Geoffrey Mason, Michael Tuite, Gaywalee Yamskulna
We develop criteria to decide if an $N=2$ or $N=4$ super conformal algebra is a subalgebra of a super vertex operator algebra in general, and of a super lattice theory in particular. We give some specific examples.
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A notion of intermediate vertex subalgebras of lattice vertex operator algebras is introduced, as a generalization of the notion of principal subspaces. Bases and the graded dimensions of such subalgebras are given.As an application, it is shown that the characters of some modules of an intermediate vertex subalgebra between $E_7$ and $E_8$ lattice vertex operator algebras satisfy some modular differential equations.This result is an analogue of the result concerning the "hole" of the Deligne dimension formulas and the intermediate Lie algebra between the simple Lie algebras $E_7$ and $E_8$.
Classification of irreducible modules of certain subalgebras of free boson vertex algebra
设M（1）是单个自由玻色子的顶点代数。对M（1）的某些顶点子代数的由两个生成元生成的不可约模进行了分类。这些子代数对应于具有中心电荷$1- 6 \frac{(p - 1) ^{2}}{p}$的W（2，2 p-1）-代数，其中p是正整数，$p \ge 2$ .我们还确定了相关的朱代数。
We study the fixed point subalgebra of a certain class of lattice vertex operator algebras by an automorphism of order 3, which is a lift of a fixed-point-free isometry of the underlying lattice. We classify the irreducible modules for the subalgebra. Moreover, the rationality and the $C_2$-cofiniteness of the subalgebra are established. Our result contains the case of the vertex operator algebra associated with the Leech lattice.
The fixed point subalgebra of a lattice vertex operator algebra by an automorphism of order three
研究了由根格$A_2$的3阶等距诱导的自同构的不动点构成的格顶点算子代数$V_{\sqrt{2}A_2}$的子代数.我们对子代数的单模进行了分类。并建立了合理性和$C_2$ -有限性.
Let $V_{L}$ be the vertex algebra associated to a non-degenerate even lattice $L$, $θ$ the automorphism of $V_{L}$ induced from the $-1$-isometry of $L$, and $V_{L}^{+}$ the fixed point subalgebra of $V_{L}$ under the action of $θ$. We show that every weak $V_{L}^{+}$-module is completely reducible.
Let $A$ be a connected commutative $\C$-algebra with derivation $D$, $G$ a finite linear automorphism group of $A$ which preserves $D$, and $R=A^G$ the fixed point subalgebra of $A$ under the action of $G$. We show that if $A$ is generated by a single element as an $R$-algebra and is a Galois extension over $R$ in the sense of M. Auslander and O. Goldman, then every finite-dimensional vertex algebra $R$-module has a structure of twisted vertex algebra $A$-module.
Conformal subalgebras of lattice vertex algebras
本文在一定的限制条件下，对格顶点超代数$V_Λ$中对应于整数格$Λ$的所有齐次共形子代数$\goth L$进行了分类.我们要求$\goth L$是由几乎有限根系$Δ\subset Λ$分次的，并且$\goth L$在Heisenberg共形代数$\goth H\subset V_Λ$的作用下是稳定的.我们还描述了这些子代数的根系。这种分类的关键成分是一个无穷型共形代数$\goth K$，它是由一个Jordan共形三元系$\goth J$通过Tits-Kantor-Koeher构造而得的。在费米子顶点超代数$V_\Z$中实现了$\goth K$的中心扩张$\^{\goth K}$，从而扩展了玻色子-费米子对应.
Let $V_{L}$ be the vertex algebra associated to a non-degenerate even lattice $L$, $θ$ the automorphism of $V_{L}$ induced from the $-1$ symmetry of $L$, and $V_{L}^{+}$ the fixed point subalgebra of $V_{L}$ under the action of $θ$. We classify the irreducible weak $V_{L}^{+}$-modules and show that any irreducible weak $V_{L}^{+}$-module is isomorphic to a weak submodule of some irreducible weak $V_{L}$-module or to a submodule of some irreducible $θ$-twisted $V_{L}$-module.
