Lagrangian Subalgebras and Quasi-trigonometric r-Matrices

It was proved by Montaner and Zelmanov that up to classical twisting Lie bialgebra structures on ${\mathfrak{g}u]}$ fall into four classes. Here ${\mathfrak{g}}$ is a simple complex finite-dimensional Lie algebra. It turns out that classical twists within one of these four classes are in a one-to-one correspondence with the so-called quasi-trigonometric solutions of the classical Yang–Baxter equation. In this paper we give a complete list of the quasi-trigonometric solutions in terms of sub-diagrams of the certain Dynkin diagrams related to ${\mathfrak{g}}$ . We also explain how to quantize the corresponding Lie bialgebra structures.