Haisheng Li
The regular representation is related to Zhu's $A(V)$-theory and an induced module from an $A(V)$-module to a $V$-module is defined in terms of the regular representation. As an application, a new proof of Frenkel and Zhu's fusion rule theorem is obtained.
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In his landmark paper, Zhu associated two associative algebras to a vertex operator algebra: what are now called Zhu's algebra and the C_2-algebra. The former has a nice interpretation in terms of the representation theory of the VOA, while the latter only serves as a finiteness condition. In this paper we undertake first steps to unravel the interpretation of the C_2 vector space. In particular we suggest that it seems and controls the twisted representations of the VOA.
Motivated by the study of indecomposable, nonsimple modules for a vertex operator algebra $V$, we study the relationship between various types of $V$-modules and modules for the higher level Zhu algebras for $V$, denoted $A_n(V)$, for $n \in \mathbb{N}$, first introduced by Dong, Li, and Mason in 1998. We resolve some issues that arise in a few theorems previously presented when these algebras were first introduced, and give examples illustrating the need for certain modifications of the statements of those theorems. We establish that whether or not $A_{n-1}(V)$ is isomorphic to a direct summa…
Genus $g$ Zhu Recursion for Vertex Operator Algebras and Their Modules
本文在 Schottky 单值化下建立 VOA 及其模的亏格 $g$ Zhu 递归，用于关联函数、Ward 恒等式、Heisenberg VOA 结构及有理格广义 VOA 配分函数。
We give a complete description of the full automorphism group of a lattice vertex operator algebra, determine the twisted Zhu's algebra for the automorphism lifted from the -1 isometry of the lattice and classify the corresponding irreducible twisted modules through the twisted Zhu algebra.
We study intertwining operators among the twisted modules for rational VOAs and show the modular invariance property of the space spanned by the trace functions concerning to some representations of rational orbifold VOAs. Our result generalizes the results obtained by Dong-Li-Mason and Miyamoto, which are basically extensions of Zhu's celebrated works.
We prove that the weak associativity for modules for vertex algebras are equivalent to a residue formula for iterates of vertex operators, obtained using the weak associativity and the lower truncation property of vertex operators, together with a known formula expressing products of components of vertex operators as linear combinations of iterates of components of vertex operators. By requiring that these two formulas instead of the commutator formula hold, we construct a functor $S$ from the category of modules for Zhu's algebra of a vertex operator algebra $V$ to the category of $\mathbb{N}…
Let V be a vertex operator algebra and G a finite automorphism group of V. For each g\in G and nonnegative rational number n\in {\mathbb Z}/|g|, a g-twisted Zhu algebra A_{g,n}(V) plays an important role in the theory of vertex operator algebras, but the given product in A_{g,n}(V) depends on the eigenspaces of g. We show that there is a uniform definition of products on V and we introduce a G-twisted Zhu algebra A_{G,n}(V) which covers all g-twisted Zhu algebras. Assume that V is simple and let {\cal S} be a finite set of inequivalent irreducible twisted V-modules which is closed under the ac…
Noetherianity of twisted Zhu algebra and bimodules
本文证明了一大类自然顶点算子代数及其模的Zhu代数、双模及其$g$-扭类似具有Noetherian性，为应用非交换代数几何方法研究其表示理论与融合规则提供了可能。
We construct two associative algebras from a vertex operator algebra $V$ and a general automorphism $g$ of $V$. The first, called $g$-twisted zero-mode algebra, is a subquotient of what we call $g$-twisted universal enveloping algebra of $V$. These algebras are generalizations of the corresponding algebras introduced and studied by Frenkel-Zhu and Nagatomo-Tsuchiya in the (untwisted) case that $g$ is the identity. The other is a generalization of the $g$-twisted version of Zhu's algebra for suitable $g$-twisted modules constructed by Dong-Li-Mason when the order of $g$ is finite. We are mainly…
On mode transition algebras for $\mathbb{Z}$-graded vertex algebras and applications to bosonic ghosts
研究$\mathbb{Z}$-分级顶点代数的模式转换代数及其在玻色鬼模块中的应用，证明其弱锁性质及构造更高层次朱代数。
We study the representation theory of the Bershadsky-Polyakov algebra $\mathcal W_k = \mathcal{W}_k(sl_3,f_θ)$. In particular, Zhu algebra of $\mathcal W_k$ is isomorphic to a certain quotient of the Smith algebra, after changing the Virasoro vector. We classify all modules in the category $\mathcal{O}$ for the Bershadsky-Polyakov algebra $\mathcal W_k$ when $k=-5/3, -9/4, -1,0$. In the case $k=0$ we show that the Zhu algebra $A(\mathcal W_k)$ has $2$--dimensional indecomposable modules.
