Some Structures Of The Classical Lie Superalgebras

The classical Lie superalgebras are divided into basic classical Lie superalgebras and strange Lie superalgebras.Basic classical Lie superalgebras include simple Lie algebras,A(m,n),A(n,n),B(m,n),c(n),D(m,n),G(3),F(4)and D(2,1,α),two series strange Lie super-algebras include P(n)and Q(n).We use the notation K to represent basic classical Lie superal-gebras except for A(n,n)and simple Lie algebras.Every local derivation or 2-local derivation of simple Lie algebras is a derivation.The definitions of the local derivation and 2-local derivation on Lie algebras are generalized to Lie superalgebras.This paper study the structures of the local superderivations and 2-local superderivations on the classical Lie superalgebras.In addition,we compute the numbers of ad-nilpotent ideals of the Borel subalgebras of some classical Lie superalgebras.The main contents are as follows:1.Using the Chevalley basis of basic classical Lie superalgebras,we prove that every local superderivation of K and A(n,n)(n>1)is a superderivation.Similarly,using the triangle decomposition and the property of the basis of Q(n),we prove that every local superderivation of Q(n)is a superderivation.In addition,we find a kind of Lie superalgebras,on which there is a local superderivation that is not a superderivation.2.Using the non-degenerate even supersymmetric invariant bilinear forms,we prove that every 2-local superderivation of K is a superderivation.Similarly,using a non-degenerate odd supersymmetric invariant bilinear form of Q(n),we prove that every 2-local superderivation of Q(n)is a superderivation.In addition,we find that a subalgebra of A(1,1)has a 2-local superderivation that is not a superderivation.3.By the method of combinatorics,we compute the numbers of ad-nilpotent ideals of the Borel subalgebras of classical Lie superalgebras A(m,n),B(m,n),C(n),D(m,n),Q(n).