mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a generalized Kac–Moody algebra is a

Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...

that is similar to a

Kac–Moody algebra In mathematics, a Kac–Moody algebra (named for Victor Kac and Robert Moody, who independently and simultaneously discovered them in 1968) is a Lie algebra, usually infinite-dimensional, that can be defined by generators and relations through a ge ...

, except that it is allowed to have imaginary

simple root Simple or SIMPLE may refer to: *Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (album), by Andy Yorke, 2008, and its title track * "Simple" (Florida Georgia Line song), 2018 * "Simple", a song by Johnn ...

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 In mathematics, the monster Lie algebra is an infinite-dimensional generalized Kac–Moody algebra acted on by the monster group, which was used to prove the monstrous moonshine conjectures. Structure The monster Lie algebra ''m'' is a ''Z2''-g ...

Motivation

Finite-dimensional

semisimple Lie algebra In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper ideals). Throughout the article, unless otherwise stated, a Lie algebra i ...

s have the following properties: * They have a nondegenerate symmetric invariant bilinear form (,). * They have a grading such that the degree zero piece (the

Cartan subalgebra In mathematics, a Cartan subalgebra, often abbreviated as CSA, is a nilpotent subalgebra \mathfrak of a Lie algebra \mathfrak that is self-normalising (if ,Y\in \mathfrak for all X \in \mathfrak, then Y \in \mathfrak). They were introduced by � ...

) is abelian. * They have a (Cartan)

involution Involution may refer to: * Involute, a construction in the differential geometry of curves * '' Agricultural Involution: The Processes of Ecological Change in Indonesia'', a 1963 study of intensification of production through increased labour inpu ...

''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 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. Given an affine Lie algebra, one can also form the associated affine Kac-Moody a ...

s. The

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 In mathematics, in particular the theory of Lie algebras, the Weyl group (named after Hermann Weyl) of a root system Φ is a subgroup of the isometry group of that root system. Specifically, it is the subgroup which is generated by reflections th ...

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

, roots, weights, and so on.

Definition

A symmetrized

Cartan matrix In mathematics, the term Cartan matrix has three meanings. All of these are named after the French mathematician Élie Cartan. Amusingly, the Cartan matrices in the context of Lie algebras were first investigated by Wilhelm Killing, whereas the ...

is a (possibly infinite) square matrix with entries

c_

such that *

c_=c_\

c_\le 0\

i\ne j\

2c_/c_\

is an integer if

c_>0.\

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

generator Generator may refer to: * Signal generator, electronic devices that generate repeating or non-repeating electronic signals * Electric generator, a device that converts mechanical energy to electrical energy. * Generator (circuit theory), an eleme ...

e_i

and

f_i

and

h_i

and relations *

_i,f_j = h_i\

i =j

, 0 otherwise *

c_e_j\

-c_f_j\

.html" ;"title="_i,\ldots,[e_i,e_j.html" ;"title="_i,[e_i,\ldots,[e_i,e_j">_i,[e_i,\ldots,[e_i,e_j">_i,\ldots,[e_i,e_j.html" ;"title="_i,[e_i,\ldots,[e_i,e_j">_i,[e_i,\ldots,[e_i,e_j [f_i,[f_i,\ldots,[f_i,f_j] = 0\

for

1-2c_/c_\

applications of

e_i\

f_i\

c_>0\

[e_i,e_j] = [f_i,f_j] = 0 \

c_ = 0.\

These differ from the relations of a (symmetrizable)

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 derivation {{Short pages monitor

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 In mathematics, a symmetric bilinear form on a vector space is a bilinear map from two copies of the vector space to the field of scalars such that the order of the two vectors does not affect the value of the map. In other words, it is a bilinear ...

such that

(e_i,f_i)=1

. * There is a character formula for

highest weight module In the mathematical field of representation theory, a weight of an algebra ''A'' over a field F is an algebra homomorphism from ''A'' to F, or equivalently, a one-dimensional representation of ''A'' over F. It is the algebra analogue of a multipli ...

s, similar to the

Weyl–Kac 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 . There is a closely related formula for the char ...

for

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

s. *

Affine Kac–Moody 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. Given an affine Lie algebra, one can also form the associated affine Kac-Moody a ...

s * Algebras with Lorentzian Cartan subalgebra whose denominator function is an

automorphic form In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group ''G'' to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup \Gamma \subset G of ...

of singular weight. There appear to be only a finite number of examples of the third type. Two examples are the

, acted on by the

monster group In the area of abstract algebra known as group theory, the monster group M (also known as the Fischer–Griess monster, or the friendly giant) is the largest sporadic simple group, having order 246320597611213317192329314147 ...

and used in the

monstrous moonshine In mathematics, monstrous moonshine, or moonshine theory, is the unexpected connection between the monster group ''M'' and modular functions, in particular, the ''j'' function. The term was coined by John Conway and Simon P. Norton in 1979. ...

conjectures, and the

fake monster Lie algebra Fake may refer to: * Deception, an act or a statement intended to deceive ** Charlatan, a person who practices deception to obtain money or other advantages ** Counterfeit, a reproduction of an item, intended to deceive ** Cover-up, an attempt to ...

. There are similar examples associated to some of the other

sporadic simple group In mathematics, a sporadic group is one of the 26 exceptional groups found in the classification of finite simple groups. A simple group is a group ''G'' that does not have any normal subgroups except for the trivial group and ''G'' itself. The ...

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

* * * {{DEFAULTSORT:Generalized Kac-Moody algebra Lie algebras