January  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
References:
[1]

T. Abe, C. Dong and H. Li, Fusion rules for the vertex operator algebra $M(1)$ and $V_L^+$,, \emph{Comm. Math. Phys.}, 253 (2005), 171. doi: 10.1007/s00220-004-1132-5.

[2]

J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, Atlas of Finite Groups. Maximal Subgroups and Ordinary Characters for Simple Groups,, Oxford University Press, (1985).

[3]

R. E. Borcherds, Vertex algebras, Kac-Moody algebras, and the Monster,, \emph{Proc. Nat. Acad. Sci. USA, 83 (1986), 3068. doi: 10.1073/pnas.83.10.3068.

[4]

R. Borcherds and A. Ryba, Modular Moonshine. II,, \emph{Duke Math. J.}, 83 (1996), 435. doi: 10.1215/S0012-7094-96-08315-5.

[5]

Borel, et. al, Seminar on Algebraic Groups and Related Finite Groups,, Springer Lecture Notes in Mathematics, (1970).

[6]

R. W. Carter, Simple Groups of Lie Type,, A Wiley-Interscience Publication, (1989).

[7]

J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups,, 3rd Edition, (1999). doi: 10.1007/978-1-4757-6568-7.

[8]

C. Y. Dong and R. L. Griess, Jr., Integral forms in vertex operator algebras which are invariant under finite groups,, \emph{J. Algebra}, 365 (2012), 184. doi: 10.1016/j.jalgebra.2012.05.006.

[9]

I. B. Frenkel, J. Lepowsky and A. Meurman, Vertex Operator Algebras and the Monster,, Academic Press, (1988).

[10]

I. B. Frenkel and Y. Zhu, Vertex operator algebras associated to representations of affine and Virasoro algebras,, \emph{Duke Math. J.}, 66 (1992), 123. doi: 10.1215/S0012-7094-92-06604-X.

[11]

D. Frohardt and R. L. Griess, Jr., Automorphisms of modular Lie algebras,, \emph{Nova J. Algebra Geom.}, 1 (1992), 339.

[12]

R. L. Griess, Jr. and G. Höhn, Virasoro frames and their stabilizers for the $E_8$ lattice type vertex operator algebra,, \emph{J. Reine Angew. Math.}, 561 (2003), 1. doi: 10.1515/crll.2003.067.

[13]

R. L. Griess and C. H. Lam, Groups of Lie type, vertex algebras and modular moonshine,, submitted, (2014).

[14]

G. M. D. Hogeweij, Almost-classical Lie algebras. I, II,, \emph{Nederl. Akad. Wetensch. Indag. Math.}, 44 (1982), 441.

[15]

I. M. Isaacs, Algebra: A Graduate Course,, Reprint of the 1994 original, (1994).

[16]

V. G. Kac, Infinite-Dimensional Lie Algebras,, 3rd edition, (1990). doi: 10.1017/CBO9780511626234.

[17]

S. Lang, Algebraic groups over finite fields,, \emph{Amer. J. Math.}, 78 (1956), 555. doi: 10.2307/2372673.

[18]

J. Lepowsky and A. Meurman, An $E_8$-approach to the Leech lattice and the Conway group,, \emph{J. Algebra}, 77 (1982), 484. doi: 10.1016/0021-8693(82)90268-X.

[19]

M. Miyamoto, A new construction of the Moonshine vertex operator algebra over the real number field,, \emph{Ann. of Math. (2)}, 159 (2004), 535. doi: 10.4007/annals.2004.159.535.

[20]

S. A. Prevost, Vertex algebras and integral bases for the enveloping algebras of affine Lie algebras,, \emph{Mem. Amer. Math. Soc.}, 96 (1992). doi: 10.1090/memo/0466.

[21]

H. Shimakura, An $E_8$-approach to the moonshine vertex operator algebra,, \emph{J. Lond. Math. Soc. (2)}, 83 (2011), 493. doi: 10.1112/jlms/jdq078.

[22]

R. Steinberg, Automorphisms of classical Lie algebras,, \emph{Pacific J. Math.}, 11 (1961), 1119. doi: 10.2140/pjm.1961.11.1119.

