Derivations Of Novikov Algebras And Rota-Baxter Operators On Hom-Novikov Algebras In Low Dimensions

Novikov algebra is closely related to Lie algebra and it is a special kind of pre-Lie algebra. For Novikov algebra, derivation is an very important definition. In the first part of this paper we mainly discuss the derivations of Novikov algebras in dimension 4. First we give the definition of Novikov algebra and its derivation, we also discuss some basic propertie of them. Then we find the classification of 4-dimensional Novikov algebras over complex number field and give the characteristic matrices of each kind of the 4-dimensional Novikov algebras over the complex number field. Then we compute the derivations of the 4-dimensional Novikov algebras over complex number field under a given basis according to the definition of derivation. At last we give all the derivations of Novikov algebras in dimension four in a table and get the derivation algebras of Novikov algebras in dimension four.Moreover, Hom-Novikov algebra is a hot spot in research and it is a special kind of the left symmetric Hom-algebras. Novikov algebras can deform into Hom-Novikov algebra along any algebras morphism. Hom-Novikov algebra can also be constructed by Hom-commutative algebras and derivations. Rota-Baxter algebras is closely related to Hom-Novikov algebras. In the Hom-Novikov algebras, Rota-Baxter operators is an important definition. In the second part of this paper we mainly discuss Rota-Baxter operator of Hom-Novikov algebras in demensions two and three. First, we give the definition of Hom-Novikov algebras and Rota-Baxter operator, and we also discuss some basic properties of them. Then we some2-dimensional Hom-Novikov algebras and some 3-dimensional Hom-Novikov algebras. First we give the characteristic matrices of each kind of 2-dimensional Novikov algebras. Then we compute the Rota-Baxter operator of the 2-dimensional Hom-Novikov algebras under a given basis according to the definition of Rota-Baxter operator. We also compute some Rota-Baxter operator of the 3-dimensional Hom-Novikov algebras in a similar way. At last we give the Rota-Baxter operators of the Hom-Novikov algebras in dimension two and the Rota-Baxter operators of the Hom-Novikov algebras in dimension three in a table.