K (n) Lie Algebra

In this paper,the Lie superalgebra K(n) on the elementary vector A(n) is constructed. By doing research on the K(n), we can know that it is a new type of Lie superalgebra and that it is not simple.When n > 4,we can get the maximal ideal of K(n) and konw that the quotient algebra K(n)/Io (the /o is the maximal ideal of K(n)) and the simple Lie superalgebra H(n-l) are isomorphic , that is to say K(n) is the extension of the simple Lie superalgebra H(n-l).The main results in this paper are the following: Theorem 1:V a 6 A(n)a, b e A(n)iS, we can define " [ , ] " [a, 6] = (2a-Ei=l1xiDi(a))xnDn(b)-(-l)^(2b-E^xiDi(b))xnDn(a)+ ?=i1(-i)QA(a)A(&) then A(n) and " [ , ] " compose Lie superalgebra Kn over field F.Theorem 2: there is not a normal Z-grading of K(n).Theorem 3: There is a maxims ideal 70 of the K(n),defining K(n)o :=?K(ri)/I0 then K(n)0 and the simple Lie superalgebra H(n-l) are isomorphic and K(n) is the extension of the simple Lie superalgebra H(n-l).