Let ${\mathscr M}(p)$ $(p=2,3,\ldots)$ be the singlet vertex operator algebra and $ω$ its conformal vector. We classify the simple weak ${\mathscr M}(p)$-modules with a non-zero element $u$ such that for some integer $s\geq 2$, $ω_i u\in{\mathbb C} u$ ($i=\lfloor s/2\rfloor+1,\lfloor s/2\rfloor+2,\ldots,s-1$), $ω_{s} u\in{\mathbb C}^{\times} u$, and $ω_i u=0$ for all $i>s$.
Yangian deformations of $\mathcal{S}$-commutative quantum vertex algebras and Bethe subalgebras
本文构造了基于标准化Yang R矩阵的新类量子顶点代数，探讨其表示论及交换性与Bethe子代数的关系。
Let $V_{L}$ be the vertex algebra associated to a non-degenerate even lattice $L$, $θ$ the automorphism of $V_{L}$ induced from the $-1$ symmetry of $L$, and $V_{L}^{+}$ the fixed point subalgebra of $V_{L}$ under the action of $θ$. In this series of papers, we classify the irreducible weak $V_{L}^{+}$-modules and show that any irreducible weak $V_{L}^{+}$-module is isomorphic to a weak submodule of some irreducible weak $V_{L}$-module or to a submodule of some irreducible $θ$-twisted $V_{L}$-module. Let $M(1)^{+}$ be the fixed point subalgebra of the Heisenberg vertex operator algebra $M(1)$ …
Let $V_{L}$ be the vertex algebra associated to a non-degenerate even lattice $L$, $θ$ the automorphism of $V_{L}$ induced from the $-1$-isometry of $L$, and $V_{L}^{+}$ the fixed point subalgebra of $V_{L}$ under the action of $θ$. In this series of papers, we classify the irreducible weak $V_{L}^{+}$-modules and show that any irreducible weak $V_{L}^{+}$-module is isomorphic to a weak submodule of some irreducible weak $V_{L}$-module or to a submodule of some irreducible $θ$-twisted $V_{L}$-module. In this paper (Part 1), we show that when the rank of $L$ is $1$, every non-zero weak $V_{L}^{…
For a vertex operator algebra $V$ with conformal vector $ω$, we consider a class of vertex operator subalgebras and their conformal vectors. They are called semi-conformal vertex operator subalgebras and semi-conformal vectors of $(V,ω)$, respectively, and were used to study duality theory of vertex operator algebras via coset constructions. Using these objects attached to $(V,ω)$, we shall understand the structure of the vertex operator algebra $(V,ω)$. At first, we define the set $\on{Sc}(V,ω)$ of semi-conformal vectors of $V$, then we prove that $\on{Sc}(V,ω)$ is an affine algebraic variety…
Let $M(1)$ be the vertex operator algebra with the Virasoro element $ω$ associated to the Heisenberg algebra of rank $1$ and let $M(1)^{+}$ be the subalgebra of $M(1)$ consisting of the fixed points of an automorphism of $M(1)$ of order $2$. We classify the simple weak $M(1)^{+}$-modules with a non-zero element $w$ such that for some integer $s\geq 2$, $ω_i w\in{\mathbb C}w$ ($i=\lfloor s/2\rfloor+1,\lfloor s/2\rfloor+2,\ldots,s-1$), $ω_{s}w\in{\mathbb C}^{\times}w$, and $ω_i w=0$ for all $i>s$. The result says that any such simple weak $M(1)^{+}$-module is isomorphic to some simple weak $M(1)…
We prove a sharpened version of a conjecture of Dong-Mason about lattice subalgebras of a strongly regular vertex operator algebra $V$, and give some applications. These include the existence of a canonical conformal subVOA $W\otimes G\otimes Z \subseteq V$, and a generalization of the theory of minimal models.
