Solutions of the Yang-Baxter equations on orthogonal groups : the case of oscillator groups

Abstract: A Lie group is called orthogonal if it carries a bi-invariant pseudo Riemannian metric. Oscillator Lie groups constitutes a subclass of the class of orthogonal Lie groups. In this paper, we determine the Lie bialgebra structures and the solutions of the classical Yang-Baxter equation on a generic class of oscillator Lie groups. On the other hand, we show that any solution of the classical Yang-Baxter equation on an orthogonal Lie group induces a metric in the dual Lie groups associated to this solution. This metric is geodesically complete if and only if the dual are unimodular. More generally, we show that any solution of the generalized Yang-Baxter equation on an orthogonal Lie group determines a left invariant locally symmetric pseudo-Riemannian metric on the corresponding dual Lie groups. Applying this result to oscillator Lie groups we get a large class of solvable Lie groups with flat left invariant Lorentzian metric.