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Polyadic Algebraic Structures

Steven Duplij

Description

The book is devoted to the thorough study of polyadic (higher arity) algebraic structures, which has a long history, starting from 19th century. The main idea was to take a single set, closed under one binary operation, and to "generalize" it by increasing the arity of the operation, called a polyadic operation. Until now, a general approach to polyadic concrete many-set algebraic structures was absent. We propose to investigate algebraic structures in the "concrete way" and provide consequent "polyadization" of each operation, starting from group-like structures and finishing with the Hopf algebra structures. Polyadic analogs of homomorphisms which change arity, heteromorphisms, are introduced and applied for constructing unusual representations, multiactions, matrix representations and polyadic analogs of direct product. We provide the polyadic generalization of the Yang-Baxter equation, find its constant solutions, and introduce polyadic tensor categories.

Suitable for university students of advanced level algebra courses and mathematical physics courses.

Key Features

  • Provides a general, unified approach
  • Widens readers perspective of the possibilities to develop standard algebraic structures
  • Provides the new kind of homomorphisms changing the arity, heteromorphisms, are introduced and applied for construction of new representations, multiactions and matrix representations
  • Presents applications of "polyadization" approach to concrete algebraic structures


About Editors

Steven Duplij (Stepan Douplii) is a theoretical and mathematical physicist from the University of Münster, Germany. Dr Duplij is the editor-compiler of "Concise Encyclopaedia of Supersymmetry" (2005, Springer), and is the author of more than a hundred scientific publications and several books. His scientific directions include supersymmetry and quantum groups, advanced algebraic structures, gravity and nonlinear electrodynamics, constrained systems and quantum computing.

Table of Contents

Contents

Preface

Acknowledgements

About the Author

Symbols

Introduction

Bibliography

Main ideas and new constructions

One-set polyadic algebraic structures

One-set algebraic structures and Hosszu-Gluskin theorem

Representations and heteromorphisms

Polyadic semigroups and higher regularity

Polyadic rings, fields and integer numbers

Two-sets polyadic algebraic structures

Polyadic algebras and deformations

Polyadic inner spaces and operators

Medial deformation of n-ary algebras

Membership deformations and obscure n-ary algebras

Polyadic quantum groups

Polyadic Hopf algebras

Solutions to higher braid equations

Polyadic categories

Polyadic tensor categories

Bibliography

Bibliographic

Hardback ISBN: 9780750326469

Ebook ISBN: 9780750326483

DOI: 10.1088/978-0-7503-2648-3

Publisher: Institute of Physics Publishing

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