Monster Lie algebra

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The monster Lie algebra is a Z²-graded Lie algebra. The piece of degree (m, n) has dimension c_mn if (m, n) ≠ (0, 0) and dimension 2 if (m, n) = (0, 0). The

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

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

The denominator formula for the monster Lie algebra is the product formula for the j-invariant:

${\displaystyle j(p)-j(q)=\left({1 \over p}-{1 \over q}\right)\prod _{n,m=1}^{\infty }(1-p^{n}q^{m})^{c_{nm}}.}$

The denominator formula (sometimes called the Koike-Norton-Zagier infinite product identity) was discovered in the 1980s. Several mathematicians, including Masao Koike, Simon P. Norton, and Don Zagier, independently made the discovery.^[1]

Construction

There are two ways to construct the monster Lie algebra.^{[citation needed]} 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 by using the Goddard–Thorn theorem of string theory. This construction is much harder, but also proves that the monster group acts naturally on it.^[1]

References