无限维李代数L(α,β)的导子李代数

本文讨论了无限维李代数Ｌ（α，β）的导子李代数的结构．分三种情况：（１）当α，β在Ｑ上线性无关时，ＤｅｒＬ（α，β）＝ＣＤｆ０ＣＤｇ０ａｄＬ（α，β），其中Ｄｆ０，Ｄｇ０是由ｆ０，ｇ０决定的导子，ｆ０，ｇ０是定义在Ｚ×Ｚ上的线性函数；（２）当α，β在Ｑ上线性相关且不同时为０时，ＤｅｒＬ（α，β）ｄｅｒＬ（α′，０）（α′≠０），ｄｅｒＬ（α，０）＝ＣＤ－α０ＣＤ－αｇ０ＣＤｆ０ａｄＬ（α，０），（α≠０），其中Ｄ－α０是某一个固定的导子，Ｄ－αｇ０，Ｄｆ０是由ｇ０，ｆ０决定的导子；（３）当α＝β＝０时，ＤｅｒＬ（０，０）＝ＣＤｆ０ＣＤｇ０ａｄＬ（０，０）．

The Derivation Algebra of Infinite Dimensional Lie Algebra L(α,β)

The paper discusses the structure of derivation algebra of infinite dimensional Lie algebra L(α,β). There are three cases:(1) If α and β are linearly independent over  Q , then  Der L(α,β)= C D f 0  C D g 0  ad L(α,β), where theD f 0  and D g 0  are derivations deterimined by the functions f 0 and g 0, f 0 and g 0 are linear functions defined on  Z × Z ;(2) When α and β are linearly dependent over  Q  and not zero at the same time we have  Der L(α,β) Der L(α′ ,0), (α′≠ 0),  Der L(α,0) = C D -α 0 C D -α g 0  C D f 0  ad L(α,0) (α≠ 0), where D -α 0 is a fixed derivation, D -α g 0  and D f 0  are derivations determined by the functions g 0 and f 0;(3) If α=β=0, then  Der L(0,0)= C D f 0  C D g 0  ad L(0,0).