A triple construction for Lie bialgebras

Pacific Journal of Mathematics, 221 (2005), no. 2, 281-301, arXiv:math.QA/0312507

We study the triple of a quasitriangular Lie bialgebra as a natural extension of the Drinfel'd double. The triple is itself a quasitriangular Lie bialgebra. We prove several results about the algebraic structure of the triple, analogous to known results for the double. Among them, we prove that in the factorisable case the triple is isomorphic to a twisting of g \oplus g \oplus g by a certain cocycle. We also consider real forms of the triple and the triangular case.