arXiv math/01120542001-12math.QA

Orbifold模型V的A_{g,n}（V）与G-扭朱代数的一致积

Uniform product of A_{g,n}(V) for an orbifold model V and G-twisted Zhu algebra

Masahiko Miyamoto, Kenichiro Tanabe

备注：: 16 pages, Latex We added a duality theorem

摘要：

设V是顶点算子代数，G是V的有限自同构群.对于任意\in和非负有理数\in { \mathbb Z}/ |g|，g-扭曲朱代数A_{g,n}（V）在顶点算子代数理论中起着重要的作用，但A_{g,n}（V）中给定的积依赖于g的特征空间.证明了V上积的统一定义，并引入了一个G-扭朱代数A_{G,n}（V），它覆盖了所有的g-扭朱代数.设V是单的，{ \cal S}是有限的不可约扭V-模集，在G的作用下是闭的.对于由{ \cal S}上的G作用自然确定的适当的2-上圈，存在有限维半单结合代数{ \cal A}_α（G，{ \cal S}）.作为A_{G,n}（V）理论的应用，我们证明了{ \cal A}_α（G，{ \cal S}）与V^G在{ \cal S}中扭V-模直和上的作用的Schur-Weyl型对偶定理成立.作为这个结果的一个自然推论，对任何g \in G，每个不可约g-扭V-模都是完全可约V^G-模。

英文摘要：

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 action of G. There is a finite dimensional semisimple associative algebra {\cal A}_α(G,{\cal S}) for a suitable 2-cocycle naturally determined by the G-action on {\cal S}. We show that a duality theorem of Schur-Weyl type holds for the actions of {\cal A}_α(G,{\cal S}) and V^G on the direct sum of twisted V-modules in {\cal S} as an application of the theory of A_{G,n}(V). It follows as a natural consequence of the result that for any g\in G every irreducible g-twisted V-module is a completely reducible V^G-module.

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