Ternary Hopf algebras. (English) Zbl 1092.17004
Nikitin, A.G. (ed.) et al., Proceedings of the fourth international conference on symmetry in nonlinear mathematical physics, Kyïv, Ukraine, July 9–15, 2001. Part 2. Dedicated to the 200th anniversary of M. Ostrohrads’kyi. Kyïv: Institute of Mathematics of NAS of Ukraine (ISBN 966-02-2486-9). Proc. Inst. Math. Natl. Acad. Sci. Ukr., Math. Appl. 43(2), 439-448 (2002).
Properties of ternary semigroups, groups and algebras are discussed in this paper. The author shows that there exist three types of ternary units. Three types of co-associativity and three kinds of co-units are given here. A classification of ternary co-algebras with respect to the property “to be derived” is presented. Ternary Hopf algebras with skew and strong antipodes are defined and concrete examples are presented. A ternary analogue of deformation, a ternary analogue of quasi-triangular Hopf algebras, a ternary “pairing” of three Hopf algebras and ternary abstract quantum Yang-Baxter equation are studied in this work. After obvious changes most of the constructions introduced in the paper are valid for the \(n\)-ary case.
For the entire collection see [Zbl 0989.00035].
For the entire collection see [Zbl 0989.00035].
Reviewer: M. O. Nesterenko (Kyïv)
MSC:
| 17A40 | Ternary compositions |
| 17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
| 81R50 | Quantum groups and related algebraic methods applied to problems in quantum theory |
| 16T05 | Hopf algebras and their applications |
