Leibniz algebras : structure and classification
著者
書誌事項
Leibniz algebras : structure and classification
Chapman & Hall/CRC, c2020
- hbk.
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注記
Includes bibliographical references and index
内容説明・目次
内容説明
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
目次
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. 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. 4-DIMENSIONAL SOLVABLE LEIBNIZ ALGEBRAS. 3.5. RIGIDITY OF LIE AND LEIBNIZ ALGEBRAS. 3.6. LEIBNIZ COHOMOLOGY COMPUTATIONS. 3.7. A RIGID LEIBNIZ ALGEBRA WITH NON-TRIVIAL HL^2. 3.8. LIE-RIGIDITY VERSUS LEIBNIZ-RIGIDITY. 4. ON SOME CLASSES OF LEIBNIZ ALGEBRAS. 5. ISOMORPHISM CRITERIA FOR FILIFORM LEIBNIZ ALGEBRAS. 5.1. ON BASE CHANGES IN COMPLEX FILIFORM LEIBNIZ ALGEBRAS. 5.2. A CRITERION OF ISOMORPHISMS OF COMPLEX FILIFORM. NON-LIE LEIBNIZ ALGEBRAS.
6. CLASSIFICATION OF FILIFORM LEIBNIZ ALGEBRAS IN LOW DIMENSIONS. 6.1. ISOMORPHISM CRITERIA FOR THE FIRST CLASS. 6.2. CLASSIFICATION OF THE FIRST CLASS IN LOW DIMENSIONS. 6.3. ISOMORPHISM CRITERIA FOR THE SECOND CLASS. 6.4. CLASSIFICATION OF THE SECOND CLASS IN LOW DIMENSIONS. 6.5. SIMPLIFICATIONS AND NOTATIONS IN THE THIRD CLASS. 6.6. CLASSIFICATION IN DIMENSION FIVE. 6.7 CLASSIFICATION IN DIMENSION SIX.
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