Let $V$ be a vertex operator subalgebra of $U$. Assume that $U$, $V$, and its commutant $V^c$ in $U$ are CFT-type, self-dual, and regular VOAs. Assume also that the double commutant $V^{cc}$ equals $V$. We prove that any intertwining operator of $V$ is a compression of intertwining operators of $U$.
We provide a complete classification of unitary subalgebras of even rank-one lattice vertex operator algebras. As a consequence of the correspondence between vertex operator algebras and conformal nets, we also obtain a complete classification of conformal subnets of even rank-one lattice conformal nets.
Filtration and gradation of quantum affine vertex algebras
该文对与双杨代数等相关的量子仿射顶点代数进行统一构造，展示其含带递增过滤的稠密量子顶点子代数，且证明相关分次代数同构于特定的\(\mathbb Z\)-分次稠密量子顶点子代数。
Given a finite-dimensional complex Lie algebra g equipped with a nondegenerate, symmetric, invariant bilinear form B, let V_k(g,B) denote the universal affine vertex algebra associated to g and B at level k. For any reductive group G of automorphisms of V_k(g,B), we show that the invariant subalgebra V_k(g,B)^G is strongly finitely generated for generic values of k. This implies the existence of a new family of deformable W-algebras W(g,B,G)_k which exist for all but finitely many values of k.
Borel-type subalgebras of the lattice vertex operator algebra
该研究提出格 VOA 的圆锥型、Borel 型与抛物型子 VOA，并对最低秩非平凡 Borel 型例子显式给出其 Zhu 代数的生成元—关系表示。
Motivated by \cite{am1}, for every finite subgroup $Γ\subset PSL(2,\mathbb{C})$ we investigate the fixed point subalgebra $\triplet^Γ$ of the triplet vertex $\mathcal {W}(p)$, of central charge $1-\frac{6(p-1)^{2}}{p}$, $p\geq2$. This part deals with the $A$-series in the ADE classification of finite subgroups of $PSL(2,\mathbb{C})$. First, we prove the $C_2$-cofiniteness of the $A_m$-fixed subalgebra $\triplet^{A_m}$. Then we construct a family of $\am$-modules, which are expected to form a complete set of irreps. As a strong support to our conjecture, we prove modular invariance of (generali…
Let $L_{D_{\ell}}(-\ell +{3/2},0)$ (resp. $L_{B_{\ell}}(-\ell +{3/2},0)$) be the simple vertex operator algebra associated to affine Lie algebra of type $D_{\ell}^{(1)}$ (resp. $B_{\ell}^{(1)}$) with the lowest admissible half-integer level $-\ell + {3/2}$. We show that $L_{D_{\ell}}(-\ell +{3/2},0)$ is a vertex subalgebra of $L_{B_{\ell}}(-\ell +{3/2},0)$ with the same conformal vector. For $\ell =4$, $L_{D_{4}}(-{5/2},0)$ is a vertex subalgebra of three copies of $L_{B_{4}}(-{5/2},0)$ contained in $L_{F_{4}}(-{5/2},0)$, and all five of these vertex operator algebras have the same conformal v…
Integrability of C_2-cofinite vertex operator algebras
证明了满足某些有限性条件（C2-上有限和CFT-型）的顶点算子代数V的可积性定理：对于正整数k，由权为1的子空间V1的单李子代数{ \frak g}生成的顶点算子子代数同构于不可约的最高权\hat{\frak g} -模L（k，0），且V是可积的\hat{\frak g} -模.还考虑了{ \frak g}被交换李子代数所取代的情形，并讨论了可积性的几个结果.
Classification of irreducible modules for the vertex operator algebra V_L^+: General case
对任意正定偶格L上的顶点算子代数V_L的不动点顶点算子子代数V_L^+在L的-1等距提升的自同构下的不可约模进行了分类.
