We study classical twists of Lie bialgebra structures on the polynomial
current algebra g[u], where g is a simple complex
finite-dimensional Lie algebra. We focus on the structures induced by the
so-called quasi-trigonometric solutions of the classical Yang-Baxter equation.
It turns out that quasi-trigonometric r-matrices fall into classes labelled
by the vertices of the extended Dynkin diagram of g. We give
complete classification of quasi-trigonometric r-matrices belonging to
multiplicity free simple roots (which have coefficient 1 in the decomposition
of the maximal root). We quantize solutions corresponding to the first root of
sl(n).Comment: 41 pages, LATE