三重顶点算子代数的ADE不变子代数

Rational vertex operator algebras are the primary objects for the representation theory of vertex operator algebras. There are many rational vertex operator algebras whose representation theory is well understood. On the other hand, C2 -cofinite irrational vertex operator algebras resemble rational vertex operator algebras in many respects, e.g., finite representation type, modular invariance theorem. Known examples about C2-cofinite irrational vertex operator algebras are very limited so far. Among them, the triplet vertex algebras W(p)（p>1）are the most important examples. In Adamovic and Milas' paper, it was pointed out that W(p) admits an action of PSL(2, C). Thus for each finite subgroups of PSL(2, C), we can consider the corresponding invariant subalgebra. According to the classification of finite subgroups of PSL(2, C), there are ,up to conjugation, three series of finite subgroups, i.e., A-series, D-series, and E-series. This project is mainly devoted to the proof of C2-cofiniteness of the corresponding ADE invariant subalgebras and their representations..

有理顶点算子代数是顶点算子代数的表示论的首要研究对象。目前很多有理顶点算子代数的表示论都已经很清楚了。另外，C2余有限的无理顶点算子代数在很多方面和有理顶点算子代数十分相似，比如，有限表示型，模不变定理。目前已知的C2余有限的无理顶点算子代数的例子是极为有限的。其中三重顶点算子代数W(p)（p>1）是最重要的例子。 Adamovic 和 Milas的论文中提到W(p)上可以自然的定义李群PSL(2, C)的作用。因此对于PSL(2, C)的每个有限子群，我们可以得到相应的不变子代数。根据PSL(2, C)的有限子群的分类，在共轭等价下PSL(2, C)只有三个系列的有限子群：A系列，D系列，和E系列。本项目主要致力于相应的不变子代数的C2余有限性的证明及其表示论。

本项目主要是关于三重顶点算子代数的表示论。通过细致地研究相应的 Zhu 代数，我们几乎完全确定了E(6), E(7)的情形时的所有不可约模。作为一个副产品，我们给出了两个例外有理顶点算子代数的刻画，这是中心电荷为1的有理顶点算子代数的分类工作的一个重要贡献。