Representations of vertex operator algebra V_L^+ for rank one lattice L
在秩为1的正定偶格上的顶点算子代数V_L的不动点顶点算子子代数V_L^+在L的-1等距提升的自同构下的不可约模的分类.
On the commutant of the principal subalgebra in the $A_1$ lattice vertex algebra
本文研究 $A_1$ 格顶点算子代数 $V_{A_1}$ 的主子代数 $W$ 的交换子 $C$，给出其无穷多个元素的显式极小强生成元集，证明 $C$ 非有限生成，并确定其 Zhu 泊松代数结构及伴随簇，同时验证 $W$ 与 $C$ 构成对偶对。
Vertex operator superalgebras are studied and various results on rational Vertex operator superalgebras are obtained. In particular, the vertex operator super subalgebras generated by the weight 1/2 and weight 1 subspaces are determined. It is also established that if the even part $V_{\bar 0}$ of a vertex operator superalgebra $V$ is rational, so is $V.$
Let L be a Lie pseudoalgebra, a in L. We show that, if a generates a (finite) solvable subalgebra S=<a>, then one may find a lifting a' in S of [a] in S/S' such that <a'> is nilpotent. We then apply this result towards vertex algebras: we show that every finite vertex algebra V admits a decomposition into a semi-direct product V = U + N, where U is a subalgebra of V whose underlying Lie conformal algebra U^lie is a nilpotent self-normalizing subalgebra of V^lie, and N is a canonically determined ideal contained in the nilradical Nil V.
We are continuing our study of ADE-orbifold subalgebras of the triplet vertex algebra W(p). This part deals with the dihedral series. First, subject to a certain constant term identity, we classify all irreducible modules for the vertex algebra $\bar{M(1)} ^+$, the $\Z_2$--orbifold of the singlet vertex algebra $\bar{M(1)}$. Then we classify irreducible modules and determine Zhu's and $C_2$--algebra for the vertex algebra $\triplet ^{D_2}$. A general method for construction of twisted $\triplet$--modules is also introduced. We also discuss classification of twisted $\bar{M(1)}$--modules includ…
Let $V\subseteq A$ be a conformal inclusion of vertex operator algebras and let $\mathcal{C}$ be a category of grading-restricted generalized $V$-modules that admits the vertex algebraic braided tensor category structure of Huang-Lepowsky-Zhang. We give conditions under which $\mathcal{C}$ inherits semisimplicity from the category of grading-restricted generalized $A$-modules in $\mathcal{C}$, and vice versa. The most important condition is that $A$ be a rigid $V$-module in $\mathcal{C}$ with non-zero categorical dimension, that is, we assume the index of $V$ as a subalgebra of $A$ is finite a…
McKay's observation and vertex operator algebras generated by two conformal vectors of central charge 1/2
本文是我们的论文math.QA/0403010的继续，在此基础上构造了格VOA $V_{\sqrt{2}E_8}$的几个陪集子代数，并讨论了这类代数与著名的McKay在扩展E_8图上的观察和Monster单群之间的关系.在本文中，我们将提供技术细节。我们完全确定了所构造的陪集子代数的结构，并证明了它们都是由两个中心荷为1/2的共形向量生成的。我们还研究了这些陪集子代数的表示论，并表明，产品的两个Miyamoto对合是在所需的共轭类的怪物单群，如果陪集子代数U实际上是包含在月光VOA。对于1A，2A，2B和4A的情况，Moonshine VOA内U的存在也被建立。此外，还讨论了3A、5A和3C的情形。
Howe Pairs in the Theory of Vertex Algebras
对于任何顶点代数V和V的任何子代数A，存在V的一个新的子代数，称为A在V中的交换子。本文将交换子解释为不变量理论中的一个顶点代数概念。通过把交换代数问题归结为交换代数中的一个问题，给出了一种在适当的顶点代数范畴中描述交换代数的方法。我们给一个有趣的例子豪对（即一对相互交换子）在顶点代数设置。
In this paper, a continuation of \cite{MPS}, we investigate the $S_3$-orbifold subalgebra of $(\mathcal{V}_c)^{\otimes 3}$, that is, we consider the $S_3$-fixed point vertex subalgebra of the tensor product of three copies of the universal Virasoro vertex operator algebras $\mathcal{V}_c$. Our main result is construction of a minimal, strong set of generators of this subalgebra for any generic values of $c$. More precisely, we show that this vertex algebra is of type $(2,4,6^2,8^2,9,10^2,11,12^3)$. We also investigate two prominent examples of simple $S_3$-orbifold algebras corresponding to ce…
