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
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
