On the General Classification of Lie Bialgebra Structures over Polynomials

TL;DR

This paper extends the classification of Lie bialgebra structures over polynomial algebras g[u], focusing on those associated with arbitrary simple roots, building on previous work that addressed maximal roots.

Contribution

It provides a comprehensive classification of Lie bialgebra structures over g[u] for all simple roots, expanding the understanding beyond maximal root cases.

Findings

Classified Lie bialgebras for arbitrary simple roots

Connected structures to extended Dynkin diagrams

Extended previous maximal root classifications

Abstract

The present paper is a continuation of [5], where Lie bialgebra structures on g[u] were studied. These structures fall into different classes labelled by the vertices of the extended Dynkin diagram of g. In [5] the Lie bialgebras corresponding to the maximal root were classified. In the present article, we investigate the Lie bialgebras corresponding to an arbitrary simple root.