[23]

J. Thompson, A simple subgroup of $E_8(3)$,, in \emph{Finite Groups} (ed. N. Iwahori), (1976), 113.

show all references

References:
[1]

T. Abe, C. Dong and H. Li, Fusion rules for the vertex operator algebra $M(1)$ and $V_L^+$,, \emph{Comm. Math. Phys.}, 253 (2005), 171. doi: 10.1007/s00220-004-1132-5.

[2]

J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, Atlas of Finite Groups. Maximal Subgroups and Ordinary Characters for Simple Groups,, Oxford University Press, (1985).

[3]

R. E. Borcherds, Vertex algebras, Kac-Moody algebras, and the Monster,, \emph{Proc. Nat. Acad. Sci. USA, 83 (1986), 3068. doi: 10.1073/pnas.83.10.3068.

[4]

R. Borcherds and A. Ryba, Modular Moonshine. II,, \emph{Duke Math. J.}, 83 (1996), 435. doi: 10.1215/S0012-7094-96-08315-5.

[5]

Borel, et. al, Seminar on Algebraic Groups and Related Finite Groups,, Springer Lecture Notes in Mathematics, (1970).

[6]

R. W. Carter, Simple Groups of Lie Type,, A Wiley-Interscience Publication, (1989).

[7]

J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups,, 3rd Edition, (1999). doi: 10.1007/978-1-4757-6568-7.

[8]

C. Y. Dong and R. L. Griess, Jr., Integral forms in vertex operator algebras which are invariant under finite groups,, \emph{J. Algebra}, 365 (2012), 184. doi: 10.1016/j.jalgebra.2012.05.006.

[9]

I. B. Frenkel, J. Lepowsky and A. Meurman, Vertex Operator Algebras and the Monster,, Academic Press, (1988).

[10]

I. B. Frenkel and Y. Zhu, Vertex operator algebras associated to representations of affine and Virasoro algebras,, \emph{Duke Math. J.}, 66 (1992), 123. doi: 10.1215/S0012-7094-92-06604-X.

[11]

D. Frohardt and R. L. Griess, Jr., Automorphisms of modular Lie algebras,, \emph{Nova J. Algebra Geom.}, 1 (1992), 339.

[12]

R. L. Griess, Jr. and G. Höhn, Virasoro frames and their stabilizers for the $E_8$ lattice type vertex operator algebra,, \emph{J. Reine Angew. Math.}, 561 (2003), 1. doi: 10.1515/crll.2003.067.

[13]

R. L. Griess and C. H. Lam, Groups of Lie type, vertex algebras and modular moonshine,, submitted, (2014).

[14]

G. M. D. Hogeweij, Almost-classical Lie algebras. I, II,, \emph{Nederl. Akad. Wetensch. Indag. Math.}, 44 (1982), 441.

[15]

I. M. Isaacs, Algebra: A Graduate Course,, Reprint of the 1994 original, (1994).

[16]

V. G. Kac, Infinite-Dimensional Lie Algebras,, 3rd edition, (1990). doi: 10.1017/CBO9780511626234.

[17]

S. Lang, Algebraic groups over finite fields,, \emph{Amer. J. Math.}, 78 (1956), 555. doi: 10.2307/2372673.

[18]

J. Lepowsky and A. Meurman, An $E_8$-approach to the Leech lattice and the Conway group,, \emph{J. Algebra}, 77 (1982), 484. doi: 10.1016/0021-8693(82)90268-X.

[19]

M. Miyamoto, A new construction of the Moonshine vertex operator algebra over the real number field,, \emph{Ann. of Math. (2)}, 159 (2004), 535. doi: 10.4007/annals.2004.159.535.

[20]

S. A. Prevost, Vertex algebras and integral bases for the enveloping algebras of affine Lie algebras,, \emph{Mem. Amer. Math. Soc.}, 96 (1992). doi: 10.1090/memo/0466.

[21]

H. Shimakura, An $E_8$-approach to the moonshine vertex operator algebra,, \emph{J. Lond. Math. Soc. (2)}, 83 (2011), 493. doi: 10.1112/jlms/jdq078.

[22]

R. Steinberg, Automorphisms of classical Lie algebras,, \emph{Pacific J. Math.}, 11 (1961), 1119. doi: 10.2140/pjm.1961.11.1119.

[23]

J. Thompson, A simple subgroup of $E_8(3)$,, in \emph{Finite Groups} (ed. N. Iwahori), (1976), 113.

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