Z_3 symmetry and W_3 algebra in lattice vertex operator algebras
利用陪集构造和轨道折叠理论，将中心荷为6/5的W3代数实现为与\sqrt{2} A2型格相关联的顶点算子代数V_{ \sqrt{2} A2}的子代数.证明了W_3是有理数。它的不可约模的分类和明确的建设。并计算了这些不可约模的特征标。
Infinite Abelian Subalgebras in Quantum W-Algebras: An Elementary Proof
给出了量子W-代数中无穷维交换子代数存在性的一个初等证明。在适当的实现中，这些子代数定义了各种量子可积系统的守恒荷。我们考虑与单李代数相联系的所有主W-代数。证明是基于更一般的结果，即对于一类顶点算子，量子算子通过等价变换与它们的经典算子相关。
Given a simple vertex algebra A and a reductive group G of automorphisms of A, the invariant subalgebra A^G is strongly finitely generated in most examples where its structure is known. This phenomenon is subtle, and is generally not true of the classical limit of A^G, which often requires infinitely many generators and infinitely many relations to describe. Using tools from classical invariant theory, together with recent results on the structure of the W_{1+\infty} algebra, we establish the strong finite generation of a large family of invariant subalgebras of βγ-systems, bc-systems, and bcβ…
For the simple Lie algebra $ \frak{so}_m$, we study the commutant vertex operator algebra of $ L_{\hat{\frak{so}}_{m}}(n,0)$ in the $n$-fold tensor product $ L_{\hat{\frak{so}}_{m}}(1,0)^{\otimes n}$. It turns out that this commutant vertex operator algebra can be realized as a fixed point subalgebra of $L_{\hat{\frak{so}}_{n}}(m,0)$ (or its simple current extension) associated with a certain abelian group. This result may be viewed as a version of level-rank duality.
Associative subalgebras of the Griess algebra and related topics
证明了Griess代数中的某些幂等元如何在Frenkel-Lepowsky-Meurman的月光模顶点算子代数中生成维拉索罗代数的离散级数表示.它还表明，每个Niemeier格确定（在许多方面）的Griess代数的某些极大结合子代数。
Given a vertex algebra $\mathcal{V}$ and a subalgebra $\mathcal{A}\subset \mathcal{V}$, the commutant $\text{Com}(\mathcal{A},\mathcal{V})$ is the subalgebra of $\mathcal{V}$ which commutes with all elements of $\mathcal{A}$. This construction is analogous to the ordinary commutant in the theory of associative algebras, and is important in physics in the construction of coset conformal field theories. When $\mathcal{A}$ is an affine vertex algebra, $\text{Com}(\mathcal{A},\mathcal{V})$ is closely related to rings of invariant functions on arc spaces. We find strong finite generating sets for a…
In this article, we study the moonshine vertex operator algebra starting with the tensor product of three copies of the vertex operator algebra $V_{\sqrt2E_8}^+$, and describe it by the quadratic space over $\F_2$ associated to $V_{\sqrt2E_8}^+$. Using quadratic spaces and orthogonal groups, we show the transitivity of the automorphism group of the moonshine vertex operator algebra on the set of all full vertex operator subalgebras isomorphic to the tensor product of three copies of $V_{\sqrt2E_8}^+$, and determine the stabilizer of such a vertex operator subalgebra. Our approach is a vertex o…
Y(z)-injective vertex superalgebras and Hopf actions
本文研究Y(z)-内射顶点超代数，证明单顶点超代数与具PBW基的顶点超代数属于该类，且内忠实作用于其上的有限维Hopf代数必为群代数，将不动点子代数研究归约为群作用情形。
In this article, we completely determine the isomorphism classes of lattice vertex operator algebras and the vertex operator subalgebras fixed by a lift of the -1-isometry of the lattice. We also provide similar results for certain even lattices associated with doubly-even binary codes.
