四维李代数的交叉模和三阶上同调群

Crossed modules and third cohomology of four dimensional Lie algebras

根据Levi定理,四维不可解李代数L可以分解为它的半单纯子代数与根的半直和L0 S.结合李代数交叉模的定义,计算出四维不可解李代数的交叉模等价类只有一个,相应的三阶上同调群是平凡的.同时对四维可解李代数,讨论了导代数一维情形的交叉模等价类与三阶上同调群.

According to Levi theorem,four dimensional unsolvable Lie algebras L can be decomposed as a semidirect sum L0S,where L0 is the semisimple subalgebra and S the radical of L.With the definition of crossed modules of Lie algebras,it is shown that there is only one equivalent class of crossed modules of four dimensional unsolvable Lie algebras,that the corresponding third cohomology is trivial.Then crossed modules and third cohomology of four dimensional solvable Lie algebras are discussed where its derivated algebra is one dimensional.