Leibniz Algebras: Structure and Classification

Description

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 Contents Preface Acknowledgments Authors Notations And Conventions 1 Introduction 1.1 Algebras 1.2 Associative Algebras 1.3 Lie Algebras 1.4 Loday Algebras 2 Structure Of Leibniz Algebras 2.1 Some Properties Of Leibniz Algebras 2.2 Nilpotent And Solvable Leibniz Algebras 2.3 On Levi’s Theorem For Leibniz Algebras 2.4 Semisimple Leibniz Algebras 2.5 On Cartan Subalgebras Of Finite-dimensional Leibniz Algebras 2.6 Some Properties Of Weight Spaces Of Leibniz Algebras And Cartan’s Criterion Of Solvability 3 Classification Problem In Low Dimensions 3.1 Algebraic Classification Of Low-dimensional Leibniz Algebras 3.2 Application 3.3 Low-dimensional Nilpotent Leibniz Algebras 3.4 Four-dimensional Solvable Leibniz Algebras 3.5 Rigidity Of Lie And Leibniz Algebras 3.6 Leibniz Cohomology Computations 3.7 Rigid Leibniz Algebra With Non-trivial 3.8 Lie-rigidity Versus Leibniz-rigidity 4 On Some Classes Of Leibniz Algebras 4.1 Irreducible Components Of A Subclass Of Nilpotent Leibniz Algebras 4.2 Classification Of Naturally Graded Complex Filiform Leibniz Algebras 4.3 Classification Of Some Solvable Leibniz Algebras 5 Isomorphism Criteria For Filiform Leibniz Algebras 5.1 On Base Changes In Complex Filiform Leibniz Algebras 5.2 Criteria Of Isomorphisms Of Complex Filiform Non-lie Leibniz Algebras 6 Classification Of Filiform Leibniz Algebras In Low Dimensions 6.1 Isomorphism Criteria For First Class 6.2 Classification Of First Class In Low Dimensions 6.3 Isomorphism Criteria For Second Class 6.4 Classification Of Second Class In Low Dimensions 6.5 Simplifications And Notations In Third Class 6.6 Classification In Dimension Five 6.7 Classification In Dimension Six Appendix A Linear Algebra A.1 Vector Spaces And Subspaces A.2 Linear Transformations A.3 Dual Vector Space A.4 Tensor Products A.5 Tensor Algebra A.6 Matrix Of A Linear Transformation A.7 Jordan Normal Form Appendix B Elements of Representation Theory Appendix C Zariski Topology C.1 Action Of A Group C.2 Algebraic Groups Bibliography Index