Leibniz Algebras: Structure and Classification is designed to introduce the reader to the theory of Leibniz algebras.
Leibniz algebra is generalization of Lie algebras. These algebras preserve a unique property of Lie algebras that the right multiplication operators are derivations. They first appeared in papers of A.M Blokh in the 1960s, under the name D-algebras, emphasizing their close relationship with derivations. The theory of D-algebras did not get as fulsome an examination as it deserve immediately after its introduction. Later the same algebras were introduced by Jean-Louis Loday in 1993, who called them Leibniz algebras due to the identity they satisfy. The main motivation for the introduction of Leibniz algebras was to study the periodicity phenomena in algebraic K-theory.
Nowadays, the theory of Leibniz algebras is one of actively developing areas of modern algebra. Along with (co)homological, structural and classification results on Leibniz algebras, some papers with various applications of the Leibniz algebras also now appear. However, the focus of this book is mainly on the classification problems of Leibniz algebras. Particularly, the authors propose a method of classification of a subclass of Leibniz algebras based on algebraic invariants. The method is applicable in the Lie algebras case as well.
Chapter 1: Introduction
Chapter 2: Structure of Leibniz Algebra
Chapter 3: Classification Problems in Low Dimensions
Chapter 4: On some Classes of Leibniz Algebra
Chapter 5: Isomorphism Criteria for Filiform Leibniz Algebra
Chapter 6: Classification of Filiform Leibniz Algebra in Low Dimensions