Classification of classical twists of the standard Lie bialgebra structure on a loop algebra

JOURNAL OF GEOMETRY AND PHYSICS

卷:164

Article

Abedin, Raschid¹  Maximov, Stepan^2,3 

[1] Univ Paderborn, Inst Math, D-33098 Paderborn, Germany

[2] Chalmers Univ Technol, Dept Math Sci, S-41296 Gothenburg, Sweden

[3] Univ Gothenburg, S-41296 Gothenburg, Sweden

关键词: Lie bialgebra;    Classical Yang-Baxter equation;    Loop algebra;    Kac-Moody algebra;    Manin triple;    Trigonometric r-matrices;   

DOI  :  10.1016/j.geomphys.2021.104149

来源: Elsevier

【 摘 要 】

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. (C) 2021 The Authors. Published by Elsevier B.V.

【 授权许可】

Free