Triangular Structures on Flat Lie Algebras

We study a large class of exact Lie bialgebras arising from noncommutative deformations of Poisson-Lie groups endowed with a left invariant Riemannian metric. We call these structures triangular metaflat Lie bialgebras. We show that given the metaflatness geometrical condition, these exact bialgebra structures arise necessarily from a solution of the classical Yang-Baxter equation. Moreover, the dual Lie bialgebra is also metaflat constituting an important kind of symmetry.