The standard Lie bialgebra structure on an affine Kac–Moody algebra induces a Lie bialgebra structure on the underlying loop algebra and its parabolic subalgebras. In this paper we classify all classical twists of the induced Lie bialgebra structures in terms of Belavin–Drinfeld quadruples up to a natural notion of equivalence. To obtain this classification we first show that the induced bialgebra structures are defined by certain solutions of the classical Yang–Baxter equation (CYBE) with two parameters. Then, using the algebro–geometric theory of CYBE, based on torsion free coherent sheaves, we reduce the problem to the well-known classification of trigonometric solutions given by Belavin and Drinfeld. The classification of twists in the case of parabolic subalgebras allows us to answer recently posed open questions regarding the so-called quasi-trigonometric solutions of CYBE.

仿射Kac-Moody代数上的标准Lie双代数结构在底层循环代数及其抛物子代数上诱导出Lie双代数结构。在本文中，我们根据Belavin–Drinfeld四倍体到自然对等概念，将诱导Lie代数结构的所有经典扭曲分类。为了获得这种分类，我们首先证明了诱导的代数结构是由具有两个参数的经典Yang–Baxter方程（CYBE）的某些解定义的。然后，基于CYBE的代数几何理论，基于无扭相干绳轮，我们将问题简化为Belavin和Drinfeld给出的著名三角解决方案分类。