參考宮本教授的方法, 我們利用二進制的密碼, L(7/10,0)和L(7/10,0)所有不同構的不可約模來構照密碼超級頂點算子代數. 主要是分類這樣的頂點算子代數表現理論與其對應的不同構模. 另外我們還證明了怎樣的頂點算子代數會與我們所架構的密碼頂點代數同構.

Following the approach of [25], we construct a class of vertex operator (super) algebra WD using some binary code D and irreducible modules of the unitary Virasoro vertex operator algebra L( 7/10 ; 0). We also introduce a notion of induced modules and show that all irreducible WD-modules are induced modules. In addition, we study the fusion rules and show that certain irreducible WD modules are simple current modules.

[1] T. Abe, G. Buhl and C. Dong, Rationality, regularity, and C2-cofiniteness, to appear
in Trans. Amer. Math. Soc. 356 (2004), 3391-3402
[2] A. Belavin, A.M. Polyakov, and A.B Zamolodchikov. Infinite conformal symmetries
in two-dimensional quantum field theory. Nucl. Phys. B, 241:333-380,1984.
[3] R. Borcherds. Vertex algebrs, Kac-Moody algebras, and the Monster. Proceeding of
National Academy of Science USA, 83:3068-3071, 1986.
[4] J. H. Conway, A simple construction for the Fischer-Griess monster group, Invent.
Math. (79) 1985. 513-540[5] C. Dong and J. Lepowsky, Generalized vertex algebras and relative vertex operators,
Progrress in Mah. Vol. 112, Birkhauser, Boston 1993.
[6] C. Dong, Vertex algebras associated with even lattices, J. Algebra 161 (1993), 245-265.
[7] C. Dong, G. Mason and Y. Zhu, Discrete series of the Virasoro algebra and the moonshine
module. Proc. Symp. Pure. Math. American Math Soc. II (1994), 295-316.
[8] C. Dong and G. Mason, On quantum Galois theory, Duke Math. J. 86 (1997), 305-321.
[9] C. Dong, H. Li, G. Mason and S.P. Norton, Associative subalgebras of Griess algebra and related topics, Proc. of the Conference on the Monster and Lie algebra at the Ohio
State University, May 1996, ed. by J. Ferrar and K. Harada, Walter de Gruyter, Berlin
- New York, 1998.
[10] C. Dong, R.L. Griess and C.H. Lam, Uniqueness results for the moonshine vertex
operator algebra, American Journal of Mathematics, 129(2007), no. 2, 583-609.
[11] C. Dong, R.L. Griess and G. Hohn, Framed vertex operator algebras, codes and the
moonshine module, Comm. Math. Phys. 193 (1998), 407-448.
[12] C. Dong and K. Nagatomo: Representations of vertex operator algebra V +
L for rank
one lattice L. Comm. Math. Phys. 202 (1999) 162-195. (math.QA/9807168)
[13] I.B. Frenkel, J. Lepowsky, and A.Meurman, A natural representation of the Fisher-
Griess Monster with the modular function J as character. Proceedings of national
Academy of Science USA, 81:3256-3260, 1984.
[14] I.B. Frenkel, J. Lepowsky, and A.Meurman, Vertex Operator Algebras and the Monster.
NewYork: Academic Press, 1988
[15] I.B. Frenkel and Y. Zhu, Vertex operator algebras associated to representation of
affine and Virasoro algebras. Duke Math. J. 66 (1992), 123-168.
[16] I.B. Frenkel, Y. Zhu Huang and J. Lepowsky, On axiomatic approaches to vertex
operator algebras and the modules, Memorirs Amer Math. Soc. 104, 1993.
[17] D. Friedan, Z. Qiu, and S. Shenker, Conformal invariace, unitarity two-dimensional
critical exponents, Math. Sci. Res. Inst. Pual 3 (1984), 419-449.
[18] P.Goddard, A. Kent, and D. OLive, Unitary representations of the Virasoro algebra
and sup-Virasoro algebras, Comm. Math. Phys., 103(1986),105-119.
[19] Gorenstein, D.:Finite group New YorK-London:Harper & Row, Publishers, 1968.
[20] R.L. Griess, Jr.: The vertex operator algebra related to E8 with automorphism group
O+(10; 2), in The Monster and Lie algebras (Columbus, Ohio, 1996),43-58 Ohio State
Univ. Mat[21] R.L. Griess, Jr.: GNAVOA, I. Studies in groups, nonassociative algebras and vertex
operator algbras. Vertex Operator Algebras in Mathematics and Physics (Toronto,
2000), pp. 71V88, Fields Ins. Commun., 39, Amer. Math. Soc., Providence, 2003.
