An exposition of generalized Kac-Moody algebras

Abstract

We present a detailed exposition of the theory of generalized Kac-Moody algebras associated to symmetrizable matrices. A proof of the character formula for a standard module is given, generalizing the argument of Garland and Lepowsky for the Kac-Moody case. A short proof of the theorem that any generalized Kac-Moody algebra can be decomposed into direct (vector space) sums of free subalgebras and a Kac-Moody subalgebra is given.

Dimension formula for graded Lie algebras and its applications

“…The Weyl-Kac-Borcherds character formula. We first recall some of the basic facts about generalized , [Ju2]). Let I be a finite or countably infinite index set.…”

Some generalized Kac-Moody algebras with known root multiplicities

“…[Jur96] therefore extends the GKM G(C) by an algebra of 'degree derivations'. This increases the dimension of the Cartan subalgebra and ensures that the simple roots will be defined linearly independent.…”

“…Let E be the subalgebra generated by the e i , i ∈ I, and F be the subalgebra generated by the f i , i ∈ I. Then the GKM G(C) has the triangular decomposition [Jur96] G(C) = E ⊕ H ⊕ F.…”

“…This increases the dimension of the Cartan subalgebra and ensures that the simple roots will be defined linearly independent. Note that, besides the approaches of [Jur96] and [Bor92], others have been proposed, such as the approach described in [Kac90] whose 'realization' of the Cartan matrix again guarantees the linear independence of the resulting simple roots.…”

“…subspace of the GKM as multiplication by the scalar n i . [Jur96] defines the extended Lie algebra G e = G e (C) as the semidirect product of the GKM G(C) with the space of all degree derivations. The extension is central.…”