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In mathematics, the monster Lie algebra is an infinite-dimensional generalized Kac–Moody algebra acted on by the monster group, which was used to prove the monstrous moonshine conjectures. == 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: :: 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Monster Lie algebra」の詳細全文を読む スポンサード リンク
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