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Ching Hung Lam

Publications and source records attributed to Ching Hung Lam.

At least 55 records · Page 3Linked to original sources
arXiv2013

Applications of vertex algebra covering procedures to Chevalley groups and modular moonshine

A vertex operator algebra of lattice type ADE has a standard integral form which extends a Chevalley basis for its degree 1 Lie algebra. This integral form may be used to define a vertex algebra over a commutative ring $R$ and to get a Chevalley group over $R$ of the same type, acting as automorphisms of this vertex algebra. We define vertex algebras of types BCFG over a commutative ring and certain reduced VAs, then get analogous results about automorphism groups. In characteristics 2 and 3, there are exceptionally large automorphism groups. A covering algebra idea of Frohardt and Griess for Lie algebras is applied to the vertex algebra situation. We use integral form and covering procedures for vertex algebras to complete the modular moonshine program of Borcherds and Ryba for proving an embedding of the sporadic group $F_3$ in $E_8(3)$.

math.QA↗
arXiv2013

Weyl groups and vertex operator algebras generated by Ising vectors satisfying $(2B,3C)$ condition

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 these vertex operator algebras are determined.

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arXiv2013

On Majorana representations of the group $3^2{:}2$ of 3C-pure type and the corresponding vertex operator algebras

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 $3^2{:}2$ of 3C-pure type.

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arXiv2012

Classification of holomorphic framed vertex operator algebras of central charge 24

This article is a continuation of our work on the classification of holomorphic framed vertex operator algebras of central charge 24. We show that a holomorphic framed VOA of central charge 24 is uniquely determined by the Lie algebra structure of its weight one subspace. As a consequence, we completely classify all holomorphic framed vertex operator algebras of central charge 24 and show that there exist exactly 56 such vertex operator algebras, up to isomorphism.

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arXiv2011

Residue codes of extremal Type II Z_4-codes and the moonshine vertex operator algebra

In this paper, we study the residue codes of extremal Type II Z_4-codes of length 24 and their relations to the famous moonshine vertex operator algebra. The main result is a complete classification of all residue codes of extremal Type II Z_4-codes of length 24. Some corresponding results associated to the moonshine vertex operator algebra are also discussed.

math.QA↗
arXiv2011

Quadratic spaces and holomorphic framed vertex operator algebras of central charge 24

In 1993, Schellekens obtained a list of possible 71 Lie algebras of holomorphic vertex operator algebras with central charge 24. However, not all cases are known to exist. The aim of this article is to construct new holomorphic vertex operator algebras using the theory of framed vertex operator algebras and to determine the Lie algebra structures of their weight one subspaces. In particular, we study holomorphic framed vertex operator algebras associated to subcodes of the triply even codes $\RM(1,4)^3$ and $\RM(1,4)\oplus \EuD(d_{16}^+)$ of length 48. These vertex operator algebras correspond to the holomorphic simple current extensions of the lattice type vertex operator algebras $(V_{\sqrt{2}E_8}^+)^{\otimes 3}$ and $V_{\sqrt{2}E_8}^+\otimes V_{\sqrt{2}D_{16}^+}^+$. We determine such extensions using a quadratic space structure on the set of all irreducible modules $R(W)$ of $W$ when $W= (V_{\sqrt{2}E_8}^+)^{\otimes 3}$ or $V_{\sqrt{2}E_8}^+\otimes V_{\sqrt{2}D_{16}^+}^+$. As our main results, we construct seven new holomorphic vertex operator algebras of central charge 24 in Schellekens' list and obtain a complete list of all Lie algebra structures associated to the weight one subspaces of holomorphic framed vertex operator algebras of central charge 24.

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arXiv2011

"McKay's E7 observation on the Baby Monster" and "McKay's E6 observation on the largest Fischer group"

E7 part: In this paper, we study McKay's E7 observation on the Baby Monster. By investigating so called derived c=7/10 Virasoro vectors, we show that there is a natural correspondence between dihedral subgroups of the Baby Monster and certain subalgebras of the Baby Monster vertex operator algebra which are constructed by the nodes of the affine E7 diagram. This allows us to reinterpret McKay's E7 observation via the theory of vertex operator algebras. 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. E6 part: In this paper, we study McKay's E6-observation on the largest Fischer 3-transposition group Fi24. We investigate a vertex operator algebra VF of central charge 23+1/5 on which the Fischer group Fi24 naturally acts. We show that there is a natural correspondence between dihedral subgroups of Fi24 and certain vertex operator subalgebras constructed by the nodes of the affine E6 diagram by investigating so called derived Virasoro vectors of central charge 6/7. This allows us to reinterpret McKay's E6-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, we identify these involutions with the 3-transpositions of the Fischer group Fi24.

math.QA↗
arXiv2011

A new existence proof of the Monster by VOA theory

We use uniqueness of a VOA (vertex operator algebra) extension of $(V_{EE_8}^+)^3$ to a Moonshine type VOA to give a new existence proof of a finite simple group of Monster type. The proof is relatively direct. Our methods depend on VOA representation theory and are free of many special calculations which traditionally occur in theory of the Monster.

