Solutions of the Yang–Baxter equations on quadratic Lie groups: The case of oscillator groups

A Lie group is called quadratic if it carries a bi-invariant semi-Riemannian metric. Oscillator Lie groups constitute a subclass of the class of quadratic 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 algebras. Moreover, we show that any solution of the generalized classical Yang–Baxter equation (resp. classical Yang–Baxter equation) on a quadratic Lie group determines a left invariant locally symmetric (resp. flat) semi-Riemannian metric on the corresponding dual Lie groups.

► A Lie group with a bi-invariant semi-Riemannian metric is called quadratique.
► A solution of the CYBE on a quadratique Lie group defines a flat metric on the dual.
► A generalized solution determines a locally symmetric metric on the dual.
► Lie bi-algebra structures on generic oscillator Lie algebras are determined.