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Vertex operator algebras, extended $ E_8$ diagram, and McKay's observation on the Monster simple group


Authors: Ching Hung Lam, Hiromichi Yamada and Hiroshi Yamauchi
Journal: Trans. Amer. Math. Soc. 359 (2007), 4107-4123
MSC (2000): Primary 17B68, 17B69, 20D08
DOI: https://doi.org/10.1090/S0002-9947-07-04002-0
Published electronically: April 6, 2007
MathSciNet review: 2309178
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Abstract: 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 $ E_8$ diagram at each time. We then construct a certain coset (or commutant) subalgebra $ U$ associated with $ L$ in the lattice VOA $ V_{\sqrt{2}E_8}$. 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 $ \vert E_8/L\vert$. Such an automorphism can be extended to the Leech lattice VOA $ V_\Lambda$, 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^\natural$, the product of two Miyamoto involutions is in the desired conjugacy class of the Monster simple group.


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Additional Information

Ching Hung Lam
Affiliation: Department of Mathematics, National Cheng Kung University, Tainan, Taiwan 701
Email: chlam@mail.ncku.edu.tw

Hiromichi Yamada
Affiliation: Department of Mathematics, Hitotsubashi University, Kunitachi, Tokyo 186-8601, Japan
Email: yamada@math.hit-u.ac.jp

Hiroshi Yamauchi
Affiliation: Graduate School of Mathematical Sciences, The University of Tokyo, Tokyo, 153-8914, Japan
Email: yamauchi@ms.u-tokyo.ac.jp

DOI: https://doi.org/10.1090/S0002-9947-07-04002-0
Received by editor(s): April 4, 2004
Received by editor(s) in revised form: March 4, 2005
Published electronically: April 6, 2007
Additional Notes: The first author was partially supported by NSC grant 91-2115-M-006-014 of Taiwan, R.O.C
The second author was partially supported by JSPS Grant-in-Aid for Scientific Research No. 15540015
Article copyright: © Copyright 2007 American Mathematical Society

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