Monster Lie algebra
Encyclopedia
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.
The piece of degree (m,n) has dimension cmn if
(m,n) is nonzero, and dimension 2 if (m,n) is (0,0).
The integers cn are the coefficients
of qn of the j-invariant
as elliptic modular function
The Cartan subalgebra is the 2-dimensional subspace of degree
(0,0), so the monster Lie algebra has rank 2.
The monster Lie algebra has just one real simple root
, given by the vector
(1,-1), and the Weyl group
has order 2, and acts by mapping
(m,n) to (n,m). The imaginary simple roots are the vectors
for n = 1,2,3,...,
and they have multiplicities cn.
The denominator formula for the monster Lie algebra is the product formula
for the j-invariant:
It can also be constructed from the monster vertex algebra
by using the no-ghost theorem of string theory
. This construction is much harder, but has the advantage of proving that the monster group
acts naturally on it.
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...
, the monster Lie algebra is an infinite
dimensional generalized Kac–Moody algebra
Generalized Kac–Moody algebra
In mathematics, a generalized Kac–Moody algebra is a Lie algebra that is similar to aKac–Moody algebra, except that it is allowed to have imaginary simple roots.Generalized Kac–Moody algebras are also sometimes called GKM algebras,...
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:...
, which was used to prove 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.
Structure
The monster Lie algebra m is a Z2-graded Lie algebra.The piece of degree (m,n) has dimension cmn if
(m,n) is nonzero, and dimension 2 if (m,n) is (0,0).
The integers cn are the coefficients
of qn of the j-invariant
J-invariant
In mathematics, Klein's j-invariant, regarded as a function of a complex variable τ, is a modular function defined on the upper half-plane of complex numbers.We haveThe modular discriminant \Delta is defined as \Delta=g_2^3-27g_3^2...
as elliptic modular function
The Cartan subalgebra is the 2-dimensional subspace of degree
(0,0), so the monster Lie algebra has rank 2.
The monster Lie algebra has just one real 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...
, given by the vector
(1,-1), and the 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...
has order 2, and acts by mapping
(m,n) to (n,m). The imaginary simple roots are the vectors
for n = 1,2,3,...,
and they have multiplicities cn.
The denominator formula for the monster Lie algebra is the product formula
for the j-invariant:
Construction
There are two ways to construct the monster Lie algebra. As it is a generalized Kac–Moody algebra whose simple roots are known, it can be defined by explicit generators and relations; however, this presentation does not give an action of the monster group on it.It can also be constructed from the monster vertex algebra
Monster vertex algebra
The monster vertex algebra is a vertex algebra acted on by the monster group that was constructed by Igor Frenkel, James Lepowsky, and Arne Meurman. R...
by using the no-ghost theorem of string theory
String theory
String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...
. This construction is much harder, but has the advantage of proving that the monster group
Monster group
In the mathematical field of group theory, the Monster group M or F1 is a group of finite order:...
acts naturally on it.