华东师范大学学报(自然科学版) ›› 2011, Vol. 2011 ›› Issue (3): 90-99.

• 应用数学与基础数学 • 上一篇    下一篇

阶化Cartan型李代数的诱导模及其扩张的一点注记

黄 慧   

  1. 江西工业贸易职业技术学院, 南昌 330038
  • 收稿日期:2010-09-01 修回日期:2010-12-01 出版日期:2011-05-25 发布日期:2011-05-25
  • 通讯作者: 黄 慧

Note on induced modules and their extensions for graded Lie algebras of Cartan type

HUANG Hui   

  1. Jiangxi Vocational Technical College of Industry and Trade, Nanchang 330038, China
  • Received:2010-09-01 Revised:2010-12-01 Online:2011-05-25 Published:2011-05-25
  • Contact: HUANG Hui

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摘要：

在广义限制李代数的意义下，证明了W, S, H型系列的阶化Cartan型李代数的”修正”诱导模为余诱导模. 得到了诱导模和余诱导模之间的关联，从而推广了Rolf Farnsteiner和 Helmut Strade 在限制李代数情形下关于诱导模与余诱导模之间的关联. 进而证明了W, S, H型系列的阶化Cartan型李代数的所有具有广义特征标高度不超过某个界的不可约非例外单模均为余诱导模. 应用此结论以及Rolf Farnsteiner关于上同调的结果，最后进一步得到了一些有关W, S, H型系列的阶化Cartan型李代数单模之间的扩张的结论.

关键词: 广义限制李代数, Cartan型李代数, 诱导模, 余诱导模, 扩张, 广义限制李代数, Cartan型李代数, 诱导模, 余诱导模, 扩张

Abstract: In the sense of generalized restricted Lie algebras, it was proved that the modified induced modules of graded Lie algebras of Cartan types W, S, H coincide with coinduced modules. The relationship between induced modules and coinduced modules was obtained, extending the corresponding result by Rolf Farnsteiner and Helmut Strade in the case of restricted Lie algebras. Therefore, it was proved that any irreducible non-exceptional modules for graded Lie algebras of Cartan types W, S, H with generalized p-character of height not more than a precise upper-bound is a coinduced module. By applying this with some results on cohomology obtained by Rolf Farnsteiner in 1990s, we finally got some further results on extensions between simple modules of graded Lie algebras of Cartan types W, S, H.

Key words: Cartan type Lie algebra, induced module, coinduced module, extension, generalized restricted Lie algebra, Cartan type Lie algebra, induced module, coinduced module, extension

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