Non-coboundary Poisson-Lie structures on the book group (Q2880352)

A systematic construction and classification of the Poisson-Lie structures on the real 3D book group is presented.NEWLINENEWLINEIt is well-known that Poisson-Lie structures on a connected and simply connected Lie group \(G\) are in correspondence with the Lie bialgebra structures on the Lie algebra \(\mathfrak{g} = \text{Lie}(G)\). If the cobracket \(\delta\) of \(\mathfrak{g}\) is induced by a solution of the classical Yang-Baxter equation, the bialgebra \((\mathfrak{g},\delta)\) is called coboundary. For semisimple Lie algebras, all Lie bialgebras structures are coboundary, but for non-semisimple Lie algebras, non-coboundary structures do appear.NEWLINENEWLINEIn these notes, all possible Poisson-Lie structures on the (solvable) book group \(G\) are obtained by analysing quadratic Poisson structures that are compatible with the coproduct on \(\mathcal{C}^{\infty}(G)\) induced from the group multiplication. The classification yields nine equivalence classes of Poisson-Lie structures and it turns out that seven of them are non-coboundaries. The Poisson dynamics of these non-coboundary structures is then studied and it is shown that two different \(q\)-deformations of \(\mathfrak{sl}_{2}\) appear as two of these equivalence classes.NEWLINENEWLINEThe paper ends with a final section that contains several remarks and future research proposals.