Vertex operator algebras, extended $E_8$ diagram, and McKay's observation on the Monster simple group
Vertex operator algebras, extended $E_8$ diagram, and McKay's observation on the Monster simple group
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DOI：
10.1090/s0002-9947-07-04002-0
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发表时间：
2004-02
影响因子：
1.3
通讯作者：
C. Lam;H. Yamada;H. Yamauchi
中科院分区：
文献类型：
--
作者：
C. Lam;H. Yamada;H. Yamauchi
利用顶点算子代数理论，研究了McKay关于Monster单群的观察，将Monster单群的2A-对合与扩展的E8图联系起来.我们首先考虑E8格的子格L，通过每次从扩展Eg图中移除一个节点而获得。然后，我们在格VOA V <$2E8中构造一个与L相关的陪集（或交换子）子代数U。U中有两个中心电荷为1/2的自然共形矢量，使得它们的内积正好是Conway（1985）所预言的值。格里斯代数的U符合代数描述他的表3。存在U的一个标准自同构，|E8/L|.这样的自同构可以推广到Leech格VOA VA，它实际上是两个Miyamoto对合的乘积。在本文的续篇（2005）中，将详细讨论U的性质。如果U实际上包含在Moonshine VOA V 1中，则两个Miyamoto对合的乘积在Monster单群的期望共轭类中。
We study McKay's observation on the Monster simple group, which relates the 2A-involutions of the Monster simple group to the extended E 8 diagram, using the theory of vertex operator algebras (VOAs). We first consider the sublattices L of the E 8 lattice obtained by removing one node from the extended Eg diagram at each time. We then construct a certain coset (or commutant) subalgebra U associated with L in the lattice VOA V √2E8 . There are two natural conformal vectors of central charge 1/2 in U such that their inner product is exactly the value predicted by Conway (1985). The Griess algebra of U coincides with the algebra described in his Table 3. There is a canonical automorphism of U of order |E 8 /L|. Such an automorphism can be extended to the Leech lattice VOA V A , and it is in fact a product of two Miyamoto involutions. In the sequel (2005) to this article, the properties of U will be discussed in detail. It is expected that if U is actually contained in the Moonshine VOA V 1 , the product of two Miyamoto involutions is in the desired conjugacy class of the Monster simple group.