On Some Lie Bialgebra Structures on Polynomial Algebras and their Quantization

Abstract

We study classical twists of Lie bialgebra structures on the polynomial current algebra \({\mathfrak{g}[u]}\), where \({\mathfrak{g}}\) 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 \({\mathfrak{g}}\). We give the 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 \({\mathfrak{sl}(n)}\).