Finite Simple Groups and Related Codes, Lattices and Vertex Operator Algebras

我们研究了与有限单群相关的码、格和顶点算子代数。主要结果如下：1.完成了Fischer单群F_&lt；22&gt；，F_&lt；23&gt；，F‘&lt；24&gt；的2-和3-根子群的分类。2.对12维么模高斯格进行了分类。3.给出了极值Z_6-码的一个充分条件，并作为应用，构造了许多与Leech格有关的Z_6-码。此外，还对长度为8的自对偶码进行了分类。4.研究了利用三元自对偶码构造么模格的方法。首次构造了44，60和68维的极值奇么模格。5.利用Z_8码及其子代数V_&lt；√&lt；2&gt；A_3&gt；构造了顶点算子代数。6.利用董梅森给出的Z_3-orbilold结构，我们给出了与嵌入到Leech格的子代数(√&lt；2&gt；A_2)^&lt；12&gt；(√&lt；2&gt；A_2)^&lt；12&gt；^&lt；12&gt；^&lt；12&gt；定义了Monster单群的某些3A-元在V^*上的显式作用。

We have studied codes, lattices and vertex operator algebras related to finite simple groups. Main results are as follows :1.The classifications of the 2-and 3-radical subgroups of Fischer's simple groups F_<22>, F_<23>, F'_<24> have been completed.2.Even unimodular Gaussian lattices of dimension 12 have been classified.3.A sufficient condition for extremal Z_6-codes, and as an application, many Z_6-code related the Leech lattice had been constructed. Moreover the self-dual Z_6-codes of length 8 are classified.4.The construction method of unimodular lattices using ternary self-dual codes has been studied. Extremal odd unimodular lattices in dimensions 44, 60 and 68 are constructed for the first time.5.The constructions of vertex operator algebras using Z_8-codes and subalgebras V_<√<2>A_3> have been considered. We also give a complete decomposition of the Moonshine VOA V^* associated with some subalgebra given by an embedding of the lattice (√<2>A_3)^8 into the Leech lattice.6.By using Z_3-orbifold construction given by Dong-Mason, we give a complete decomposition of the Moonshine VOA V^* associated with some subalgebra given by an embedding of the lattice (√<2>A_2)^<12> the Leech lattice. We also give the explicit actions of certain 3A-elements of the Monster simple group on V^* are defined.