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
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:
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