Some Properties Of Novikov Superalgebras

Novikov algebra which was named by Osborn is a new algebra structure. It has a great development at present, and we have attained some good results.In this paper, we chiefly study the properties of Novikov superalgebra and Novikov-poisson superalgebra. We attaine these results by generalizing Novikov algebras properties to Novikov superalgebra. Moreover, we also construct a glass of Novikov superalgebras and Novikov-poisson superalgebras. Impotent Results:(1)We present the definations of subalgebra , ideal , quotient algebra ,modules and representation.(2)Define the derivation of Novikov superalgebra A and construct new Novikov superalgebras.THEOREM Let ( A , o ) be a Novikov superalgebra and let W(A) = W(A)₀ + W(A)₁ be the space of derivations of A , Define the operation [ · , · ] on W(A) by :for Then ( W(A) ,[·,·]) forms a Lie superalgebra.THEOREM Let W(A) = W(A)₀ + W(A)₁ be the space of derivations of Novikov superalgebra A .Define a new operation " o " on W(A) by:Then ( W(A) , o ) forms a Novikov superalgebra with respect to the action:THEOREM Let ( A , · ) be a super commutative associative algebra and D(A) = D(A)_i + D(A)_j be the space of derivations of A. DenoteDefine a operation " o " on N by:(d2for di + Ci e ^ d2 + ï¿¡2 € NjThen ( N , o ) forms a Novikov superalgebra.(3)PROPOSITION Let ( A , ? ) be a commutative associative algebra and let d be aeven derivation of ( A , ? ). we define the operation " o " as follows:x o y = x â–  d(y)Then ( A , o ) is a Novikov superalgebra , and ( A , o ) is simple if only if A is even<9-simple.(4) Construct a glass of new Novikov-poisson superalgebras.