Journal of the Mathematical Society of Japan

Fixed point subalgebras of lattice vertex operator algebras by an automorphism of order three

Kenichiro TANABE and Hiromichi YAMADA

Full-text: Open access

Abstract

We study the fixed point subalgebra of a certain class of lattice vertex operator algebras by an automorphism of order 3, which is a lift of a fixed-point-free isometry of the underlying lattice. We classify the irreducible modules for the subalgebra. Moreover, the rationality and the $C_2$-cofiniteness of the subalgebra are established. Our result contains the case of the vertex operator algebra associated with the Leech lattice.

Article information

Source
J. Math. Soc. Japan, Volume 65, Number 4 (2013), 1169-1242.

Dates
First available in Project Euclid: 24 October 2013

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1382620191

Digital Object Identifier
doi:10.2969/jmsj/06541169

Mathematical Reviews number (MathSciNet)
MR3127822

Zentralblatt MATH identifier
1342.17021

Subjects
Primary: 17B69: Vertex operators; vertex operator algebras and related structures
Secondary: 17B68: Virasoro and related algebras

Keywords
vertex operator algebra orbifold Leech lattice

Citation

TANABE, Kenichiro; YAMADA, Hiromichi. Fixed point subalgebras of lattice vertex operator algebras by an automorphism of order three. J. Math. Soc. Japan 65 (2013), no. 4, 1169--1242. doi:10.2969/jmsj/06541169. https://projecteuclid.org/euclid.jmsj/1382620191


