Vertex operator algebras, extended 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), 41074123
MSC (2000):
Primary 17B68, 17B69, 20D08
Published electronically:
April 6, 2007
MathSciNet review:
2309178
Fulltext PDF Free Access
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Additional Information
Abstract: We study McKay's observation on the Monster simple group, which relates the involutions of the Monster simple group to the extended diagram, using the theory of vertex operator algebras (VOAs). We first consider the sublattices of the lattice obtained by removing one node from the extended diagram at each time. We then construct a certain coset (or commutant) subalgebra associated with in the lattice VOA . There are two natural conformal vectors of central charge in such that their inner product is exactly the value predicted by Conway (1985). The Griess algebra of coincides with the algebra described in his Table 3. There is a canonical automorphism of of order . Such an automorphism can be extended to the Leech lattice VOA , and it is in fact a product of two Miyamoto involutions. In the sequel (2005) to this article, the properties of will be discussed in detail. It is expected that if is actually contained in the Moonshine VOA , 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 1868601, Japan
Email:
yamada@math.hitu.ac.jp
Hiroshi Yamauchi
Affiliation:
Graduate School of Mathematical Sciences, The University of Tokyo, Tokyo, 1538914, Japan
Email:
yamauchi@ms.utokyo.ac.jp
DOI:
http://dx.doi.org/10.1090/S0002994707040020
PII:
S 00029947(07)040020
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 912115M006014 of Taiwan, R.O.C
The second author was partially supported by JSPS GrantinAid for Scientific Research No. 15540015
Article copyright:
© Copyright 2007
American Mathematical Society
