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On isomorphism problems for vertex operator algebras associated with even lattices

Author: Hiroki Shimakura
Journal: Proc. Amer. Math. Soc.
MSC (2010): Primary 17B69; Secondary 11H06, 11H71
Posted: October 6, 2011
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Abstract: In this article, we completely determine the isomorphism classes of lattice vertex operator algebras and the vertex operator subalgebras fixed by a lift of the -isometry of the lattice. We also provide similar results for certain even lattices associated with doubly-even binary codes.

References

[Ab01] Toshiyuki Abe, Fusion rules for the charge conjugation orbifold, J. Algebra 242 (2001), no. 2, 624–655. MR 1848962 (2002f:17044), http://dx.doi.org/10.1006/jabr.2001.8838
[AD04] Toshiyuki Abe and Chongying Dong, Classification of irreducible modules for the vertex operator algebra 𝑉⁺_{𝐿}: general case, J. Algebra 273 (2004), no. 2, 657–685. MR 2037717 (2005c:17040), http://dx.doi.org/10.1016/j.jalgebra.2003.09.043
[ADL05] Toshiyuki Abe, Chongying Dong, and Haisheng Li, Fusion rules for the vertex operator algebra 𝑀(1) and 𝑉⁺_{𝐿}, Comm. Math. Phys. 253 (2005), no. 1, 171–219. MR 2105641 (2005i:17033), http://dx.doi.org/10.1007/s00220-004-1132-5
[Bo86] Richard E. Borcherds, Vertex algebras, Kac-Moody algebras, and the Monster, Proc. Nat. Acad. Sci. U.S.A. 83 (1986), no. 10, 3068–3071. MR 843307 (87m:17033), http://dx.doi.org/10.1073/pnas.83.10.3068
[CS99] J. H. Conway and N. J. A. Sloane, Sphere packings, lattices and groups, 3rd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 290, Springer-Verlag, New York, 1999. With additional contributions by E. Bannai, R. E. Borcherds, J. Leech, S. P. Norton, A. M. Odlyzko, R. A. Parker, L. Queen and B. B. Venkov. MR 1662447 (2000b:11077)
[Do93] Chongying Dong, Vertex algebras associated with even lattices, J. Algebra 161 (1993), no. 1, 245–265. MR 1245855 (94j:17023), http://dx.doi.org/10.1006/jabr.1993.1217
[DGH98] Chongying Dong, Robert L. Griess Jr., and Gerald Höhn, Framed vertex operator algebras, codes and the Moonshine module, Comm. Math. Phys. 193 (1998), no. 2, 407–448. MR 1618135 (99g:17050), http://dx.doi.org/10.1007/s002200050335
[DM04] Chongying Dong and Geoffrey Mason, Rational vertex operator algebras and the effective central charge, Int. Math. Res. Not. 56 (2004), 2989–3008. MR 2097833 (2005k:17034), http://dx.doi.org/10.1155/S1073792804140968
[DM04b] Chongying Dong and Geoffrey Mason, Holomorphic vertex operator algebras of small central charge, Pacific J. Math. 213 (2004), no. 2, 253–266. MR 2036919 (2005c:17044), http://dx.doi.org/10.2140/pjm.2004.213.253
[DN99] Chongying Dong and Kiyokazu Nagatomo, Representations of vertex operator algebra 𝑉_{𝐿}⁺ for rank one lattice 𝐿, Comm. Math. Phys. 202 (1999), no. 1, 169–195. MR 1686535 (2000b:17037), http://dx.doi.org/10.1007/s002200050578
[Eb02] Wolfgang Ebeling, Lattices and codes, Second revised edition, Advanced Lectures in Mathematics, Friedr. Vieweg & Sohn, Braunschweig, 2002. A course partially based on lectures by F. Hirzebruch. MR 1938666 (2003i:11093)
[FHL93] Igor B. Frenkel, Yi-Zhi Huang, and James Lepowsky, On axiomatic approaches to vertex operator algebras and modules, Mem. Amer. Math. Soc. 104 (1993), no. 494, viii+64. MR 1142494 (94a:17007)
[FLM88] Igor Frenkel, James Lepowsky, and Arne Meurman, Vertex operator algebras and the Monster, Pure and Applied Mathematics, vol. 134, Academic Press Inc., Boston, MA, 1988. MR 996026 (90h:17026)
[Hö95] G. Höhn, Selbstduale Vertexoperatorsuperalgebren und das Babymonster, dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, 1995.
[Hö03a] Gerald Höhn, Self-dual codes over the Kleinian four group, Math. Ann. 327 (2003), no. 2, 227–255. MR 2015068 (2004i:94063), http://dx.doi.org/10.1007/s00208-003-0440-y
[Hö03b] Gerald Höhn, Genera of vertex operator algebras and three-dimensional topological quantum field theories, Vertex operator algebras in mathematics and physics (Toronto, ON, 2000), Fields Inst. Commun., vol. 39, Amer. Math. Soc., Providence, RI, 2003, pp. 89–107. MR 2029792 (2005m:11077)
[Hö08] Gerald Höhn, Conformal designs based on vertex operator algebras, Adv. Math. 217 (2008), no. 5, 2301–2335. MR 2388095 (2009e:17055), http://dx.doi.org/10.1016/j.aim.2007.11.003
[KKM91] Masaaki Kitazume, Takeshi Kondo, and Izumi Miyamoto, Even lattices and doubly even codes, J. Math. Soc. Japan 43 (1991), no. 1, 67–87. MR 1082423 (92b:11041), http://dx.doi.org/10.2969/jmsj/04310067
[Sh04] Hiroki Shimakura, The automorphism group of the vertex operator algebra 𝑉⁺_{𝐿} for an even lattice 𝐿 without roots, J. Algebra 280 (2004), no. 1, 29–57. MR 2081920 (2005d:17038), http://dx.doi.org/10.1016/j.jalgebra.2004.05.018
[Sh06] Hiroki Shimakura, The automorphism groups of the vertex operator algebras 𝑉⁺_{𝐿}: general case, Math. Z. 252 (2006), no. 4, 849–862. MR 2206630 (2007d:17040), http://dx.doi.org/10.1007/s00209-005-0890-x
[Sh07] Hiroki Shimakura, Lifts of automorphisms of vertex operator algebras in simple current extensions, Math. Z. 256 (2007), no. 3, 491–508. MR 2299567 (2008f:17044), http://dx.doi.org/10.1007/s00209-006-0080-5
[Sh11] H. Shimakura, An -approach to the moonshine vertex operator algebra, J. London Math. Soc. 83 (2011), 493-516.
[Ve79] B. B. Venkov, Odd unimodular lattices, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 86 (1979), 40–48, 189 (Russian). Algebraic numbers and finite groups. MR 535479 (81a:10043)
[Wi41] Ernst Witt, Eine Identität zwischen Modulformen zweiten Grades, Abh. Math. Sem. Hansischen Univ. 14 (1941), 323–337 (German). MR 0005508 (3,163d)

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