Representations of the vertex algebra $W_{1+\infty}$ with a negative integer central charge
设$\D$是${\C} \setminus \{0\}$上正则微分算子的李代数，${\hD}= {\D} + {\C} C$是${\D}$的中心扩张.设$W_{1+\infty,-N}$是与具有中心电荷$c=-N$的不可约真空$\hD$ -模相关联的顶点代数。证明了$W_{1+\infty,-N}$是具有$2 N$生成元的Heisenberg顶点代数M（1）的子代数，并构造了2N维不可约$W_{1+\infty,-N}$ -模族.考虑这些模块作为$\hD$ -模块，我们确定相应的最高权重。
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 $\mathrm{Fi}_{23}$ and $\mathrm{Fi}_{22}$ and $c=25/28$ and $c=11/12$ Virasoro vectors of subalgebras of the moonshine vertex operator algebra.
Lie subalgebras of differential operators on the super circle
本文对超圆$S^{1|1}$上的微分算子李超代数的中心扩张$\hsd$的保主阶的反对合进行了分类.阐明了由这些反对合固定的相应子代数与$OSP$和$P$型的$\hat{gl}_{\infty|\infty}$的子代数之间的关系.我们得到了这些子代数上的不可约最高权模是拟有限模的一个判别准则，并构造了这类模的自由域实现。我们进一步建立了它们与Fock空间上某些有限维经典李群之间的对偶。
In this paper, the irreducible modules for the $\mathbb{Z}_{2}$-orbifold vertex operator subalgebra of the parafermion vertex operator algebra associated to the irreducible highest weight modules for the affine Kac-Moody algebra $A_1^{(1)}$ of level $k$ are classified and constructed.
Ghost Systems: A Vertex Algebra Point of View
费米子和玻色子鬼系统定义在一个单顶点代数，其中承认一个单参数家庭的共形结构。这些结构相互关联的观察结果提供了一种简单的方法来获得鬼系统的一般扭曲模的特征公式。扭曲模下的U（1）对称及其子群也定义了不变顶点子代数的无限集合。从W-代数的角度详细研究了它们的结构，特别着重研究了费米子鬼系统的Z_N-不变子代数。
A unitary vertex operator algebra arising from the 3C-algebra
本文通过3C代数的共轭表示证明了特定顶点算子代数的单位性，并推导了其模的融合规则。
OPE-Algebras
在第七/0010293号论文中，Kapustin和Orlov引入了一个OPE-algebra的概念，并提出它形式化了共形域论，就像顶点代数形式化了手征代数一样，即共形域论的全纯域的子代数。本文主要研究顶点代数一般理论中的概念和结果可以推广到OPE-algebra上的问题。
Every isometry $σ$ of a positive-definite even lattice $Q$ can be lifted to an automorphism of the lattice vertex algebra $V_Q$. An important problem in vertex algebra theory and conformal field theory is to classify the representations of the $σ$-invariant subalgebra $V_Q^σ$ of $V_Q$, known as an orbifold. In the case when $σ$ is an isometry of $Q$ of order two, we classify the irreducible modules of the orbifold vertex algebra $V_Q^σ$ and identify them as submodules of twisted or untwisted $V_Q$-modules. The examples where $Q$ is a root lattice and $σ$ is a Dynkin diagram automorphism are pr…
Decomposition of the vertex operator algebra V_{\sqrt{2}A_3}
对于与A_1型根格\sqrt{2} × \sqrt{2} A_1有关的顶点算子代数V_{ \sqrt{2} A_1}，Dong-Li-Maosn-Norton证明了Virasoro向量是1 +1个相互正交的共形向量之和，其中中心电荷c_i=1-6/（i+2）（i+3），i=1，.，l和c_{l+1} =2l/（l+3），由这些向量生成的子代数T是Virasoro顶点算子代数L（c_i，0）的张量积.本文确定了V_{ \sqrt{2} A_3}完全分解为不可约T-模的和。
In this paper, we study a certain deformation $D$ of the Virasoro algebra that was introduced and called $q$-Virasoro algebra by Nigro,in the context of vertex algebras. Among the main results, we prove that for any complex number $\ell$, the category of restricted $D$-modules of level $\ell$ is canonically isomorphic to the category of quasi modules for a certain vertex algebra of affine type. We also prove that the category of restricted $D$-modules of level $\ell$ is canonically isomorphic to the category of $\mathbb{Z}$-equivariant $ϕ$-coordinated quasi modules for the same vertex algebra.…
