2014, 21: 167-176. doi: 10.3934/era.2014.21.167

Groups of Lie type, vertex algebras, and modular moonshine

1. 

Department of Mathematics, University of Michigan, Ann Arbor, MI, 48109, United States

2. 

Institute of Mathematics, Academia Sinica, Taipei, 10617, Taiwan

Received  May 2014 Published  November 2014

We use recent work on integral forms in vertex operator algebras to construct vertex algebras over general commutative rings and Chevalley groups acting on them as vertex algebra automorphisms. In this way, we get series of vertex algebras over fields whose automorphism groups are essentially those Chevalley groups (actually, an exact statement depends on the field and involves upwards extensions of these groups by outer diagonal and graph automorphisms). In particular, given a prime power $q$, we realize each finite simple group which is a Chevalley or Steinberg variations over $\mathbb{F}_q$ as "most of'' the full automorphism group of a vertex algebra over $\mathbb{F}_q$. These finite simple groups are \[ A_n(q), B_n(q), C_n(q), D_n(q), E_6(q), E_7(q), E_8(q), F_4(q), G_2(q) \] \[ \text{and } ^{2}A_n(q), ^{2}D_n(q), ^{3}D_4(q), ^{2}E_6(q), \] where $q$ is a prime power.
    Also, we define certain reduced VAs. In characteristics 2 and 3, there are exceptionally large automorphism groups. A covering algebra idea of Frohardt and Griess for Lie algebras is applied to the vertex algebra situation.
    We use integral form and covering procedures for vertex algebras to complete the modular moonshine program of Borcherds and Ryba for proving an embedding of the sporadic group $F_3$ of order $2^{15}3^{10}5^3 7^2 13{\cdot }19{\cdot} 31$ in $E_8(3)$.
Citation: Robert L. Griess Jr., Ching Hung Lam. Groups of Lie type, vertex algebras, and modular moonshine. Electronic Research Announcements, 2014, 21: 167-176. doi: 10.3934/era.2014.21.167
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show all references

References:
[1]

Comm. Math. Phys., 253 (2005), 171-219. doi: 10.1007/s00220-004-1132-5.  Google Scholar

[2]

Oxford University Press, Eynsham, 1985.  Google Scholar

[3]

Proc. Nat. Acad. Sci. USA, 83 (1986), 3068-3071. doi: 10.1073/pnas.83.10.3068.  Google Scholar

[4]

Duke Math. J., 83 (1996), 435-459. doi: 10.1215/S0012-7094-96-08315-5.  Google Scholar

[5]

Springer Lecture Notes in Mathematics, 131, Springer-Verlag, Berlin, 1970. Google Scholar

[6]

A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1989.  Google Scholar

[7]

3rd Edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 290, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4757-6568-7.  Google Scholar

[8]

J. Algebra, 365 (2012), 184-198. doi: 10.1016/j.jalgebra.2012.05.006.  Google Scholar

[9]

Academic Press, Inc., Boston, MA, 1988.  Google Scholar

[10]

Duke Math. J., 66 (1992), 123-168. doi: 10.1215/S0012-7094-92-06604-X.  Google Scholar

[11]

Nova J. Algebra Geom., 1 (1992), 339-345. Available from: http://www.math.lsa.umich.edu/~rlg/griesspublicationlist.html.  Google Scholar

[12]

J. Reine Angew. Math., 561 (2003), 1-37. doi: 10.1515/crll.2003.067.  Google Scholar

[13]

submitted, 2014, about 40 pp. Google Scholar

[14]

Nederl. Akad. Wetensch. Indag. Math., 44 (1982), 441-452, 453-460.  Google Scholar

[15]

Reprint of the 1994 original, Graduate Studies in Mathematics, 100, American Mathematical Society, Providence, RI, 2009.  Google Scholar

[16]

3rd edition, Cambridge University Press, Cambridge, 1990. doi: 10.1017/CBO9780511626234.  Google Scholar

[17]

Amer. J. Math., 78 (1956), 555-563. doi: 10.2307/2372673.  Google Scholar

[18]

J. Algebra, 77 (1982), 484-504. doi: 10.1016/0021-8693(82)90268-X.  Google Scholar

[19]

Ann. of Math. (2), 159 (2004), 535-596. doi: 10.4007/annals.2004.159.535.  Google Scholar

[20]

Mem. Amer. Math. Soc., 96 (1992), viii+97 pp. doi: 10.1090/memo/0466.  Google Scholar

[21]

J. Lond. Math. Soc. (2), 83 (2011), 493-516. doi: 10.1112/jlms/jdq078.  Google Scholar

[22]

Pacific J. Math., 11 (1961), 1119-1129. doi: 10.2140/pjm.1961.11.1119.  Google Scholar

[23]

in Finite Groups (ed. N. Iwahori), Japan Society for Promotion of Science, Tokyo, 1976, 113-116. Google Scholar

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