Leibniz Algebras: Structure and Classification, 1st Edition (Hardback) book cover

Leibniz Algebras

Structure and Classification, 1st Edition

By Shavkat Ayupov, Bakhrom Omirov, Isamiddin Rakhimov

CRC Press

320 pages | 6 B/W Illus.

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Hardback: 9780367354817
pub: 2019-11-26
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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.


  • Provides a systematic exposition of the theory of Leibniz algebras and recent results on Leibniz algebras
  • Suitable for final year bachelor students, master's degree students and PhD students who are going to do research in the structural theory of finite-dimensional algebras, particularly, Lie and Leibniz algebras
  • Covers important and more general parts of the structural theory of Leibniz algebras that are not addressed in other texts

Table of Contents

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

About the Authors

Shavkat Ayupov

Shavkat Ayupov is the Director of the Institute of Mathematics of Uzbekistan Academy of Sciences, Professor at the National University of Uzbekistan and, Doctor of Sciences in Physics and Mathematics and fellow of TWAS (The Academy of Sciences for the Developing World). His research interests include Functional analysis, Algebra and Topology. Sh. Ayupov is recipient of several international titles and awards and his main research deals with the study of Operator algebras, Jordan and Lie structures on von Neumann algebras, Derivations and automorphisms on operator algebras, Structure theory of Leibniz algebras and Superalgebras and other non-associative algebras. Sh. Ayupov is an organizer of CIMPA research school workshops and International Conferences on Nonassociative Algebras and Applications and on Operator Algebras and Quantum Probability. He has spoken at numerous plenary sessions and has been invited to numerous talks in various international conferences and workshops. Sh. Ayupov is also the Chief Editor of the Uzbek Mathematical Journal and has authored 4 textbooks, 5 monographs, and more than 150 research papers which have been published in several international journals.

Bakhrom Omirov

Bakhrom Omirov is Professor at the National University of Uzbekistan, Doctor of Sciences in Physics and Mathematics, and research fellow at the Institute of Mathematics of the Uzbekistan Academy of Sciences. His research interests include Non associative algebras, Lie (super)algebras, Leibniz (super)algebras, n-Leibniz algebras, structure theory of algebras, p-adic analysis, evolution algebras and their applications. B. Omirov has received several local and international awards. He is currently leading several international research projects and collaborations and has been invited as a speaker to many workshops and universities abroad. He is also authored more than 100 research papers in high impact international journals.

Isamiddin Rakhimov

Isamiddin Rakhimov is Professor at the Universiti Putra, Malaysia, Doctor of Sciences in Physics and Mathematics, and research fellow at the Institute for Mathematical Research (INSPEM), Universiti Putra Malaysia. His research interests focus on the theory of finite-dimensional algebras and its applications. He has been invited to give speak in few international conferences and workshops. I. Rakhimov received his Ph.D. degree in Algebra from the Sankt Petersburg University of Russia. He has organized several international mathematical events and is on the Editorial Board of a few international journals. He has also authored 1 textbook, 2 monographs and more than 70 scientific papers published in international cited journals.

Subject Categories

BISAC Subject Codes/Headings:
MATHEMATICS / Algebra / General
MATHEMATICS / Number Theory