Polyadic systems, representations and quantum groups

TL;DR

This paper reviews polyadic systems, introduces heteromorphisms connecting systems of different arities, and explores their representations, including matrix forms, and extends concepts like quantum groups to ternary structures.

Contribution

It introduces heteromorphisms for polyadic systems with different arities and develops their representations, including matrix forms and generalizations of quantum groups.

Findings

Defined heteromorphisms connecting polyadic systems of different arities

Presented matrix representations for ternary groups

Extended quantum group concepts to ternary structures

Abstract

Polyadic systems and their representations are reviewed and a classification of general polyadic systems is presented. A new multiplace generalization of associativity preserving homomorphisms, a 'heteromorphism' which connects polyadic systems having unequal arities, is introduced via an explicit formula, together with related definitions for multiplace representations and multiactions. Concrete examples of matrix representations for some ternary groups are then reviewed. Ternary algebras and Hopf algebras are defined, and their properties are studied. At the end some ternary generalizations of quantum groups and the Yang-Baxter equation are presented.