Polyadic systems, representations and quantum groups

A review of polyadic systems and their representations is given. The classification of general polyadic systems is done. The multiplace generalization of homomorphisms, preserving associativity, is presented. The multiplace representations and multiactions are defined, concrete examples of matrix representations for some ternary groups are given. The ternary algebras and Hopf algebras are defined, their properties are studied. At the end some ternary generalizations of quantum groups and the Yang-Baxter equation are presented.