Classification of quasi-trigonometric solutions of the classical Yang-Baxter equation

Details Classification of quasi-trigonometric solutions of the classical Yang-Baxter equation Classification of quasi-trigonometric solutions of the classical Yang-Baxter equation

2008-06-12

arxiv.org/abs/0806.2053v1

Mathematics

Quantum Algebra

Scientific paper

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.

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