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).
Several mathematicians, including Masao Koike, Simon P. Norton, and Don Zagier, independently made the discovery.
[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.
This construction is much harder, but also proves that the monster group acts naturally on it.