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…
This is the first in a series of papers where we study logarithmic intertwining operators for various vertex subalgebras of Heisenberg vertex operator algebras. In this paper we examine logarithmic intertwining operators associated with rank one Heisenberg vertex operator algebra $M(1)_a$, of central charge $1-12a^2$. We classify these operators in terms of {\em depth} and provide explicit constructions in all cases. Furthermore, for $a=0$ we focus on the vertex operator subalgebra L(1,0) of $M(1)_0$ and obtain logarithmic intertwining operators among indecomposable Virasoro algebra modules. I…
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.
Graded Satake diagrams and super-symmetric pairs
本文列出了基本矩阵李超代数中可量子化为标准量子超群中余理想子代数的经典球面子代数，分类了相应的 Satake 型图，并证明每个图定义了一族真球面子代数。
We consider unitary simple vertex operator algebras whose vertex operators satisfy certain energy bounds and a strong form of locality and call them strongly local. We present a general procedure which associates to every strongly local vertex operator algebra V a conformal net A_V acting on the Hilbert space completion of V and prove that the isomorphism class of A_V does not depend on the choice of the scalar product on V. We show that the class of strongly local vertex operator algebras is closed under taking tensor products and unitary subalgebras and that, for every strongly local vertex …
Regularity of certain vertex operator algebras with two generators
对于每个$m \in {\C} \setminus \{0, -2\}$和每个非负整数$k$，我们定义了顶点算子（超）代数$D_{m,k}$，它有两个生成元和秩$ \frac{3 m}{m + 2}$。如果$m$是正整数，则$D_{m,k}$可以实现为格顶点代数的子代数。在这种情况下，我们证明了$D_{m,k}$是一个正则顶点算子（超）代数，并找到了不可约模的个数。
6-transposition property of $τ$-involutions of vertex operator algebras
本文研究了顶点算子代数的Griess代数中由两个Ising向量生成的子代数。我们证明了它的结构是由伊辛向量的某些内积唯一决定的。证明了两个$τ$ -对合的乘积的阶小于等于6，并确定了两个Ising向量的内积.
We generalize Conway-Sloane's constructions of the Leech lattice from Niemeier lattices using Lorentzian lattice to holomorphic vertex operator algebras (VOA) of central charge 24. It provides a tool for analyzing the structures and relations among holomorphic VOAs related by orbifold construction. In particular, we are able to get some useful information about certain lattice subVOAs associated with the Cartan subalgebra of the weight one Lie algebra. We also obtain a relatively elementary proof that any strongly regular holomorphic VOA of central charge $24$ with a non-trivial weight one sub…
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