Non-coboundary Poisson-Lie structures on the book group

TL;DR

This paper classifies all Poisson-Lie structures on a specific 3D Lie group, highlighting the prevalence of non-coboundary structures and their applications to integrable systems and quantum group quantization.

Contribution

It provides a complete classification of Poisson-Lie structures on the book group, emphasizing non-coboundary cases and their relevance to Poisson dynamics and quantization.

Findings

Most Poisson-Lie structures on the book group are non-coboundary.

Two q-deformed Poisson algebras of sl(2,R) are identified.

A quadratic Poisson structure underpins integrability of 3D Lotka-Volterra equations.

Abstract

All possible Poisson-Lie (PL) structures on the 3D real Lie group generated by a dilation and two commuting translations are obtained. Its classification is fully performed by relating these PL groups with the corresponding Lie bialgebra structures on the corresponding "book" Lie algebra. By construction, all these Poisson structures are quadratic Poisson-Hopf algebras for which the group multiplication is a Poisson map. In contrast to the case of simple Lie groups, it turns out that most of the PL structures on the book group are non-coboundary ones. Moreover, from the viewpoint of Poisson dynamics, the most interesting PL book structures are just some of these non-coboundaries, which are explicitly analysed. In particular, we show that the two different q-deformed Poisson versions of the sl(2,R) algebra appear as two distinguished cases in this classification, as well as the quadratic Poisson structure that underlies the integrability of a large class of 3D Lotka-Volterra equations. Finally, the quantization problem for these PL groups is sketched.