[22] K. Harada and M.-L. Lang,modular forms associated with the Monster module in The
Monster and Lie algebras (Columbus, Ohio, 1996), Ohio State Univ. Math. Res. Inst.
Publ. 7, de Gruyter, Berlin, 1998, 59-83.
[23] G. Höhn, Conformal designs based on vertex operator algebras, Adv.Math. 217 (2008),
2301-2335.
[24] G. Höhn, Ching Hung Lam, Hiroshi Yamauchi, McKay’s observations on the Baby
Monster, Inter. Math. Res. Not., 2012(2012), 166-212.
[25] G. Höhn, Ching Hung Lam, Hiroshi Yamauchi, McKay’s E7 and E6 observations on
the Babymonster and the largest Fischer group, Inter. Math Res. notice. 2011.
[26] Y.-Z. Huang and J. Lopowsky, A theory of the tensor products for module categories
for a vertex operator algebra, III, J. Pure Appl. Algebra. 100 (1995), 141-171.
[27] Y.-Z. Huang, Differential equations and intertwining operators, Comm. Contemp.
Math. 7 (2005), 275-299.
[28] G. Kac and A. K. Raina, Bombay lectures on highest weight representations of infinite
dimensional Lie algebras. World Scientific Publishing Co Pte Ltd, 1988
[29] M. Kitazume, M. Miyamoto and H. Yamada, Ternary codes and vertex operator
algebras, J. Algebra, 223 (2000), 379-395.
[30] C.H Lam and H. Yamada, Z2 Z2-codes and vertex operator algebras, J. Algebra
224 (2000), 268-291.
[31] C.H Lam, Some twisted modules for framed vertex operator algebras, J. Algebra, 231
(2000), 331-341.h. Res. Inst. Publ. 7, de Gruyter, Berlin, 1998,
[32] C.H Lam, N. Lam and H. Yamauchi, Extension of unitary Virasoro vertex operator
algebra by a simple module. Internat. Math. Res. Notices 11 (2003), 577-611.
[33] C.H Lam and H.Yamauchi, On the structure of the framed vertex operator algebras
and their pointwise frame stabilizers, Comm. math. Physics 277(2008), no. 1, 237-285.
[34] 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. Internat. Math. Res.
Papers 3 (2005), 117–181.
[35] H. Li: Symmetric invariant bilinear forms on vertex operator algebras, J. Pure Appl.
Algebra 96 (1994) 279-297.
[36] H. Li, Local systems of vertex operators, vertex superalgebras and modules, J. Pure
Appl. Algebra, 109(1996), 143-195.
[37] H. Li, Local systems of twisted vertex operators, vertex operator superalgebras and
twisted modules, in Moonshine, the Monster, and Related Topics (South Hadley, MA,
1994), Editors C. Dong and G. Mason, Contemporary Mathematics, Vol. 193, American
Mathematical Society, Providence, RI, 1996, 203-236
[38] A. Matsuo, Norton’s trace formula for the Griess algebra of a vertex operator algebra
with large symmetry, Commun.Math.Phys. 224 (2001), 565-591.
[39] M. Miyamoto, Griess algebras and conformal vectors in vertex operator algebras, J.
Algebra 179(1996), 523-548.
[40] M. Miyamoto, Representation theory of code vertex operator algebras, J. Algebra 201
(1998), 115-150.
[41] M. Miyamoto, Modular invariance of vertex operator algebras satysfying C2-
cofiniteness, to appear in Duke Math. J.
[42] G. Moore and N. Seiberg, Classical and quantum conformal field theory. Comm.
Math. Phys., 123:177-254,1989.
[43] M.P. Tuite, Exceptional vertex operator algebras and the Virasoro algebra, Contemp.
Math. 497 213-225 (2009).
[44] M.P. Tuite, The Virasoro algebra and some exceptional Lie and finite groups, SIGMA
3 (2007), 008.
[45] W. Wang, Rationality of Virasoro vertex operator algebras. Internat. Math. Res.
Notices 71 (1993), 197-211.
[46] H. Yamauchi, Module categories of simple current extensions of vertex operator algebras,
J. Pure Appl. Algebra 189 (2004), 315-328.
[47] H. Yamauchi, A theory of simple current extensions of vertex operator algebras
and applications to the moonshine vertex operator algebra, Ph.D. thesis, University of
Tsukuba, 2004.; available on the author’s web site: http://www.ms.u-tokyo.ac.jp/ yamauchi