山内 博

<p>In this paper, we present a general construction of 3-transposition groups as automorphism groups of vertex operator algebras. Applying to the moonshine vertex operator algebra, we establish the Conway-Miyamoto correspondences between Fischer 3-transposition groups and and and Virasoro vectors of subalgebras of the moonshine vertex operator algebra.</p>

We study simple current extensions of tensor products of two vertex operator<br /> algebras satisfying certain conditions. We establish the relationship between<br /> the fusion rule for the simple current extension and the fusion rule for a<br /> tensor factor. In some special case, we construct a chain of simple current<br /> extensions. As an example, we obtain a chain of simple current extensions<br /> starting from the simple affine vertex operator algebra associated with<br /> $\widehat{sl}_2$ at level a positive integer $k$. The irreducible modules are<br /> classified and the fusion rules are determined for those simple current<br /> extensions.

We prove the uniqueness of the simple vertex operator algebra of OZ-type<br /> generated by Ising vectors of $\sigma$-type. We also prove that the simplicity<br /> can be omitted if the Griess algebra is isomorphic to the Matsuo algebra<br /> associated with the root system of type $A_n$.

In this paper, we present a general construction of 3-transposition groups as<br /> automorphism groups of vertex operator algebras. Applying to the moonshine<br /> vertex operator algebra, we establish the Conway-Miyamoto correspondences<br /> between Fischer 3-transposition groups $\mathrm{Fi}_{23}$ and<br /> $\mathrm{Fi}_{22}$ and $c=25/28$ and $c=11/12$ Virasoro vectors of subalgebras<br /> of the moonshine vertex operator algebra.

We consider a series of VOAs generated by 3-dimensional Griess algebras. We<br /> will show that these VOAs can be characterized by their 3-dimensional Griess<br /> algebras and their structures are uniquely determined. As an application, we<br /> will determine the groups generated by the Miyamoto involutions associated to<br /> Virasoro vectors of our VOAs.

We construct a vertex operator algebra associated with a Z/kZ-code of length n for an integer k ≥ 2.We realize it inside a lattice vertex operator algebra as the commutant of a certain subalgebra. The vertex operator algebra is isomorphic to a known one in the cases k = 2,3.

In this paper, we show that $\sigma$-involutions associated to extendable<br /> c=4/5 Virasoro vectors generate a 3-transposition group in the automorphism<br /> group of a vertex operator algebra (VOA). Several explicit examples related to<br /> lattice VOA are also discussed in details. In particular, we show that the<br /> automorphism group of the VOA $V_{K_{12 } }^{\hat{\nu } }$ associated to the<br /> Coxeter Todd lattice $K_{12}$ contains a subgroup isomorphic to<br /> ${}^+\Omega^{-}(8,3)$.

We introduce the Z_2-extended Griess algebra of a vertex operator<br /> superalgebra with an involution and derive the Matsuo-Norton trace formulae for<br /> the extended Griess algebra based on conformal design structure. We illustrate<br /> an application of our formulae by reformulating the one-to-one correspondence<br /> between 2A-elements of the Baby-monster simple group and N=1 c=7/10 Virasoro<br /> subalgebras inside the Baby-monster vertex operator superalgebra.

In this paper, we study McKay's E-6-observation on the largest Fischer 3-transposition group Fi(24). We investigate a vertex operator algebra VF(sic) of central charge 2315 on which the Fischer group Fi(24) naturally acts. We show that there is a natural correspondence between dihedral subgroups of Fi(24) and certain vertex operator subalgebras constructed by the nodes of the affine E-6 diagram by investigating so-called derived Virasoro vectors of central charge 6/7. This allows us to reinterpret McKay's E-6-observation via the theory of vertex operator algebras. It is also shown that the product of two non-commuting Miyamoto involutions of sigma-type associated to derived c = 6/7 Virasoro vectors is an element of order 3, under certain general hypotheses on the vertex operator algebra. For the case of VF(sic), we identify these involutions with the 3-transpositions of the Fischer group Fi(24).

For a class of vertex operator algebras including the Moonshine module, we will show that the product of two Miyamoto involutions associated to derived c=7/10 Virasoro vectors in certain commutant vertex operator algebras is an element of order at most 4. For the case of the Moonshine module, we obtain the Baby monster vertex operator algebra as the commutant and we can identify the group generated by these Miyamoto involutions with the Baby Monster and recover the {3,4}-transposition property of the Baby Monster in terms of vertex operator algebras.

In this article we review the proof of a weak version of the FLM conjecture in [LY2]. Namely, we prove that a holomorphic c = 24 framed vertex operator algebra is isomorphic to the moonshine VOA if and only if its weight one subspace is trivial. A brief review on the theory of framed VOA is also given.

