The Construction Of Novikov Superalgebra And The Derivations Of The Nilpotent Lie Algebras

Novikov algebra is introduced in Hamiltonian operators about inegrability of certainnonlinear partial differential equations, and it is a special case of a left-symmetric algebra.And Novikov superalgebra is related to the quadrstic conformal superalgebra whichcorresponds to the Hamiltonian pairs and play an important role in the completely integrablesystem. Novikov superalgebra appears also in close relation to operator Yang-Baxter superequation and Lie superalgebra. People researched many problems about Novikovsuperalgebra before, such as its derivations, properties and its constructions. In this paper wediscuss the existence of Novikov superalgebra structure on solvable Lie superalgebras,nilpotent Lie superalgebras and filiform Lie superalgebras in low dimension on the base ofstudying Novikov algebra. We give a method to construct Novikov superalgebra structure byclassical r-matrices on Lie superalgebra modules. At the same time we discuss the derivationof the nilpotent Lie algebra. Scholars calculated the derivations of the nilpotent Lie algebrasthat less than or equal to five-dimensional before, but the classification of the six-dimensionalnilpotent Lie algebra and the seven-dimensional nilpotent Lie algebras are too complex andtheir derivations are not be calculated. In the third chapter, we calculate the derivations ofsix-dimensional nilpotent Lie algebras over fields of characteristic not2and indecomposableseven-dimensional nilpotent Lie algebras over algebraically closed fields of characteristic not2in the third chapter of this paper, and we list the results in table3.1and3.2. Since the latterhalf of the80s, the study of the complete Lie algebras become active, and get someachievements. Recently years, more and more people notice the complete Lie algebras. Manypapers research its properties, classification and some other problems. In the forth chapter ofthis paper, we discuss a method to construct complete Lie algebras by a certain complexsemisimple Lie algebras and its representation. We take the three-dimension simple Liealgebras s sl(2,C)and its natural representation and adjoint representation to construct twocomplete Lie algebras respectively, and at the same time we prove that these complete Liealgebras are not solvable.