Polyadization of algebraic structures

Abstract

A generalization of the semisimplicity concept for polyadic algebraic structures is proposed. If semisimple structures can be presented in the block-diagonal matrix form (Wedderburn decomposition), a general form of polyadic structures is given by block-shift matrices. We combine these forms in a special way to get a general shape of semisimple nonderived polyadic structures.We then introduce the polyadization concept (a "polyadic constructor") according to which one can construct a nonderived polyadic algebraic structure of any arity from a given binary structure. The polyadization of supersymmetric structures is also discussed. The "deformation" by shifts of operations on the direct power of binary structures is defined and used to obtain a nonderived polyadic multiplication. Illustrative concrete examples for the new constructions are given. CONTENTS 1. INTRODUCTION 2. PRELIMINARIES 3. POLYADIC SEMISIMPLICITY 3.1. Simple polyadic structures 3.2. Semisimple polyadic structures 4. POLYADIZATION CONCEPT 4.1. Polyadization of binary algebraic structures 4.2. Concrete examples of the polyadization procedure 4.2.1. Polyadization of GL p2, Cq 4.2.2. Polyadization of SO p2, Rq 4.3. "Deformation" of binary operations by shifts 4.4. Polyadization of binary supergroups 4.4.1.