Export citation

References

  • T. Abe, Fusion rules for the charge conjugation orbifold, J. Algebra, 242 (2001), 624–655.
  • T. Abe, G. Buhl and C. Dong, Rationality, regularity, and $C_2$-cofiniteness, Trans. Amer. Math. Soc., 356 (2004), 3391–3402.
  • T. Abe and C. Dong, Classification of irreducible modules for the vertex operator algebra $V_L^+$: general case, J. Algebra, 273 (2004), 657–685.
  • T. Abe, C. Dong and H. S. Li, Fusion rules for the vertex operator algebra $M(1)$ and $V_L^+$, Comm. Math. Phys., 253 (2005), 171–219.
  • G. Buhl, A spanning set for VOA modules, J. Algebra, 254 (2002), 125–151.
  • 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, Clarendon Press, Oxford, 1985.
  • R. Dijkgraaf, C. Vafa, E. Verlinde and H. Verlinde, The operator algebra of orbifold models, Comm. Math. Phys., 123 (1989), 485–526.
  • C. Dong, Vertex algebras associated with even lattices, J. Algebra, 161 (1993), 245–265.
  • C. Dong, Twisted modules for vertex algebras associated with even lattices, J. Algebra, 165 (1994), 91–112.
  • C. Dong, C. H. Lam, K. Tanabe, H. Yamada and K. Yokoyama, ${\mathbb Z}_3$ symmetry and $W_3$ algebra in lattice vertex operator algebras, Pacific J. Math., 215 (2004), 245–296.
  • C. Dong and J. Lepowsky, Generalized Vertex Algebras and Relative Vertex Operators, Progr. Math., 112, Birkhäuser, Boston, 1993.
  • C. Dong and J. Lepowsky, The algebraic structure of relative twisted vertex operators, J. Pure Appl. Algebra, 110 (1996), 259–295.
  • C. Dong, H. S. Li and G. Mason, Simple currents and extensions of vertex operator algebras, Comm. Math. Phys., 180 (1996), 671–707.
  • C. Dong, H. S. Li and G. Mason, Twisted representations of vertex operator algebras, Math. Ann., 310 (1998), 571–600.
  • C. Dong, H. S. Li and G. Mason, Modular-invariance of trace functions in orbifold theory and generalized moonshine, Comm. Math. Phys., 214 (2000), 1–56.
  • C. Dong, H. S. Li, G. Mason and S. P. Norton, Associative subalgebras of the Griess algebra and related topics, In: The Monster and Lie Algebras, The Ohio State University, 1996, (ed. J. Ferrar and K. Harada), Ohio State Univ. Math. Res. Inst. Publ., 7, Walter de Gruyter, Berlin, 1998, pp.,27–42.
  • C. Dong and G. Mason, On quantum Galois theory, Duke Math. J., 86 (1997), 305–321.
  • C. Dong, G. Mason and Y. Zhu, Discrete series of the Virasoro algebra and the moonshine module, Proc. Sympos. Pure Math., 56 (1994), 295–316.
  • C. Dong and K. Nagatomo, Representations of vertex operator algebra $V_L^{+}$ for rank one lattice $L$, Comm. Math. Phys., 202 (1999), 169–195.
  • C. Dong and G. Yamskulna, Vertex operator algebras, generalized doubles and dual pairs, Math. Z., 241 (2002), 397–423.
  • I. B. Frenkel, Y.-Z. Huang and J. Lepowsky, On Axiomatic Approaches to Vertex Operator Algebras and Modules, Mem. Amer. Math. Soc., 104, 1993.
  • I. B. Frenkel, J. Lepowsky and A. Meurman, Vertex Operator Algebras and the Monster, Pure Appl. Math., 134, Academic Press, Boston, MA, 1988.
  • I. B. Frenkel and Y. Zhu, Vertex operator algebras associated to representations of affine and Virasoro algebras, Duke Math. J., 66 (1992), 123–168.
  • M. Kitazume, C. H. Lam and H. Yamada, Decomposition of the moonshine vertex operator algebra as Virasoro modules, J. Algebra, 226 (2000), 893–919.
  • M. Kitazume, C. H. Lam and H. Yamada, $3$-state Potts model, moonshine vertex operator algebra, and $3A$ elements of the monster group, Int. Math. Res. Not., 2003 (2003), 1269–1303.
  • M. Kitazume, M. Miyamoto and H. Yamada, Ternary codes and vertex operator algebras, J. Algebra, 223 (2000), 379–395.
  • C. H. Lam and H. Yamada, $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ codes and vertex operator algebras, J. Algebra, 224 (2000), 268–291.
  • C. H. Lam, H. Yamada and H. Yamauchi, McKay's observation and vertex operator algebras generated by two conformal vectors of central charge 1/2, IMRP Int. Math. Res. Pap., 2005 (2005), 117–181.
  • J. Lepowsky, Calculus of twisted vertex operators, Proc. Nat. Acad. Sci. U.S.A., 82 (1985), 8295–8299.
  • J. Lepowsky and H. S. Li, Introduction to Vertex Operator Algebras and Their Representations, Progr. Math., 227, Birkhäuser Boston, Inc., Boston, MA, 2004.
  • H. S. Li, Determining fusion rules by $A(V)$-modules and bimodules, J. Algebra, 212 (1999), 515–556.
  • H. S. Li, The regular representation, Zhu's $A(V)$-theory, and induced modules, J. Algebra, 238 (2001), 159–193.
  • M. Miyamoto, 3-state Potts model and automorphism of vertex operator algebra of order 3, J. Algebra, 239 (2001), 56–76.
  • M. Miyamoto and K. Tanabe, Uniform product of $A_{g,n}(V)$ for an orbifold model $V$ and $G$-twisted Zhu algebra, J. Algebra, 274 (2004), 80–96.
  • K. Tanabe, On intertwining operators and finite automorphism groups of vertex operator algebras, J. Algebra, 287 (2005), 174–198.
  • K. Tanabe and H. Yamada, The fixed point subalgebra of a lattice vertex operator algebra by an automorphism of order three, Pacific J. Math., 230 (2007), 469–510.
  • K. Tanabe and H. Yamada, Representations of a fixed-point subalgebra of a class of lattice vertex operator algebras by an automorphism of order three, European J. Combin., 30 (2009), 725–735.
  • G. Yamskulna, $C_2$-cofiniteness of the vertex operator algebra $V_L^+$ when $L$ is a rank one lattice, Comm. Algebra, 32 (2004), 927–954.
  • Y. Zhu, Modular invariance of characters of vertex operator algebras, J. Amer. Math. Soc., 9 (1996), 237–302.