Double Lie algebras, semidirect product, and integrable systems

TL;DR

This paper explores integrable systems on double Lie algebras by constructing a semidirect product with the τ-representation, enabling the use of AKS theory and Lie bialgebra structures for analysis.

Contribution

It introduces a method to establish an Ad-invariant bilinear form on double Lie algebras via semidirect products, facilitating integrable systems analysis.

Findings

Existence of a natural Ad-invariant bilinear form on the semidirect product.

Application of AKS theory to double Lie algebras.

Connection to Lie bialgebras and Poisson-Lie groups.

Abstract

We study integrable systems on double Lie algebras in absence of Ad-invariant bilinear form by passing to the semidirect product with the $\tau $-representation. We show that in this stage a natural Ad-invariant bilinear form does exist, allowing for a straightforward application of the AKS theory, and giving rise to Manin triple structure, thus bringing the problem to the realm of Lie bialgebras and Poisson-Lie groups.