Monster Lie algebra

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In mathematics, the monster Lie algebra is an infinite dimensional generalized Kac-Moody algebra acted on by the monster group, that was used to prove the monstrous moonshine conjectures.

[edit] 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 as elliptic modular function

j(q) -744 = {1 \over q} + 196884 q + 21493760 q^2 + \cdots

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

(1,n) 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:

j(p)-j(q) = \left({1 \over p} - {1 \over q}\right) \prod_{n,m=1}^{\infty}(1-p^n q^m)^{c_{nm}}

[edit] 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. It can also be constructed from the monster vertex operator algebra 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.

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