Natural Filtrations Of Infinite Dimensional Modular Lie Superalgebras Of Cartan Type

In this paper, the infinite dimensional Cartan type Lie superalgebras over an algebraic closed field of positive characteristic are considered. In particular, the natural filtrations of Lie superalgebras of types H, K, KO, SHO and SKO are proved to be invariant under automorphisms, thus generalizing the known results of W, S and HO given in.First, we recall the definitions of Lie superalgebras W, S, H, K, HO, KO. Then, similar to the case of characteristic zero, modular Lie superalgebras SHO and SKO are constructed, and their simplicity is determined.By studying ad-nilpotent elements, especially the ones in the even part, we determine subalgebras generated by certain ad-nilpotent elements, then we prove that the natural filtrations of infinite dimensional Cartan type Lie superalgebras H, K, KO, SHO, SKO are invariant under automorphisms. Thereby, a necessary condition for two Lie superal-gebras of same type to be isomorphic is given, by which Lie superalgebras of each type of H, K, KO, SHO are classified up to isomorphism. Finally, applying the results, we prove that an automorphism of infinite dimensional Cartan type Lie superalgebras over an algebraic closed field of positive characteristic is determined by its action on the -1 component in the principal gradations.