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A Lie group is called orthogonal if it carries a bi-invariant pseudoRiemannian metric. Oscillator Lie groups constitutes a subclass of the class oforthogonal Lie groups. In this paper, we determine the Lie bialgebra structuresand the solutions of the classical Yang-Baxter equation on a generic class ofoscillator Lie groups. On the other hand, we show that any solution of theclassical Yang-Baxter equation on an orthogonal Lie group induces a metric inthe dual Lie groups associated to this solution. This metric is geodesicallycomplete if and only if the dual are unimodular. More generally, we show thatany solution of the generalized Yang-Baxter equation on an orthogonal Lie groupdetermines a left invariant locally symmetric pseudo-Riemannian metric on thecorresponding dual Lie groups. Applying this result to oscillator Lie groups weget a large class of solvable Lie groups with flat left invariant Lorentzianmetric.

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