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), 4107-4123

MSC (2000):
Primary 17B68, 17B69, 20D08

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 -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 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:
http://dx.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