Triangular structures on flat Lie algebras

TL;DR

This paper classifies a broad class of exact Lie bialgebras called metaflat Lie bialgebras, showing they originate from solutions to the classical Yang-Baxter equation and exhibit dual flatness, enriching the understanding of Lie bialgebra structures.

Contribution

It provides a complete classification of exact metaflat Lie bialgebras and links their structure to solutions of the classical Yang-Baxter equation, highlighting duality properties.

Findings

Metaflatness implies origin from classical Yang-Baxter solutions.

Dual bialgebras are also flat and metaflat.

Complete classification of these structures is achieved.

Abstract

In this work we study a large class of exact Lie bialgebras arising from noncommutative deformations of Poisson-Lie groups endowed with a left invariant Riemannian metric. We call these structures \emph{exact metaflat Lie bialgebras}. We give a complete classification of these structures. We show that given the metaflatness geometrical condition, these exact bialgebra structures arise necessarily from a nontrivial solution of the classical Yang-Baxter equation. Moreover, the dual Lie bialgebra is also flat and metaflat constituting an important kind of symmetry.