derivations. Some authors give a more general definition by removing the condition that the Cartan matrix should be symmetric. Not much is known about these non-symmetrizable generalized Kac–Moody algebras, and there seem to be no interesting examples. It is also possible to extend the definition to superalgebras. Structure A generalized Kac–Moody algebra can be graded by giving ei degree 1, fi degree −1, and hi degree 0. The degree zero piece is an abelian subalgebra spanned by the elements hi and is called the Cartan subalgebra. Properties Most properties of generalized Kac–Moody algebras are straightforward extensions of the usual properties of (symmetrizable) Kac–Moody algebras. A generalized Kac–Moody algebra has an invariant symmetric bilinear form such that . There is a character formula for highest weight modules, similar to the Weyl–Kac character formula for Kac–Moody algebras except that it has correction terms for the imaginary simple roots. Examples Most generalized Kac–Moody algebras are thought not to have distinguishing features. The interesting ones are of three types: Finite-dimensional semisimple Lie algebras. Affine Kac–Moody algebras Algebras with Lorentzian Cartan subalgebra whose denominator function is an automorphic form of singular weight. There appear to be only a finite number of examples of the third type. Two examples are the monster Lie algebra, acted on by the monster group and used in the monstrous moonshine conjectures, and the fake monster Lie algebra. There are similar examples associated to some of the other sporadic simple groups. It is possible to find many examples of generalized Kac–Moody algebras using the following principle: anything that looks like a generalized Kac–Moody algebra is a generalized Kac–Moody algebra. More precisely, if a Lie algebra is graded by a Lorentzian lattice and has an invariant bilinear form and satisfies a few other easily checked technical conditions, then it is a generalized Kac–Moody algebra. In particular one can use vertex algebras to construct a Lie algebra from any even lattice. If the lattice is positive definite it gives a finite-dimensional semisimple Lie algebra, if it is positive semidefinite it gives an affine Lie algebra, and if it is Lorentzian it gives an algebra satisfying the conditions above that is therefore a generalized Kac–Moody algebra. When the lattice is the even 26 dimensional unimodular Lorentzian lattice the construction gives the fake monster Lie algebra; all other Lorentzian lattices seem to give uninteresting algebras. References Lie algebras