Classical r-matrices via semidualisation

TL;DR

This paper explores how double cross sum decompositions of Lie algebras relate to classical r-matrices of their semiduals, with applications to three-dimensional gravity and connections to Bianchi classifications.

Contribution

It provides a new method to derive classical r-matrices for semidual Lie bialgebras using decompositions, especially for Lie algebras from generalized complexification.

Findings

Derived explicit r-matrices for semidual Lie bialgebras.

Connected r-matrices with Bianchi classification of 3D Lie algebras.

Applied framework to local isometry algebras in 3D gravity.

Abstract

We study the interplay between double cross sum decompositions of a given Lie algebra and classical r-matrices for its semidual. For a class of Lie algebras which can be obtained by a process of generalised complexification we derive an expression for classical r-matrices of the semidual Lie bialgebra in terms of the data which determines the decomposition of the original Lie algebra. Applied to the local isometry Lie algebras arising in three-dimensional gravity, decomposition and semidualisation yields the main class of non-trivial r-matrices for the Euclidean and Poincare group in three dimensions. In addition, the construction links the r-matrices with the Bianchi classification of three dimensional real Lie algebras.