Yangians, quantum groups and solutions of the quantum dynamical Yang-Baxter equation*) D. ARNAUDON, L. FRAPPAT, E. RAGOUCY Laboratoire d'Annecy-le-Vieux de Physique Th~orique LAPTH CNRS, UMR 5108, associ~e ~ l'Universit~ de Savoie LAPP, BP 110, 1:-74941 Annecy-le-Vieux Cedex, France J. AVAN Laboratoire de Physique Thd~orique et Hautes /~nerg/es, CNRS, UMR 7589, Universit$s Paris VI/VII, France M. RossI Dept. of Math., Heriot-Watt University, Edinburgh, United Kingdom Received 22 August 2001 We construct Drinfel'd twists that define deformed Hopf structures. In particular, we obtain deformed double Yangians and dynamical double Yangians. 1 Introduction In the work presented here, we show how several dynamical or deformed struc- tures (of either Yangian or quantum group type) can be interpreted as Drinfel'd twist transformations of known Hopf algebras. This therefore shows the Quasi- Triangular Quasi-Hopf algebra (QTQHA) structure of these objects. For the details in the construction of these twists, we mainly refer to [1-6]. This construction completes the scheme presented in [7, 8] where the algebraic structures were linked mostly by limits of their parameters. All the examples we show here are based on the algebra s/(2), but in most cases the construction has been shown to extend to more general cases. After an introduction to (quasi)-Hopf algebras and to Drinfel'd twists, we will show how to obtain dynamical quantum groups. Then we will construct deformed double Yangians, and finally dynamical double Yangians. 2 Quasi-triangular (quasi)-Hopf algebras (QTQHA) A Quasi-Hopf algebra (A, A, ~, e, S) is an algebra endowed with a coproduct A, a counit e and an antipode, together with an element ~ E A ®3 such that (id ® A)A = ~(A ® id)A~5 -1, (id @ e) o A = (e ® id) o A = id, (1) (id @ id @ ZI)4~ • (A @ id @ id)~ = (1 @ 4~). (id @ Zl @ id)~. (q5 @ 1), (2) (id ® E ® id)~ = 1. (3) • ) Presented by D. Arnaudon at the 10th Colloquium on Quantum Groups: "Quantum Groups and Integrable Systems", Prague, 21-23 June, 2001. 1254 Czechoslovak Journal of Physics, Vol. 51 (2001), No. 12