Finite-dimensional Lie superalgebras of Cartan type over fields of prime characteristic

Abstract

IN this note the infinite-dimensional Lie superalgebras of Cartan type X(m, n ) ( X = W, S, H or K ) over field F of prime characteristic are constructed. Then the second class of finitedimensional Lie superalgebras of Cartan type over F is defined. Their simplicity and restrictability are discussed. Finally a conjecture about classification of the finite-dimensional simple Lie superalgebras over F is given. Let F be a field and charF = p >2. I3t n be a positive integer and n > 1. A ( n ) denotes the exterior algebra over F with generators E l , . . a , En. If u = ( i l , i 2 , ..., i r ), where a 1 < i l < i 2 < . . . < i , < n , thenweset Q = ~ i , [ i 2 . . . ~ i r , and / P I = r . LetDi=-, i = l , . . , n . a Ei Imitating the situation that the characteristic number of basic field is zero, we can get the