In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a generalized Kac–Moody algebra is a Lie algebra

Lie algebra

In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...

 that is similar to a
Kac–Moody algebra

Kac–Moody algebra

In mathematics, a Kac–Moody algebra is a Lie algebra, usually infinite-dimensional, that can be defined by generators and relations through a generalized Cartan matrix...

, except that it is allowed to have imaginary simple root

Simple root

in mathematics the term simple root can refer to one of two unrelated notions:*A simple root of a polynomial is a root of multiplicity one*A simple root in a root system is a member of a subset determined by a choice of positive roots...

s.
Generalized Kac–Moody algebras are also sometimes called GKM algebras,
Borcherds–Kac–Moody algebras, BKM algebras, or Borcherds algebras. The best known example is the monster Lie algebra

Monster Lie algebra

In mathematics, the monster Lie algebra is an infinitedimensional generalized Kac–Moody algebra acted on by the monster group, which was used to prove the monstrous moonshine conjectures.- Structure :...

.

Motivation

Finite dimensional semisimple Lie algebras have the following properties:

• They have a symmetric invariant bilinear form .
• They have a grading such that the degree zero piece (the Cartan subalgebra) is abelian.
• They have a (Cartan) involution w.
• (a, w(a)) is positive if a is nonzero.

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 algebra

Affine Lie algebra

In mathematics, an affine Lie algebra is an infinite-dimensional Lie algebra that is constructed in a canonical fashion out of a finite-dimensional simple Lie algebra. It is a Kac–Moody algebra for which the generalized Cartan matrix is positive semi-definite and has corank 1...

s.

The monster Lie algebra

Monster Lie algebra

In mathematics, the monster Lie algebra is an infinitedimensional generalized Kac–Moody algebra acted on by the monster group, which was used to prove the monstrous moonshine conjectures.- Structure :...

 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 Kac–Moody algebras.
They are essentially the same as algebras given by certain generators and relations (described below).

Informally, generalized Kac–Moody algebras are the Lie algebras that
behave like
finite dimensional
semisimple Lie algebras. In particular they have a Weyl group

Weyl group

In mathematics, in particular the theory of Lie algebras, the Weyl group of a root system Φ is a subgroup of the isometry group of the root system. Specifically, it is the subgroup which is generated by reflections through the hyperplanes orthogonal to the roots, and as such is a finite reflection...

, Weyl character formula

Weyl character formula

In mathematics, the Weyl character formula in representation theory describes the characters of irreducible representations of compact Lie groups in terms of their highest weights. It was proved by ....

, Cartan subalgebra,
roots, weights, and so on.

Definition

A symmetrized Cartan matrix

Cartan matrix

In mathematics, the term Cartan matrix has three meanings. All of these are named after the French mathematician Élie Cartan. In fact, Cartan matrices in the context of Lie algebras were first investigated by Wilhelm Killing, whereas the Killing form is due to Cartan.- Lie algebras :A generalized...

 is a (possibly infinite) square matrix
with entries such that

• 
• if
• is an integer if

The universal generalized Kac–Moody algebra with given symmetrized Cartan matrix is defined by generator

Generator matrix

In coding theory, a generator matrix is a basis for a linear code, generating all its possible codewords.If the matrix is G and the linear code is C,where w is a codeword of the linear code C, c is a row vector, and a bijection exists between w and c. A generator matrix for an q-code has...

s and and and relations

• if , 0 otherwise
• ,
• for applications of or if
• if

These differ from the relations of a (symmetrizable) Kac-Moody 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 Kac-Moody algebra simple roots are always real.

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. 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
e_i degree 1, f_i degree -1,
and h_i degree 0.

The degree zero piece is an abelian subalgebra spanned by the elements h_i 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
  Symmetric bilinear form
  A symmetric bilinear form is a bilinear form on a vector space that is symmetric. Symmetric bilinear forms are of great importance in the study of orthogonal polarity and quadrics....
  
   such that .

• There is a character formula
  Weyl character formula
  In mathematics, the Weyl character formula in representation theory describes the characters of irreducible representations of compact Lie groups in terms of their highest weights. It was proved by ....
  
   for highest weight modules, similar to the Weyl–Kac character formula for Kac–Moody algebra
  Kac–Moody algebra
  In mathematics, a Kac–Moody algebra is a Lie algebra, usually infinite-dimensional, that can be defined by generators and relations through a generalized Cartan matrix...
  
  s 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
  Automorphic form
  In mathematics, the general notion of automorphic form is the extension to analytic functions, perhaps of several complex variables, of the theory of modular forms...
  
   of singular weight.

There appear to be only a finite number of examples of the third type.
Two examples are the monster Lie algebra

Monster Lie algebra

In mathematics, the monster Lie algebra is an infinitedimensional generalized Kac–Moody algebra acted on by the monster group, which was used to prove the monstrous moonshine conjectures.- Structure :...

,
acted on by the monster group

Monster group

In the mathematical field of group theory, the Monster group M or F1 is a group of finite order:...

 and used in the monstrous moonshine

Monstrous moonshine

In mathematics, monstrous moonshine, or moonshine theory, is a term devised by John Horton Conway and Simon P. Norton in 1979, used to describe the connection between the monster group M and modular functions .- History :Specifically, Conway and Norton, following an initial observationby John...

 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.