Representation theory and tensor product theory for vertex operator algebras
Representation theory and tensor product theory for vertex operator algebras
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发表时间：
1994-07
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通讯作者：
Haisheng Li
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作者：
Haisheng Li
首先根据一个泛性质给出了顶点算子代数中两个模的张量积的定义，然后给出了张量积的构造。证明了伴随模的单位性质和张量积的交换性，直至模同构。我们将这种张量积构造与Frenkel和Zhu的$A(M)$-理论联系起来。我们给出了Frenkel和Zhu关于融合规则的一个公式的证明。通过引入广义缠结算符的概念，给出了经典李代数理论中“Hom”-函子对顶点算子代数理论的类比。证明了顶点算子代数中从一个模到另一个模的广义缠结算子空间是一个广义模。从这个结果，我们得到了对于任意有理顶点算子代数的土屋和卡尼的“核民主定理”的一般形式。这证明了我们构造张量积得到的融合规则与使用Tsuchiya和Kanie方法得到的融合规则是相同的，对于WZW模型和最小模型都是如此。证明了如果$V$满足一定的“有限性”和“半简单性”条件，则在广义模内存在唯一的极大子模。进一步证明了该极大子模与某张量积模的合模同构。这给出了张量积模的另一种构造，这个结果与Huang和Lepowsky的构造密切相关。
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 ``generalized intertwining operator.'' We prove that the space of generalized intertwining operators from one module to another for a vertex operator algebra is a generalized module. From this result we derive a general form of Tsuchiya and Kanie's ``nuclear democracy theorem'' for any rational vertex operator algebra. This proves that the fusion rules obtained from our construction of tensor products are the same as the fusion rules obtained by using Tsuchiya and Kanie's method, for both WZW models and minimal models. We prove that if $V$ satisfies certain ``finiteness'' and ``semisimplicity'' conditions, then there exists a unique maximal submodule inside the generalized module. Furthermore, we prove that this maximal submodule is isomorphic to the contragredient module of a certain tensor product module. This gives another construction of tensor product modules and this result turns out to be closely related to Huang and Lepowsky's construction.