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

有限简单群及相关代码、格和顶点算子代数

• 批准号: 12440003
• 负责人: KITAZUME Masaaki
• 金额: $ 4.16万
• 依托单位: Chiba University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 2000
• 资助国家: 日本
• 起止时间: 2000 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-12440003/
• 关键词: Finite Group; Simple Group; Sporadic Simple Group; Monster Simple Group; Code; Lattice; Design; Vertex Operator Algebra; モンスター純群; W代数; グラフ

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.

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。

项目成果

M.Kitazume, M.Miyamoto: "3-transposition automorphism groups of VOA"Advanced Studies in Pure Mathematics. 32. 315-324 (2001)

M.Kitazume，M.Miyamoto：“美国之音的3-转置自同构群”纯数学高级研究。

M.Kitazume, C.H.Lam, H.Yamada: "Moonshine Vertex Operator Algebra as L(1/2,0)【cross product】L(7/10,0)【cross product】L(4/5,0)【cross product】L(1,0)-modules"Journal of Pure and Applied Algebra. (発表予定). (2002)

M.Kitazume、C.H.Lam、H.Yamada：“Moonshine 顶点算子代数为 L(1/2,0)【叉积】L(7/10,0)【叉积】L(4/5,0)【叉积】L(1,0)-模《纯粹与应用代数杂志》。（待出版）。（2002）

M.Kitazume, C.H.Lam, H.Yamada: "Moonshine Vertex Operator Algebra as L(1/2,0)【cross product】(7/1O,0)【cross product】(4/5,0)【cross product】(1,0)-modules"Journal of Pure and Applied Algebra. 173. 15-48 (2002)

M.Kitazume、C.H.Lam、H.Yamada：“Moonshine 顶点算子代数为 L(1/2,0)【叉积】(7/1O,0)【叉积】(4/5,0)【叉积” 】(1,0)-模》纯粹与应用代数学报. 173. 15-48 (2002)

M.Kitazume,C.H.Lam,H.Yamada: "A class of vertex operator algebras constructed from Z_8 codes"Journal of Algebra. (発表予定). (2001)

M.Kitazume、C.H.Lam、H.Yamada：“由 Z_8 代码构造的一类顶点算子代数”《代数杂志》（即将出版）。

M.Kitazume, C.H.Lam, H.Yamada: "Moonshine Vertex Operator Algebra as L(1/2, 0) 【cross product】 L(7/10, 0) 【cross product】 L(4/5, 0) 【cross product】 L(1, 0)-modules"Journal of Pure and Applied Algebra. 173. 15-48 (2002)

M.Kitazume、C.H.Lam、H.Yamada：“Moonshine 顶点算子代数为 L(1/2, 0) [叉积] L(7/10, 0) [叉积] L(4/5, 0) [叉积】 L(1, 0)-模》纯粹与应用代数学报. 173. 15-48 (2002)