On Some Lie Bialgebra Structures on Polynomial Algebras and their Quantization

TL;DR

This paper classifies certain Lie bialgebra structures on polynomial algebras related to simple Lie algebras, focusing on quasi-trigonometric solutions of the classical Yang-Baxter equation and their quantization.

Contribution

It provides a complete classification of quasi-trigonometric r-matrices associated with multiplicity free roots and quantizes solutions for the first root of sl(n).

Findings

Classified quasi-trigonometric r-matrices by Dynkin diagram vertices.

Quantized solutions for the first root of sl(n).

Connected Lie bialgebra structures to extended Dynkin diagram vertices.

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 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 $\mathfrak{sl}(n)$.