Derivations of low-dimensional Leibniz Algebras

A derivation is a function on an algebra which generalizes certain features of the derivative operator. Specifically, given an algebra A over a ring or field K, an K-derivation is an K-linear map D from A to itself that satisfies Leibniz's law: D(ab)=(Da)b+a(Db). More generally, an K-linear map D of A into an A-module M, satisfying the Leibniz law is also called a derivation. The collection of all K-derivation of A to itself is denoted by Der(A). The collection of K-derivations of A into an A-module M is denoted by Der(A,M). Derivations occur in many different contexts in diverse areas of mathematics. If the algebra A is noncommutative, then the commutator with respect to an element of the algebra A defines a linear endomorphism of A to itself, which is a derivation over K. Furthermore, the K-module Der(A) forms a Lie algebra with respect to Lie bracket defined by the commutator: [D1,D2]=D1 D2 - D2 D1. In this book we deal with the derivations of Leibniz algebras. The Leibniz algebra is a generalization of Lie algebra, so it makes sense to study the problems related to Lie algebras for the class of Leibniz algebras.

Autorentext

Isamiddin S.Rakhimov graduated from the Leningrad State University (now Saint Petersburg State University), Russia in 1979. He received his PhD degree from the same university in 1986 and started his teaching in the Tashkent State University, Uzbekistan. Since 2004 Dr. Rakhimov is an associate professor of the Universiti Putra Malaysia, Malaysia.