In this paper, we study the structure of a general framed vertex operator algebra (VOA). We show that the structure codes (C, D) of a framed VOA V satisfy certain duality conditions. As a consequence, we prove that every framed VOA is a simple current extension of the associated binary code VOA V-C. This result suggests the feasibility of classifying framed vertex operator algebras, at least if the central charge is small. In addition, the pointwise frame stabilizer of V is studied. We completely determine all automorphisms in the pointwise stabilizer, which are of order 1, 2 or 4. The 4A-twisted sector and the 4A-twisted orbifold theory of the famous moonshine VOA V are also constructed explicitly. We verify that the top module of this twisted sector is of dimension 1 and of weight 3/4 and the VOA obtained by 4A-twisted orbifold construction of V is isomorphic to V itself.

In this article we study and obtain a classification of Ising vectors in vertex operator algebras (VOAs) associated to binary codes and root 2 times root lattices, where an Ising vector is a conformal vector with central charge 1/2 generating a simple Virasoro sub-VOA. Then we apply our results to study certain commutant subalgebras related to root systems. We completely classify all Ising vectors in such commutant subalgebras and determine their full automorphism groups.

We study McKay's observation on the Monster simple group, which relates the 2A-involutions of the Monster simple group to the extended E-8 diagram, using the theory of vertex operator algebras (VOAs). We first consider the sublattices L of the E8 lattice obtained by removing one node from the extended E8 diagram at each time. We then construct a certain coset (or commutant) subalgebra U associated with L in the lattice VOA V (root 2E8). There are two natural conformal vectors of central charge 1/2 in U such that their inner product is exactly the value predicted by Conway (1985). The Griess algebra of U coincides with the algebra described in his Table 3. There is a canonical automorphism of U of order vertical bar E-8/L vertical bar. Such an automorphism can be extended to the Leech lattice VOA V(sic), and it is in fact a product of two Miyamoto involutions. In the sequel (2005) to this article, the properties of U will be discussed in detail. It is expected that if U is actually contained in the Moonshine VOA V, the product of two Miyamoto involutions is in the desired conjugacy class of the Monster simple group.

In this article, we show that a framed vertex operator algebra (VOA) V satisfying the conditions: (i) V is holomorphic (i.e., V is the only irreducible V-module); (ii) V is of rank 24; and (iii) V-1 = 0; is isomorphic to the moonshine VOA V-(sic) constructed by Frenkel-Lepowsky-Meurman [12].

In this article we give a new proof of the determination of the full automorphism group of the baby-monster vertex operator superalgebra based on a theory of simple current extensions. As a corollary, we also prove that the Z(2)-orbifold construction with respect to a 2A-involution of the Monster applied to the moonshine vertex operator algebra V-b yields V-b itself again. (C) 2004 Elsevier Inc. All rights reserved.

This paper is a continuation of our paper math.QA/0403010 at which several<br /> coset subalgebras of the lattice VOA $V_{\sqrt{2}E_8}$ were constructed and the<br /> relationship between such algebras with the famous McKay observation on the<br /> extended E_8 diagram and the Monster simple group were discussed. In this<br /> article, we shall provide the technical details. We completely determine the<br /> structure of the coset subalgebras constructed and show that they are all<br /> generated by two conformal vectors of central charge 1/2. We also study the<br /> representation theory of these coset subalgebras and show that the product of<br /> two Miyamoto involutions is in the desired conjugacy class of the Monster<br /> simple group if a coset subalgebra U is actually contained in the Moonshine<br /> VOA. The existence of U inside the Moonshine VOA for the cases of 1A, 2A, 2B<br /> and 4A is also established. Moreover, the cases for 3A, 5A and 3C are<br /> discussed.

We study module categories of simple current extensions of rational C-cofinite vertex operator algebras of CFT-type and prove that they are semnisimple. We also develop a method of induced modules for the simple current extensions. (C) 2003 Elsevier B.V. All rights reserved.

In this article, using an idea of the physics superselection principal, we study a modularity on vertex operator algebras arising from semisimple primary vectors. We generalizes the theta functions on vertex operator algebras and prove that the internal automorphisms do not change the genus one twisted conformal blocks.

In this article we study a VOA with two Miyamoto involutions generating S-3. In [math.GR/ 0112031], Miyamoto showed that a VOA generated by two conformal vectors whose Miyamoto involutions generate an automorphism group isomorphic to S3 is isomorphic to one of the four candidates he listed. We construct one of them and prove that our VOA is actually the same as VA(e, f) studied by Miyarnoto. We also show that there is an embedding into the moonshine VOA. Using our VOA, we can define the 3A-triality of the Monster. (C) 2003 Elsevier Inc. All rights reserved.

We study intertwining operators among the twisted modules for rational VOAs and show a new modular invariance of the space spanned by the trace functions associated to intertwining operators. Our result generalizes the results obtained by Zhu [J. Amer. Math. Soc. 9 (1996) 237-302], Dong, Li, and Mason [Commun. Math. Phys. 214 (2000) 1-56], and Miyamoto [q-alg/0010180]. (C) 2003 Elsevier Inc. All rights reserved.