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arXiv2011

Diagonal lattices and rootless $EE_8$ pairs

Let E be an integral lattice. We first discuss some general properties of an SDC lattice, i.e., a sum of two diagonal copies of E in E \bot E. In particular, we show that its group of isometries contains a wreath product. We then specialize this study to the case of E = E_8 and provide a new and fairly natural model for those rootless lattices which are sums of a pair of EE_8-lattices. This family of lattices was classified in [7]. We prove that this set of isometry types is in bijection with the set of conjugacy classes of rootless elements in the isometry group O(E_8), i.e., those h \in O(E_8) such that the sublattice (h - 1)E_8 contains no roots. Finally, our model gives new embeddings of several of these lattices in the Leech lattice.

math.GR↗
arXiv2010

A moonshine path for 5A and associated lattices of ranks 8 and 16

We continue the program, begun in \cite{gl3cpath}, to make a moonshine path between a node of the extended $E_8$-diagram and the Monster simple group. Our goal is to provide a context for observations of McKay, Glauberman and Norton by realizing their theories in a more concrete form. In this article, we treat the 5A-node. Most work in this article is a study of certain lattices of ranks 8, 16 and 24 whose determinants are powers of 5.

math.GR↗
arXiv2010

On the structure of framed vertex operator algebras and their pointwise frame stabilizers

In this paper, we study the structure of a general framed vertex operator algebra. 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 would give a prospect on the classification of framed vertex operator algebras. In addition, the pointwise frame stabilizer of V is studied. We completely determine all automorphisms in this 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 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 the moonshine VOA is isomorphic to the moonshine VOA itself.

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arXiv2009

A moonshine path from $E_8$ to the monster

One would like an explanation of the provocative McKay and Glauberman-Norton observations connecting the extended $E_8$-diagram with pairs of 2A involutions in the Monster sporadic simple group. We propose a down-to-earth model for the 3C-case which exhibits a logic to these connections.

math.GR↗
arXiv2009

Frame Stabilizers for framed vertex operator algebras associated to lattices having 4-frames

In this paper, we study certain Virasoro frames for lattice vertex operator algebras and their $\Z_2$-orbifolds using linear codes over $\Z_4$. We also compute the corresponding frame stabilizer from the view point of binary codes and $\Z_4$-codes. As an application, we determine the frame stabilizers of several Virasoro frames of the vertex operator algebra $V_{E_8}$ and the moonshine vertex operator algebra $V^\natural$.

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arXiv2008

Ising vectors in the vertex operator algebra $V_Λ^+$ associated with the Leech lattice $Λ$

In this article, we study the Ising vectors in the vertex operator algebra $V_Λ^+$ associated with the Leech lattice $Λ$. The main result is a characterization of the Ising vectors in $V_Λ^+$. We show that for any Ising vector $e$ in $V_Λ^+$, there is a sublattice $E\cong \sqrt{2}E_8$ of $Λ$ such that $e\in V_E^+$. Some properties about their corresponding $τ$-involutions in the moonshine vertex operator algebra $V^\natural$ are also discussed. We show that there is no Ising vector of $σ$-type in $V^\natural$. Moreover, we compute the centralizer $C_{\aut V^\natural}(z, τ_e)$ for any Ising vector $e\in V_Λ^+$, where $z$ is a 2B element in $\aut V^\natural$ which fixes $V_Λ^+$. Based on this result, we also obtain an explanation for the 1A case of an observation by Glauberman-Norton (2001), which describes some mysterious relations between the centralizer of $z$ and some 2A elements commuting $z$ in the Monster and the Weyl groups of certain sublattices of the root lattice of type $E_8$ .

math.QA↗
arXiv2008

$EE_8$-lattices and dihedral groups

We classify integral rootless lattices which are sums of pairs of $EE_8$-lattices (lattices isometric to $\sqrt 2$ times the $E_8$-lattice) and which define dihedral groups of orders less than or equal to 12. Most of these may be seen in the Leech lattice. Our classification may help understand Miyamoto involutions on lattice type vertex operator algebras and give a context for the dihedral groups which occur in the Glauberman-Norton moonshine theory.

math.RT↗
arXiv2008

Rootless pairs of $EE_8$-lattices

We describe a classification of pairs $M, N$ of lattices isometric to $EE_8:=\sqrt 2 E_8$ such that the lattice $M + N$ is integral and rootless and such that the dihedral group associated to them has order at most 12. It turns out that most of these pairs may be embedded in the Leech lattice. Complete proofs will appear in another article. This theory of integral lattices has connections to vertex operator algebra theory and moonshine.

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