Algebraic geometry of Lie bialgebras defined by solutions of the classical Yang-Baxter equation

TL;DR

This paper explores the algebraic geometry underlying infinite-dimensional Lie bialgebras derived from solutions of the classical Yang-Baxter equation, revealing their geometric origins and providing concrete examples.

Contribution

It introduces an algebro-geometric framework for understanding trigonometric solutions of the classical Yang-Baxter equation as twists of standard Lie bialgebras.

Findings

Trigonometric solutions are twists of standard Lie bialgebras.

Any such solution corresponds to an algebro-geometric datum.

Concrete examples illustrate the theory.

Abstract

This paper is devoted to algebro-geometric study of infinite dimensional Lie bialgebras, which arise from solutions of the classical Yang-Baxter equation. We regard trigonometric solutions of this equation as twists of the standard Lie bialgebra cobracket on an appropriate affine Lie algebra and work out the corresponding theory of Manin triples, putting it into an algebro-geometric context. As a consequence of this approach, we prove that any trigonometric solution of the classical Yang-Baxter equation arises from an appropriate algebro-geometric datum. The developed theory is illustrated by some concrete examples.