## Journal of the Mathematical Society of Japan

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

#### 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

https://projecteuclid.org/euclid.jmsj/1382620191

Digital Object Identifier
doi:10.2969/jmsj/06541169

Mathematical Reviews number (MathSciNet)
MR3127822

Zentralblatt MATH identifier
1342.17021

#### 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

#### 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.