Proceedings of Institute of Mathematics of NAS of Ukraine 2002, Vol. 43, Part 2, 439–448 Ternary Hopf Algebras Steven DUPLIJ Kharkov National University, Kharkov 61001, Ukraine E-mail: Steven.A.Duplij@univer.kharkov.ua http://www-home.univer.kharkov.ua/duplij Properties of ternary semigroups, groups and algebras are briefly reviewed. It is shown that there exist three types of ternary units. A ternary analog of deformation is shortly discussed. Ternary coalgebras are defined in the most general manner, their classification with respect to the property “to be derived” is made. Three types of coassociativity and three kinds of counits are given. Ternary Hopf algebras with skew and strong antipods are defined. Concrete examples of ternary Hopf algebras, including the Sweedler example (which has two ternary generalizations), are presented. A ternary analog of quasitriangular Hopf algebras is constructed, and ternary abstract quantum Yang–Baxter equation (together with its classical counterpart) is obtained. A ternary “pairing” of three Hopf algebras is built. I would like to report about the work done in part together with Andrzej Borowiec and Wieslaw Dudek, and I am grateful to them for fruitful collaboration. Firstly ternary algebraic operations were introduced already in the XIX-th century by A. Cay- ley. As the development of Cayley’s ideas it were considered n-ary generalization of matrices and their determinants [1] and general theory of n-ary algebras [2, 3] and ternary rings [4] (for physical applications in Nambu mechanics, supersymmetry, Yang–Baxter equation, etc. see [5] as surveys). The notion of an n-ary group was introduced in 1928 by W. D¨ ornte [6]. From another side, Hopf algebras [7] and their generalizations [8, 9, 10, 11] play a basic role in the quantum group theory (also see e.g. [12, 13]). We note that the derived ternary Hopf algebras are used as an intermediate tool in obtaining the Drinfeld’s quantum double [14]. Here we first present necessary material on ternary semigroups, groups and algebras [15] in the abstract arrow language. Then using systematic reversing order of arrows [7], we define ternary bialgebras and Hopf algebras, investigate their properties and give some examples 1 . Most of the constructions introduced below are valid for n-ary case as well after obvious changes. A non-empty set G with one ternary operation [ ] : G × G × G → G is called a ternary groupoid and is denoted by (G, [ ]) or ( G, m (3) ) . If on G there exists a binary operation ⊙ (or m (2) ) such that [xyz ]=(x ⊙ y) ⊙ z or m (3) = m (3) der = m (2) ◦  m (2) × id  (1) for all x, y, z ∈ G, then we say that [ ] or m (3) der is derived from ⊙ or m (2) and denote this fact by (G, [ ]) = der(G, ⊙). If [xyz ] = ((x ⊙y) ⊙z ) ⊙b holds for all x, y, z ∈ G and some fixed b ∈ G, then a groupoid (G, [ ] is b-derived from (G, ⊙). In this case we write (G, [ ]) = der b (G, ⊙) [16, 17]. A ternary isotopy is a set of functions f,g,h,w : G → G such that f ([xyz ]) = [g (x) ,h (y) ,w (z )] for all x, y, z ∈ G. If g = h = w = f , then f is ternary isomorphism. A ternary semigroup is (G, [ ]) (or ( G, m (3) ) ) where the operation [ ] (m (3) ) is associative [[xyz ] uv]=[x [yzu] v]=[xy [zuv]] (for all x, y, z, u, v ∈ G) or m (3) ◦  m (3) × id × id  = m (3) ◦  id ×m (3) × id  = m (3) ◦  id × id ×m (3)  (2) 1 Due to the lack of place in the Proceedings we present only important results and constructions omitting most proofs and detailed derivations which will appear elsewhere.