Hopf algebras and the quantum Yang-Baxter equation. (English. Russian original) Zbl 0588.17015

Sov. Math., Dokl. 32, 256-258 (1985); translation from Dokl. Akad. Nauk SSSR 283, 1060-1064 (1985).
The paper represents an announcement of results relating Hopf algebras to quantum Yang-Baxter equations. Starting from Yang’s classical solution, the author develops - in the context of quantizations of Poisson Hopf co- algebras - more general solutions with the aid of representation theory. If the Lie bi-algebra is a polynomial Lie algebra over a simple Lie algebra these representations are explicitly determined for all simple Lie algebras, including the exceptional ones. It is shown that only for sl(n), solutions to the quantum Yang-Baxter equations exist for all representations. In addition to these representations, the author introduces the class of quantized Kac-Moody Lie algebras which are apparently related to trigonometric solutions of the quantum Yang-Baxter equations.
Reviewer: T.Ratiu


17B65 Infinite-dimensional Lie (super)algebras
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
81T08 Constructive quantum field theory