The monster Lie algebra 
Richard Borcherds
Adv. Math. Vol 83, No. 1, September 1990, p. 30-47. 

We calculate the multiplicities of all the roots of the ``monster Lie 
algebra''. This gives an example of a Lie algebra all of whose simple roots 
and root multiplicities are known and which is not finite dimensional or an 
affine Kac-Moody algebra. There seem to be several similar infinite 
dimensional Lie algebras, which to a sympathetic eye appear to correspond 
to some of the sporadic simple groups.