导子单结合代数及其相联的李代数

Lie algebras associated with devivation-Simple associative algebras

介绍二元组(A,D)及其分类结果,其中A为特征零的代数闭域F上具有单位元的交换结合代数,D为由A的交换的局部有限导子组成的有限维空间,并且A为D-单的,即A没有D-不变的非平凡子理想(称这样的A为导子单结合代数);介绍了由导子单结合代数构造的李代数,特别是Cartan型、Weyl型及Block型无限维非阶化单李代数的结构和表示理论的一些结果,即确定这些李代数的同构类、导子代数、2-上同调群并给出某些不可约表示特别是quasifinite表示的分类.

The main suppose of the article is:ⅰ)to introduce the pairs(A,D),where A is a commutative associative algebra with an identity element,over an algebracally closed field F of characteristic zero,D is a finte dimensional space consisting of commuting locally finite derivations of A such that A is D_simple,i.e.,A does not have a proper D_stable ideal (such an algebra A is called a derivation_simple algebra);ⅱ)to introduce Lie algebras related to derivation_simple algebras,especially,non_graded infinite dimensional simple Lie algebras of Cartan type,Weyl type and Block type,and to present some results on the structure theory and representation theory of these Lie algebras,mainly the determinations of isomorphism classes,derivation algebras,the second cohomology groups of these Lie algebras,and some classifications of irreducible modules especially quasfinite modules.