Algebraic Geometry of Lie Bialgebras Defined by Solutions of the Classical Yang–Baxter Equation

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.

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

“…In this section we give a brief summary of the results in [7], prove the extension property for formal equivalences between geometric r-matrices (see Theorem 5.5) and, finally, combining this property with the results in [1] on geometrization of σ-trigonometric r-matrices we verify Theorem 4.1.…”

“…also defines the standard Lie bialgebra structure δ σ 0 on L σ . The geometric nature of this Manin triple is revealed in [1]: the sheaves used for construction of σ-trigonometric r-matrices can be viewed as formal gluing of twisted versions of W 0 with L σ ∼ = ∆ over the nodal Weierstraß cubic. ♦…”

“…Therefore, we need sheaves which give rise to the σ-trigonometric r-matrices from (5.13). These were constructed in [1].…”

“…We want to show that ϑ defines a regular equivalence φ on L σ , such that φ(L σ (α,k) ) = L σ ϑ(α,k) for any root (α, k) of L σ . Let φ ′ be the automorphism defined by φ ′ (z ±s i X ± i (1)) := z ±s ϑ(i) X ± ϑ(i) (1). Then it maps the root spaces onto each other in the desired way.…”

“…It is shown in [1] that all σ-trigonometric r-matrices arise as geometric r-matrices from coherent sheaves of Lie algebras on the nodal Weierstraß cubic with section L σ on the set of smooth points. Since holomorphic equivalences are formal, this result and Theorem A give the desired classification: Theorem B.…”