Poisson Lie Sigma Models

Abstract

A Manin triples (D; g; ~ g) is a bialgebra (g; ~ g which don't intersect each others and a direct sum of this bialgebra D = g ~ g). If the corresponding Lie groups have a Poisson structure, they are called Poisson-Lie groups. A Poisson-Lie sigma models is an action (3.13) calculated by a Poisson vector eld matrix. [3] have deduced the extremal eld which minimize the action of this models, which gives the motion equation (3.19). We calculate here the action and the equations of motion for some 6-dimensionals Manin triples and we give a general formula for each 4-dimensional Manin triples. The 6-dimensional Manin triples are (sl(2; C) sl(2; C) ; sl(2; C); sl(2; C)),(sl(2; C) sl(2; C) ; sl(2; C)); sl(2; C),(sl(2; C); su(2; C); sb(2; C)) and (sl(2; C); sb(2; C); su(2; C)).