In mathematics, a generalized KacâÂÂMoody algebra is a Lie algebra that is similar to a KacâÂÂMoody algebra, except that it is allowed to have imaginary simple roots. Generalized KacâÂÂMoody algebras are also sometimes called GKM algebras, BorcherdsâÂÂKacâÂÂMoody algebras, BKM algebras, or Borcherds algebras. The best known example is the monster Lie algebra.
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. The Lie algebras satisfying these weaker conditions are more or less generalized KacâÂÂMoody algebras. They are essentially the same as algebras given by certain generators and relations (described below).
Informally, generalized KacâÂÂMoody algebras are the Lie algebras that behave like finite-dimensional semisimple Lie algebras. In particular they have a Weyl group, Weyl character formula, Cartan subalgebra, roots, weights, and so on.
A symmetrized Cartan matrix is a (possibly infinite) square matrix with entries such that
The universal generalized KacâÂÂMoody algebra with given symmetrized Cartan matrix is defined by generators and and and relations
These differ from the relations of a (symmetrizable) KacâÂÂMoody algebra mainly by allowing the diagonal entries of the Cartan matrix to be non-positive. In other words, we allow simple roots to be imaginary, whereas in a KacâÂÂMoody algebra simple roots are always real.
A generalized KacâÂÂMoody 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. Not much is known about these non-symmetrizable generalized KacâÂÂMoody algebras, and there seem to be no interesting examples.
It is also possible to extend the definition to superalgebras.
A generalized KacâÂÂMoody algebra can be graded by giving e<sub>i</sub> degree 1, f<sub>i</sub> degree âÂÂ1, and h<sub>i</sub> degree 0.
The degree zero piece is an abelian subalgebra spanned by the elements h<sub>i</sub> and is called the Cartan subalgebra.
Most properties of generalized KacâÂÂMoody algebras are straightforward extensions of the usual properties of (symmetrizable) KacâÂÂMoody algebras.
Most generalized KacâÂÂMoody algebras are thought not to have distinguishing features. The interesting ones are of three types:
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 KacâÂÂMoody algebras using the following principle: anything that looks like a generalized KacâÂÂMoody algebra is a generalized KacâÂÂMoody 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 KacâÂÂMoody 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 KacâÂÂMoody 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.