Monster Lie algebra
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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.
- (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:
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.
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; 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.
- Borcherds, Richard E. (October 2002). "What Is ... the Monster?" (PDF). Notices of the American Mathematical Society. 49 (2): 1076–1077. (See p. 1077).
- Richard Borcherds, "Vertex algebras, Kac-Moody algebras, and the Monster", Proc. Natl. Acad. Sci. USA. 83 (1986) 3068-3071
- Igor Frenkel, James Lepowsky, Arne Meurman, "Vertex operator algebras and the Monster". Pure and Applied Mathematics, 134. Academic Press, Inc., Boston, MA, 1988. liv+508 pp. ISBN 0-12-267065-5
- Victor Kac, "Vertex algebras for beginners". University Lecture Series, 10. American Mathematical Society, 1996. viii+141 pp. ISBN 0-8218-0643-2; Kac, Victor G (1998). revised and expanded, 2nd edition. ISBN 0-8218-1396-X; pbk. vi+201 pp.
- R. W. Carter, "Lie Algebras of Finite and Affine Type", Cambridge Studies No. 96, 2005, ISBN 0-521-85138-6 (Introductory study text with a brief account of Borcherds algebra in Ch. 21)