Journal of the Mathematical Society of Japan

3-transposition groups of symplectic type and vertex operator algebras

Atsushi MATSUO

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The 3-transposition groups that act on vertex operator algebras in the way described by Miyamoto in [Mi] are classified under the assumption that the group is centerfree and the VOA carries a positive-definite invariant Hermitian form.

Article information

J. Math. Soc. Japan, Volume 57, Number 3 (2005), 639-649.

First available in Project Euclid: 14 September 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 51E30: Other finite incidence structures [See also 05B30]
Secondary: 17B69: Vertex operators; vertex operator algebras and related structures

3-transposition group Fischer space vertex operator algebra


MATSUO, Atsushi. 3-transposition groups of symplectic type and vertex operator algebras. J. Math. Soc. Japan 57 (2005), no. 3, 639--649. doi:10.2969/jmsj/1158241926.

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  • M. Aschbacher, 3-transposition groups, Cambridge Tracts in Math., 124, Cambridge Univ. Press, 1997.
  • R. E. Borcherds, Vertex algebras, Kac-Moody algebras, and the monster, Proc. Natl. Acad. Sci. USA, 83 (1986), 3068–3071.
  • R. C. Bose, Strongly regular graphs, partial geometries, and partially balanced designs, Pacific J. Math., 13 (1963), 389–419.
  • H. Cuypers and J. I. Hall, The $3$-transposition Groups with Trivial Center, J. Algebra, 178 (1995), 149–193.
  • J. H. Conway, A simple construction for the Fischer-Griess monster group, Invent. Math., 79 (1985), 513–540.
  • I. Frenkel, Y.-Z. Huang and J. Lepowsky, On axiomatic approaches to vertex operator algebras and modules, Mem. Amer. Math. Soc., 104, Amer. Math. Soc., 1993.
  • C.-Y. Dong, H.-S. Li, G. Mason and S. P. Norton, Associative subalgebras of the Griess algebra and related topics, The Monster and Lie algebras, Ohio State Univ. Math. Res. Inst. Publ., 7, de Gruyter, Berlin, 1998, 27–42.
  • C.-Y. Dong, G. Mason and Y.-C. Zhu, Discrete series of the Virasoro algebra and the moonshine module, Algebraic groups and their generalizations: quantum and infinite-dimensional methods, Proc. Sympos. Pure Math., 56, Part 2, Amer. Math. Soc., Providence, RI, 1994, 295–316.
  • B. Fischer, Finite Groups Generated by $3$-Transpositions. I, Invent. Math., 13 (1971), 232–246.
  • I. B. Frenkel, J. Lepowsky and A. Meurman, Vertex operator algebras and the Monster, Pure and Appl. Math., 134, Academic Press, Boston, 1988.
  • D. Friedan, Z. Qiu and S. Shenker, Details of non-unitarity proof for highest weight representations of the Virasoro algebra, Comm. Math. Phys., 107 (1986), 535–542.
  • P. Ginsparg, Applied conformal field theory, Champs, cordes, et phénomènes critiques, Les Houches Session XLIX, 1988, North-Holland, 1990, 1–168.
  • R. L. Griess, Jr., A vertex operator algebra related to $E_8$ with automorphism group ${\rm O}^+(10,2)$, The Monster and Lie algebras, Ohio State Univ. Math. Res. Inst. Publ., 7, de Gruyter, Berlin, 1998, 43–58.
  • J. I. Hall, Graphs, geometry, 3-transpositions, and symplectic $\FF_2$-transvection groups, Proc. London Math. Soc. (3), 58 (1989), 89–111.
  • J. I. Hall, Some 3-transposition groups with normal 2-subgroups, Proc. London Math. Soc. (3), 58 (1989), 112–136.
  • D. G. Higman, Finite permutation groups of rank $3$, Math. Z., 86 (1964), 145–156.
  • M. Kitazume and M. Miyamoto, 3-transposition automorphism groups of VOA, Groups and combinatorics,–-,in memory of Michio Suzuki, Adv. Stud. Pure Math., 32, Math. Soc. Japan, 2001, 315–324.
  • A. Matsuo and M. Matsuo, The automorphism group of the Hamming code vertex operator algebra, J. Algebra, 228 (2000), 204–226.
  • A. Matsuo and K. Nagatomo, Axioms for a vertex algebra and the locality of quantum fields, MSJ Mem., 4, Math. Soc. Japan, 1999.
  • M. Miyamoto, Griess algebras and conformal vectors in vertex operator algebras, J. Algebra, 179 (1996), 523–548.
  • H. Shimakura, The automorphism group of the vertex operator algebra $V^+_L$ for an even lattice $L$ without roots, J. Algebra, 280 (2004), 29–57.
  • R. Weiss, A uniqueness lemma for groups generated by 3-transpositions, Math. Proc. Cambridge Philos. Soc., 97 (1985), 421–431.