Vertex operator algebras with group action

1997年发现了与A型二阶根格相关的顶点算子代数具有三阶对称性的子代数，并研究了它的性质。特别地，确定了其自同构群，并对其单模进行了分类。这些结果被写在一篇联合论文《三元码和顶点算子代数》中。这种顶点算子代数与Monster模之间的关系还有待研究。此外，还从顶点算子代数的角度研究了Borwein型恒等式。这些结果被写在一篇名为Borwein恒等式和顶点算子代数的联合论文中。1998年，我们对与A型根格相关的顶点算子代数的所有最高权向量进行了分类。最高权向量很重要，因为一旦知道它们，顶点算子代数就可以分解成Virasoro顶点算子代数的张量积的简单模的直和。虽然不是所有的最高权向量都是已知的，但从最高权向量中可以获得许多关于顶点算子代数结构的信息，最高权向量至多为两个。例如，利用这些最高权向量构造了一个具有4阶对称性的顶点算子代数。最近，所有最高权向量都被归类为A型三阶根格。为了将这一结果推广到任意阶的根格，需要一个新的想法。

In 1997 a subalgebra having order three symmetry of a vertex operator algebra associated with a rank two root lattice of type A was discovered and its properties were studied. In particular its automorphism group was determined and its simple modules were classified. The results were written in a joint paper Ternary codes and vertex operator algebras. Relations between this vertex operator algebra and the Monster module still remain to be studied. Moreover, Borwein type identity was studied from a point of view of vertex operator algebra. The results were written in a joint paper Borwein identity and vertex opertor algebras.In 1998 all highest weight vectors with weight at most two of a vertex operator algebra associated with a root lattice of type A were classified. Highest weight vectors are important, because once they are known then a vertex operator algebra can be decomposed into a direct sum of simple modules for a tensor product of Virasoro vertex operator algebras. Although not all highest weight vectors are known, much information about the structure of the vertex operator algebra can be obtained from highest weight vectors with weight at most two. For example, a vertex operator algebra having a symmetry of order 4 was constructed by using these highest weight vectors. Recently all highest weight vectors were classified in the case of rank three root lattice of type A.In order to generalize this result to a root lattice of arbitrary rank, a new idea would be required.