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| In [[mathematics]], the '''monster Lie algebra''' is an infinite
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| dimensional [[generalized Kac–Moody algebra]] acted on by the [[monster group]], which was used to prove the [[monstrous moonshine]] conjectures.
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| == Structure ==
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| The monster Lie algebra ''m'' is a ''Z<sup>2</sup>''-graded Lie algebra.
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| The piece of degree ''(m,n)'' has dimension ''c<sub>mn</sub>'' if
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| ''(m,n)'' is nonzero, and dimension 2 if ''(m,n)'' is (0,0).
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| The integers ''c<sub>n</sub>'' are the coefficients
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| of ''q<sup>n</sup>'' of the [[j-invariant]] as [[elliptic modular function]]
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| ::<math>j(q) -744 = {1 \over q} + 196884 q + 21493760 q^2 + \cdots.</math> | |
| <!-- To do : picture of root spaces-->
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| The [[Cartan subalgebra]] is the 2-dimensional subspace of degree
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| (0,0), so the monster Lie algebra has rank 2.
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| The monster Lie algebra has just one real [[Simple root (root system)|simple root]], given by the vector
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| (1,-1), and the [[Weyl group]] has order 2, and acts by mapping
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| ''(m,n)'' to ''(n,m)''. The imaginary simple roots are the vectors
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| :(1,''n'') for ''n'' = 1,2,3,...,
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| and they have multiplicities ''c<sub>n</sub>''.
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| The [[denominator formula]] for the monster Lie algebra is the product formula
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| for the ''j''-invariant:
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| ::<math>j(p)-j(q) = \left({1 \over p} - {1 \over q}\right) \prod_{n,m=1}^{\infty}(1-p^n q^m)^{c_{nm}}.</math>
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| == Construction ==
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| 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.
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| 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 has the advantage of proving that the [[monster group]] acts naturally on it.
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| == References == | |
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| * Richard Borcherds, "Vertex algebras, Kac-Moody algebras, and the Monster", ''Proc. Natl. Acad. Sci. USA.'' '''83''' (1986) 3068-3071
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| * 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
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| * [[Victor Kac]], "Vertex algebras for beginners". ''University Lecture Series, 10.'' American Mathematical Society, 1998. viii+141 pp. ISBN 0-8218-0643-2
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| * 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)
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| [[Category:Lie algebras]]
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| [[Category:Moonshine theory]]
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Hi, everybody!
I'm Vietnamese female ;=).
I really love Leaf collecting and pressing!
My web site - Fifa 15 coin Hack