Compound annual growth rate

2014-01-05T18:34:37Z

174.48.172.164: /* Example */

In [[mathematics]], a '''generalized Kac–Moody algebra''' is a [[Lie algebra]] that is similar to a
[[Kac–Moody algebra]], except that it is allowed to have imaginary [[Simple root (root system)|simple root]]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]].

== Motivation ==
Finite dimensional [[semisimple Lie algebra]]s 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 (mathematics)|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]]s.

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 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 character formula]], [[Cartan subalgebra]],
roots, weights, and so on.

== Definition ==
A symmetrized [[Cartan matrix]] is a (possibly infinite) square matrix
with entries <math>c_{ij}</math> such that
* <math>c_{ij}=c_{ji}\ </math>
* <math>c_{ij}\le 0\ </math> if <math>i\ne j\ </math>
* <math>2c_{ij}/c_{ii}\ </math> is an integer if <math>c_{ii}>0.\ </math>

The universal generalized Kac–Moody algebra with given symmetrized Cartan matrix is defined by [[Generator matrix|generator]]s <math>e_i</math> and <math>f_i</math> and <math>h_i</math> and relations
* <math>[e_i,f_j] = h_i\ </math> if <math>i =j</math>, 0 otherwise
* <math>[h_i,e_j]= c_{ij}e_j\ </math>, <math>[h_i,f_j]=-c_{ij}f_j\ </math>
* <math>[e_i,[e_i,\ldots,[e_i,e_j]]]= [f_i,[f_i,\ldots,[f_i,f_j]]] = 0\ </math> for <math>1-2c_{ij}/c_{ii}\ </math> applications of <math>e_i\ </math> or <math>f_i\ </math> if <math>c_{ii}>0\ </math>
* <math>[e_i,e_j] = [f_i,f_j] = 0 \ </math> if <math> c_{ij} = 0.\ </math>

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 [[Group extension%23Central extension|central extension]], or adding [[outer derivation]]s.

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 ''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 <math>(e_i,f_i)=1</math>.

* There is a [[Weyl character formula|character formula]] for [[highest weight module]]s, similar to the [[Weyl–Kac character formula]] for [[Kac–Moody algebra]]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 algebra]]s.
* [[Affine Kac–Moody algebra]]s
* Algebras with [[Lorentzian algebra|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 group]]s.

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 ==
*{{cite book |title=Infinite dimensional Lie algebras |authorlink=Victor Kac | last=Kac |first=Victor G. |coauthors= |year=1994 |edition=3rd edition |publisher=Cambridge University Press |location=New York |isbn=0-521-46693-8 |pages= |url= }}
*{{cite book |title=Infinite dimensional Lie algebras |last=Wakimoto |first=Minoru |authorlink= |coauthors= |year=2001 |publisher=American Mathematical Society |location=Providence, RI |isbn=0-8218-2654-9 |pages= |url= }}
*{{cite book |title=Automorphic Forms and Lie Superalgebras |last=Ray |first=Urmie |authorlink= |coauthors= |year=2006 |publisher=Springer |location=Dordrecht |isbn=1-4020-5009-7 |pages= |url= }}

{{DEFAULTSORT:Generalized Kac-Moody algebra}}
[[Category:Lie algebras]]

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