Steven Duplij. POLYADIC SYSTEMS, REPRESENTATIONS AND QUANTUM GROUPS.

POLYADIC SYSTEMS, REPRESENTATIONS AND QUANTUM GROUPS STEVEN DUPLIJ A BSTRACT . Polyadic systems and their representations are reviewed and a classification of general polyadic systems is presented. A new multiplace generalization of associativity preserving homomor- phisms, a ’heteromorphism’ which connects polyadic systems having unequal arities, is introduced via an explicit formula, together with related definitions for multiplace representations and multiac- tions. 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 gen- eralizations of quantum groups and the Yang-Baxter equation are presented. C ONTENTS 1. I NTRODUCTION 2 2. P RELIMINARIES 3 3. S PECIAL ELEMENTS AND PROPERTIES OF POLYADIC SYSTEMS 6 4. H OMOMORPHISMS OF POLYADIC SYSTEMS 11 5. M ULTIPLACE MAPPINGS OF POLYADIC SYSTEMS AND HETEROMORPHISMS 13 6. A SSOCIATIVITY QUIVERS AND HETEROMORPHISMS 18 7. M ULTIPLACE REPRESENTATIONS OF POLYADIC SYSTEMS 24 8. M ULTIACTIONS AND G - SPACES 29 9. R EGULAR MULTIACTIONS 31 10. M ULTIPLACE REPRESENTATIONS OF TERNARY GROUPS 33 11. M ATRIX REPRESENTATIONS OF TERNARY GROUPS 37 12. T ERNARY ALGEBRAS AND H OPF ALGEBRAS 40 13. T ERNARY QUANTUM GROUPS 44 14. C ONCLUSIONS 47 R EFERENCES 47 Date : 20 May 2012. 2010 Mathematics Subject Classification. 16T05, 16T25, 17A42, 20N15, 20F29, 20G05, 20G42, 57T05. Published : Journal of Kharkov National University, (ser. Nuclei, Particles and Fields), Vol. 1017, 3(55) (2012) 28-59. 1