Extending Structures: Fundamentals and Applications, 1st Edition (Hardback) book cover

Extending Structures

Fundamentals and Applications, 1st Edition

By Ana Agore, Gigel Militaru

CRC Press

240 pages

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Hardback: 9780815347842
pub: 2019-10-14
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Description

Extending Structures: Fundamentals and Applications treats the extending structures (ES) problem in the context of groups, Lie/Leibniz algebras, associative algebras and Poisson/Jacobi algebras. This concisely written monograph offers the reader an incursion into the extending structures problem which provides a common ground for studying both the extension problem and the factorization problem.

Features

  • Provides a unified approach to the extension problem and the factorization problem
  • Introduces the classifying complements problem as a sort of converse of the factorization problem and in the case of groups it leads to a theoretical formula for computing the number of types of isomorphisms of all groups of finite order which arises from a minimal set of data
  • Describes a way of classifying a certain class of finite Lie/Leibniz/Poison/Jacobi/associative algebras etc. using flag structures
  • Introduces new (non)abelian cohomological objects for all the aforementioned categories
  • As an application to the approach used for dealing with the classification part of the ES problem, the Galois groups associated with extensions of Lie algebras and associative algebras are described

Table of Contents

Introduction

Generalities: basic notions and notation

1 Extending structures: the group case

1.1 Crossed product and bicrossed product of groups

1.2 Group extending structures and unified products

1.3 Classifying complements

1.4 Examples: applications to the structure of finite groups

2 Leibniz algebras

2.1 Unified products for Leibniz algebras

2.2 Flag extending structures of Leibniz algebras. Examples

2.3 Special cases of unified products for Leibniz algebras

2.4 Classifying complements for extensions of Leibniz algebras

2.5 Itˆo’s theorem for Leibniz algebras

3 Lie algebras

3.1 Unified products for Lie algebras

3.2 Flag extending structures. Examples

3.3 Special cases of unified products for Lie algebras

3.4 Matched pair deformations and the factorization index for Lie

algebras: the case of perfect Lie algebras

3.5 Matched pair deformations and the factorization index for Lie

algebras: the case of non-perfect Lie algebras

3.6 Application: Galois groups and group actions on Lie algebras

4 Associative algebras

4.1 Unified products for algebras

4.2 Flag and supersolvable algebras. Examples

4.3 Special cases of unified products for algebras

4.4 The Galois group of algebra extensions

4.5 Classifying complements for associative algebras

5 Jacobi and Poisson algebras

5.1 (Bi)modules, integrals and Frobenius Jacobi algebras

5.2 Unified products for Jacobi algebras

5.3 Flag Jacobi algebras. Examples

5.4 Classifying complements for Poisson algebras

Bibliography

Index

About the Authors

Ana Agore is a senior researcher at the Institute of Mathematics of the Romanian Academy, Romania. Her research interests include Hopf algebras and quantum groups, category theory and (non)associative algebras.

Gigel Militaru is a professor at the University of Bucharest, Romania. His primary research interests are non-commutative algebra, non-associative (Lie, Leibniz, Jacobi/Poisson) algebras, Hopf algebras and quantum groups.

About the Series

Chapman & Hall/CRC Monographs and Research Notes in Mathematics

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Subject Categories

    BISAC Subject Codes/Headings:
    MAT000000
    MATHEMATICS / General