In 1997 , pro zhang constructed four class of simple finite dimension cartan - type modular lie superalgebras : w , s , h , k . and gave a supposition of the classification of modular lie superalgebras of finite dimension . now someone found the fifth class simple finite dimension cartan - type modular lie superalgebras ho 在1997年，张永正教授构造了四类有限维单cartan型模李超代数： w ， s ， h ， k ，并提出一个关于有限维单模李超代数分类的猜想。
Let f be a field and charf = p = 3 . in this paper the generators of modular simple lie superalgebras w , h , s , k and ho of cartan - type are determined . their derivation algebras and their dimensions are determined , also . the main results are following : ( l ) let t = { xki | 1 < i < m , 0 < ki < 3ti } , s = { xi | m + 1 < i < s } , then t sgenerates k ( m , n , t ) 设f是特征p = 3的域，本文决定了f上有限维cartan型单李超代数w型， h型， s型， k型和ho型的生成元和它们的导子代数及其维数
For a given negatively graded lie superalgebra k - , the universal graded lie su - peralgebra u of type k - is defined and its existence proved . by posing additional conditions on k - , other types of universal graded lie superalgebras are defined and discussed . the concept of universal graded lie superalgebra leads naturally to the graded cartan type lie superalgrebras and it is proved that the graded cartan type lie superalgrebras k ( m , n , o > a ) , s ( m , n ) and h ( m , n ) can be characterized as certain types of universal graded lie superalgrebras 通过给k ~ -添加条件，又定义并讨论其它类型的泛阶化李超代数，进而，由泛阶化李超代数的定义自然地引出了阶化的cartan型李超代数，而且证得阶化的cartan型李超代数k ( m ， n ， _ a ) ， s ( m ， n )和h ( m ， n )分别可以被某种泛阶化李超代数所刻划。
