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In this note, we summarize the theory of twisted standard Lie bialgebra structures on loop algebras. We explain how these structures can be defined using trigonometric solutions to the classical Yang-Baxter equation. The trigonometric solutions were completely classified by Belavin and Drinfeld in terms of the so-called BD quadruples. The geometrization of trigonometric solutions through Manin triples allows to pullback the holomorphic equivalences of trigonometric solutions to equivalences of twisted standard structures. In this way one obtains a classification of twisted standard structures in terms of the same BD quadruples.

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Twisted Standard Lie Bialgebra Structures on Loop Algebras

  • 会议论文

  • 发表日期：

    2025年

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  • Raschid Abedin,
  • Stepan Maximov

摘要

In this note, we summarize the theory of twisted standard Lie bialgebra structures on loop algebras. We explain how these structures can be defined using trigonometric solutions to the classical Yang-Baxter equation. The trigonometric solutions were completely classified by Belavin and Drinfeld in terms of the so-called BD quadruples. The geometrization of trigonometric solutions through Manin triples allows to pullback the holomorphic equivalences of trigonometric solutions to equivalences of twisted standard structures. In this way one obtains a classification of twisted standard structures in terms of the same BD quadruples.