with Serre relations (ad e_i)^{1−a_ij}(e_j)=0 and (ad f_i)^{1−a_ij}(f_j)=0 for i ≠ j. The resulting algebra admits a triangular decomposition and a root space decomposition with a root system that includes real and imaginary roots. Its Weyl group is generated by reflections corresponding to real roots.

Classification by the generalized Cartan matrix A leads to several types. If A is positive definite, the algebra is finite-dimensional and coincides with a simple Lie algebra. If A is positive semi-definite but not definite (the affine case), one obtains affine Kac–Moody algebras, which are central extensions of loop algebras with a derivation; these algebras play a central role in conformal field theory and string theory. More general indefinite types, including hyperbolic and beyond, yield a wider class with rich representation theory.

Representations of Kac–Moody algebras include highest-weight modules, many of which are infinite-dimensional but retain structure analogous to finite-dimensional cases. Affine algebras admit integrable highest-weight modules with connections to vertex algebras, modular forms, and mathematical physics.