New $r$-Matrices for Lie Bialgebra Structures over Polynomials

TL;DR

This paper develops an explicit algorithm to generate r-matrices associated with Lie bialgebra structures over polynomial algebras, extending the classification of such structures for simple complex Lie algebras.

Contribution

It introduces a new algorithm for constructing r-matrices corresponding to Lie bialgebra structures over polynomial rings, providing explicit tools for their analysis.

Findings

Explicit algorithm for r-matrix construction

Extension of classification to polynomial Lie bialgebras

New examples of r-matrices for simple Lie algebras

Abstract

For a finite dimensional simple complex Lie algebra $\mathfrak{g}$, Lie bialgebra structures on $\mathfrak{g}[[u]]$ and $\mathfrak{g}[u]$ were classified by Montaner, Stolin and Zelmanov. In our paper, we provide an explicit algorithm to produce $r$-matrices which correspond to Lie bialgebra structures over polynomials.