We first formulate a definition of tensor product for two modules for a vertex operator algebra in terms of a certain universal property and then we give a construction of tensor products. We prove the unital property of the adjoint module and the commutativity of tensor products, up to module isomorphism. We relate this tensor product construction with Frenkel and Zhu's $A(M)$-theory. We give a proof of a formula of Frenkel and Zhu for fusion rules. We also give the analogue of the ``Hom''-functor of classical Lie algebra theory for vertex operator algebra theory by introducing a notion of ``…
Twisted $ϕ$-coordinated modules for vertex algebras and Zhu's correspondence theorem
本文研究了顶点代数中扭曲ϕ-协调模的构造及其与朱对应定理的关系，证明了简单模与不可约模之间的双射关系。
We summarize interactions between vertex operator algebras and number theory through the lens of Zhu theory. The paper begins by recalling basic facts on vertex operator algebras (VOAs) and modular forms, and then explains Zhu's theorem on characters of VOAs in a slightly new form. We then axiomatize the desirable properties of modular forms that have played a role in Zhu's theorem and related classification results of VOAs. After this we summarize known classification results in rank two, emphasizing the geometric theory of vector-valued modular forms as a means for simplifying the discussion…
Nodal degeneration of chiral algebras II: Local structure and chiral Zhu algebras
本文通过几何构造将顶点代数的Zhu代数推广到因子化代数，定义了一个导出结合代数及其手性双模，并证明其零阶同调恢复节点处顶点代数的模式转移代数。
Bimodules over twisted Zhu algebras and twisted fusion rules theorem for vertex operator algebras
本文研究了扭曲Zhu代数上的双模构造，并基于此推导出顶点算子代数的扭曲融合规则定理，通过构造特定模并证明其子模的universal性质。
Twisted regular representations and bimodules in vertex operator algebra theory
本文利用顶点算子代数的扭状正则表示理论，构建扭状Zhu代数上的双模，扩展了Li在非扭情况下的工作，并证实了Dong和Jiang关于双模理论的猜想。
Let $N_{k} (\g)$ be a vertex operator algebra (VOA) associated to the generalized Verma module for affine Lie algebra of type $A_{\ell -1} ^{(1)}$ or $C_{\ell} ^{(1)}$. We construct a family of ideals $J_{m,n} (\g)$ in $N_{k} (\g)$, and a family $V_{m,n} (\g)$ of quotient VOAs. These families include VOAs associated to the integrable representations, and VOAs associated to admissible representations at half-integer levels investigated in q-alg/9502015. We also explicitly identify the Zhu's algebras $A(V_{m,n} (\g))$ and find a connection between these Zhu's algebras and Weyl algebras.
In this paper, we prove the categories of lower bounded twisted modules of positive integer levels for simple vertex operator algebras associated with affine Lie algebras and general automorphisms are semisimple, using the twisted generalization of Zhu's algebra for these vertex operator algebras, constructed in \cite{HY}. We also show that the category of lower bounded twisted modules for a general automorphism is equivalent to the category of lower bounded twisted modules for the corresponding diagram automorphism.
This paper is an exposition of the representation theory of vertex operator algebras in terms of associative algebras A_n(V) and their bimodules. A new result on the rationality is given. That is, a simple vertex operator algebra V is rational if and only if its Zhu algebra A(V) is a semisimple associative algebra and each irreducible admissible $V$-module is ordinary.
We discuss our recent results on the representation theory of $\mathcal{W}$--algebras relevant to Logarithmic Conformal Field Theory. First we explain some general constructions of $\mathcal{W}$-algebras coming from screening operators. Then we review the results on $C_2$--cofiniteness, the structure of Zhu's algebras, and the existence of logarithmic modules for triplet vertex algebras. We propose some conjectures and open problems which put the theory of triplet vertex algebras into a broader context. New realizations of logarithmic modules for $\mathcal{W}$-algebras defined via screenings a…
It is proved that if any Z-graded weak module for vertex operator algebra V is completely reducible, then V is rational and C_2-cofinite. That is, V is regular. This gives a natural characterization of regular vertex operator algebras.
We prove a conjecture stated in a previous paper by the author about the existence of canonical filtrations for a family of vertex operator algebras in rational levels.
We study the algebra of regular functions on the big cell of the Gauss decomposition of a simple complex Lie group G. We prove that it is spanned by the matrix elements of big projective modules in the BGG category O, and admits a decomposition of Peter-Weyl type. The subspace of Whittaker vectors in the regular representation on the big cell gives a realization of the big projective modules in O, similar to the Borel-Weil theorem. These results can be extended to the quantum groups.
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