1st Edition

# Leibniz Algebras Structure and Classification

324 Pages 6 B/W Illustrations
by Chapman & Hall

323 Pages 6 B/W Illustrations
by Chapman & Hall

323 Pages 6 B/W Illustrations
by Chapman & Hall

Also available as eBook on:

Leibniz Algebras: Structure and Classification is designed to introduce the reader to the theory of Leibniz algebras.

Leibniz algebra is the 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 thorough an examination as it deserved immediately after its introduction. Later, the same algebras were introduced in 1993 by Jean-Louis Loday , 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 the more 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 appear now.  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.

Features:

• Provides a systematic exposition of the theory of Leibniz algebras and recent results on Leibniz algebras
• Suitable for final year bachelor's students, master's students and PhD students going into 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

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

### Biography

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 Technology MARA (UiTM), 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.

"Leibniz algebras have been a main focus of researchers for the past few decades. The authors, Ayupov, Omirov and Rakhimov, are principal developers of this theory. As such, they are able to provide a complete picture, from foundations to recent advances. They do so in a highly motivating treatise. Their writing is both illuminating and cutting edge. Whether wanting to get into the subject or see the most recent important results, this book is the place to go for it. It will be a must for all people working in the area."

— Professor Ernest Stitzinger, NC State University

"This reference book is a self-contained exposition of the state of the art in the theory of Leibniz algebras. It develops the classical theory of Leibniz algebras as an extension of the classical one for Lie algebras. [. . . .] The present book can be used as a textbook for graduate students or as a reference for researchers in Leibniz algebras and neighboring areas."

— Professor Antonio J. Calderon Martin, Universidad de Cádiz

"This book provides an excellent and detailed exposition on the structural theory of Leibniz algebras and a very clear description of algebraic and geometric classification techniques for Leibniz algebras."

—Professor José Manuel Casas Mirás, Universidade de Vigo

"This new book is an essential resource for scientists in the area of non-associative algebras. It contains clear explanations and useful mathematical techniques in this area. Moreover, this book is the first Leibniz algebra book. This book is going to act as a reference book for PhD students or young researchers."

—Professor Luisa Camacho Santana, Universidad de Sevilla

"The theory of Leibniz algebras is one of the most actively developing areas of modern algebra, and until now there were no books devoted to the structural description of Leibniz algebras. Leibniz Algebras: Structure and Classification fills that gap, and provides an introduction to the theory of Leibniz algebras. This book is an original, and the first one in its field. It is well organized and methodically written. The book could be of great interest for students specializing in Algebra."

—Professor Farrukh Mukhamedov, United Arab Emirates University