Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1606.05961

$\Z_3$-orbifold construction of the Moonshine vertex operator algebra and some maximal $3$-local subgroups of the Monster

Hsian-Yang Chen·Ching Hung Lam·Hiroki Shimakura
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this article, we describe some maximal $3$-local subgroups of the Monster simple group using vertex operator algebras (VOA). We first study the holomorphic vertex operator algebra obtained by applying the orbifold construction to the Leech lattice vertex operator algebra and a lift of a fixed-point free isometry of order $3$ of the Leech lattice. We also consider some of its special subVOAs and study their stabilizer subgroups using the symmetries of the subVOAs. It turns out that these stabilizer subgroups are $3$-local subgroups of its full automorphism group. As one of our main results, we show that its full automorphism group is isomorphic to the Monster simple group by using a $3$-local characterization and that the holomorphic VOA is isomorphic to the Moonshine VOA. This approach allows us to obtain relatively explicit descriptions of two maximal $3$-local subgroups of the shape $3^{1+12}.2.\Suz{:}2$ and $3^8.Ω^-(8,3).2$ in the Monster simple group.

Explore related subjects

math.QA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Hsian-Yang Chen, Ching Hung Lam, Hiroki Shimakura. 2017-02-13. $\Z_3$-orbifold construction of the Moonshine vertex operator algebra and some maximal $3$-local subgroups of the Monster. https://arxiv.org/abs/1606.05961

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Birational Equivalences for Kac--Moody Borel Enveloping Algebras

A Coxeter ordering of the simple roots of a finite-rank Kac--Moody algebra determines a finite family of commuting real-root vectors. We prove that $U^{\geq0}(\mathfrak g)$ is birationally equivalent to $Z\otimes\mathbb A_n$, where $Z$ is the residual Coxeter centralizer, by identifying the Coxeter localization $U^{\geq0}(\mathfrak g)[\mathbf X^{-1}]$ with $Z\otimes\mathbb A_n[\mathbf x^{-1}]$. For symmetrizable Cartan matrices the residual algebra is generated by finite Coxeter windows and is finitely presented. For the generic quantum Borel with torus dual to the root lattice, we prove the analogous birational equivalence.

math.QA↗
arXiv2026

A diagrammatic presentation for every pivotal pointed fusion category

We provide a generators and relations presentation of pivotal pointed fusion categories, $Vec(G,ω,π)$. Unlike the well-known skeletal model, our presentation is strict and allows multiple isomorphic objects. Our main tool is skein theory, which allows us to apply topological tools to understand the relations of morphisms in the category.

math.QA↗
arXiv2026

The Kazhdan-Lusztig category of $\mathfrak{osp}_{1|2n}$ at irrational levels

We prove the Kazhdan-Lusztig correspondence for the Lie superalgebra $\mathfrak{osp}_{1|2n}$ at irrational levels, that is, we show the category $\mathrm{KL}_k^{\rm ev}(\mathfrak{osp}_{1|2n})$ of finite-length even ordinary modules for the affine vertex operator superalgebra of $\mathfrak{osp}_{1|2n}$ at level $k \in \mathbb{C} \setminus \mathbb{Q}$ is braided tensor equivalent to the category of finite-dimensional even weight modules for the quantum group of $\mathfrak{osp}_{1|2n}$ at parameter $q = e^{πi/(2k+2n+1)}$. We also prove that ${\rm KL}_k^{\rm ev}(\mathfrak{osp}_{1|2n})$ is braided tensor equivalent to the category ${\rm KL}_\ell^{\rm ns}(\mathfrak{so}_{2n+1})$ of finite-length ordinary modules with non-spinorial top level for the affine vertex operator algebra of $\mathfrak{so}_{2n+1}$ at level $\ell$ such that $ \frac{1}{\ell+ 2n-1} = \frac{1}{2k+2n+1} + 1 \ \ ({\rm mod}\ 2\mathbb Z).$ Consequently, by gluing vertex operator (super)algebras via tensor categories, we construct a few new families of simple conformal vertex (super)algebras, including the mixed kernel VOAs that were the missing ingredient for proving certain Feigin-Frenkel type dualities in previous work of the first-named author with Linshaw, Nakatsuka, and Sato.

math.QA↗