Ching Hung Lam, Hiroshi Yamauchi
We consider a series of VOAs generated by 3-dimensional Griess algebras. We will show that these VOAs can be characterized by their 3-dimensional Griess algebras and their structures are uniquely determined. As an application, we will determine the groups generated by the Miyamoto involutions associated to Virasoro vectors of our VOAs.
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In this article we study a VOA with two Miyamoto involutions generating $S_3$. In \cite{M3}, Miyamoto showed that a VOA generated by two conformal vectors whose Miyamoto involutions generate an automorphism group isomorphic to $S_3$ 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 $\mathrm{VA}(e,f)$ studied by Miyamoto. We also show that there is an embedding into the moonshine VOA. Using our VOA, we can define the 3A-triality in the Monster.
Let $r \in \BC$ be a complex number, and $d \in \BZ_{\ge 2}$ a positive integer greater than or equal to 2. Ashihara and Miyamoto introduced a vertex operator algebra $\Vam$ of central charge $dr$, whose Griess algebra is isomorphic to the simple Jordan algebra of symmetric matrices of size $d$. In this paper, we prove that the vertex operator algebra $\Vam$ is simple if and only if $r$ is not an integer. Further, in the case that $r$ is an integer (i.e., $\Vam$ is not simple), we give a generator system of the maximal proper ideal $I_{r}$ of the VOA $\Vam$ explicitly.
We introduce the Z_2-extended Griess algebra of a vertex operator superalgebra with an involution and derive the Matsuo-Norton trace formulae for the extended Griess algebra based on conformal design structure. We illustrate an application of our formulae by reformulating the one-to-one correspondence between 2A-elements of the Baby-monster simple group and N=1 c=7/10 Virasoro subalgebras inside the Baby-monster vertex operator superalgebra.
In this article, we study Griess algebras and vertex operator subalgebras generated by Ising vectors in a moonshine type VOA such that the subgroup generated by the corresponding Miyamoto involutions has the shape $3^2{:}2$ and any two Ising vectors generate a 3C subVOA $U_{3C}$. We show that such a Griess algebra is uniquely determined, up to isomorphisms. The structure of the corresponding vertex operator algebra is also discussed. In addition, we give a construction of such a VOA inside the lattice VOA $V_{E_8^3}$, which gives an explicit example for Majorana representations of the group $3…
We determined the inner products of two conformal vectors with central charge 1/2 whose τ-involutions generates S_3 if none of τ-involutions are trivial. We also see that a subVA generated by such conformal vectors is a VOA with central charge 1/2+20/21 or $4/5+6/7 and has a Griess algebra of dimension three or four, respectively.
In this paper, we study the subalgebra generated by two Ising vectors in the Griess algebra of a vertex operator algebra. We show that the structure of it is uniquely determined by some inner products of Ising vectors. We prove that the order of the product of two $τ$-involutions is less than or equal to 6 and we determine the inner product of two Ising vectors.
In this article, we construct explicitly certain moonshine type vertex operator algebras generated by a set of Ising vectors $I$ such that (1) for any $e\neq f\in I$, the subVOA $\mathrm{VOA}(e,f)$ generated by $e$ and $f$ is isomorphic to either $U_{2B}$ or $U_{3C}$; and (2)the subgroup generated by the corresponding Miyamoto involutions $\{τ_e|\,e\in I\}$ is isomorphic to the Weyl group of a root system of type $A_n$, $D_n$, $E_6$, $E_7$ or $E_8$. The structures of the corresponding vertex operator algebras and their Griess algebras are also studied. In particular, the central charge of thes…
Formulae expressing the trace of the composition of several (up to five) adjoint actions of elements of the Griess algebra of a vertex operator algebra are derived under certain assumptions on the action of the automorphism group. They coincide, when applied to the moonshine module $V^\natural$, with the trace formulae obtained in a different way by S.Norton, and the spectrum of idempotents related to 2A, 2B, 3A and 4A element of the Monster is determined by the representation theory of Virasoro algebra at $c=1/2$, $W_3$ algebra at $c=4/5$ or $W_4$ algebra at $c=1$. The generalization to the t…
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 E_8 lattice obtained by removing one node from the extended E_8 diagram at each time. We then construct a certain coset (or commutant) subalgebra U associated with L in the lattice VOA V_{\sqrt{2}E_8}. 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. The Griess alge…
The Order of the Monster Finite Simple Group
本文通过基于Griess代数轴幂等元计数的计算方法，借助mmgroup软件完成了首个自包含的魔群阶计算，还证明了魔群的自同构群性质、对合共轭类数量并确定了小魔群的阶。
This paper is a continuation of our paper math.QA/0403010 at which several coset subalgebras of the lattice VOA $V_{\sqrt{2}E_8}$ were constructed and the relationship between such algebras with the famous McKay observation on the extended E_8 diagram and the Monster simple group were discussed. In this article, we shall provide the technical details. We completely determine the structure of the coset subalgebras constructed and show that they are all generated by two conformal vectors of central charge 1/2. We also study the representation theory of these coset subalgebras and show that the p…
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