General Theory of Leibniz Algebras

This book discusses many interesting results have been obtained in Leibniz algebras over the past two decades. The authors not only summarize recent results and methods successfully used in Leibniz algebras, but also show new prospective horizons. Any mathematical theories have a number of natural problems that arise in the process of its development, and these problems quite often have analogues in other areas such as differential geometry, homological algebra, classical algebraic topology, noncommutative geometry, etc. With this in mind the authors describe the general structure of Leibniz algebras that have already been discovered. This approach allows readers to see which parts of the theory should be developed further and also shows the significant differences of Leibniz algebras from Lie algebras. Recent results that constitute the naturally evolving general theory of the subject are then explored.

Elementary Properties and Connections.- Structure of Cyclic Leibniz Algebras.- Leibniz Algebras whose Proper Sub-algebras are Lie Algebras.- Leibniz Algebras whose Proper Sub-algebras are Ideals.- Leibniz Algebras in which the Relation “to be an ideal” is Transitive.- Leibniz Algebras whose Proper Sub-algebras are either Ideals or Contra-ideals.- Influence of Ideals and Self-idealizing Sub-algebras on the Structure of Leibniz Algebras.- Influence of Anti-commutativity on the Structure of Leibniz Algebras.- Analogue of Schur's Theorem and its Generalizations.

Leonid A. Kurdachenko is a Distinguished Professor in the Department of Algebra and Geometry at Oles Honchar Dnipro National University, Ukraine. He is a prominent leading researcher in algebra and has authored more than 200 peer reviewed journal articles and more than a dozen books dedicated to contemporary topics in modern algebra published by Wiley, CRC, Word Scientific, and others. He was an invited speaker for numerous international algebra conferences and served as visiting professor in many universities in different countries including Germany, Greece, Italy, Spain, USA, and others.