Finite-dimensional semisimple Lie algebras have the following properties: For example, for the algebras of n by n matrices of trace zero, the bilinear form is (a, b) = Trace(ab), the Cartan involution is given by minus the transpose, and the grading can be given by "distance from the diagonal" so that the Cartan subalgebra is the diagonal elements.
Conversely one can try to find all Lie algebras with these properties (and satisfying a few other technical conditions).
The answer is that one gets sums of finite-dimensional and affine Lie algebras.
The monster Lie algebra satisfies a slightly weaker version of the conditions above: (a, w(a)) is positive if a is nonzero and has nonzero degree, but may be negative when a has degree zero.
such that The universal generalized Kac–Moody algebra with given symmetrized Cartan matrix is defined by generators
A generalized Kac–Moody algebra is obtained from a universal one by changing the Cartan matrix, by the operations of killing something in the center, or taking a central extension, or adding outer derivations.
Some authors give a more general definition by removing the condition that the Cartan matrix should be symmetric.
Most generalized Kac–Moody algebras are thought not to have distinguishing features.
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.