非退化可解李代数模的结构

设g是带有非退化对称不变双线性型的有限维可解非幂零李代数,证明了g的极大环面子代数H作用在g的有限维模上是可对角化;给出g的Casimir算子Ω的概念,并证明了Ω作用在g的不可分解模上是一个纯量0。

Let g be a finite-dimensional solvable nonnilpotent Lie algebra equipped with a nondegenerate symmetric invariant bilinear form 〈 ＊ ,＊〉.In this paper, we prove that the maximal torus subalgebra H of g is diagonalizable on any finite-dimensional g？module,and we give the notion of Casimir operator of g and prove that acts on W as a scalar O,where W is a finite-dimensional nondecomposable module of g.