AI 中文总结
本文严格分析AdS/CFT可积性中仿射双代数的奇点结构，提出三点和四点仿射代数，证明其构成一致拟三角李双代数。
AI 中文摘要
在这项工作中，我们重新审视了经典仿射李双代数的解析结构，该代数支撑着AdS/CFT世界sheet散射和一维Hubbard模型的可积结构。早期工作表明，代数关系在评估参数平面上暗示了两个奇点位置，除了构成环代数和仿射代数基础的Laurent多项式的常见展开点无穷大和零之外。我们的目标是使用亚纯函数而非几何级数展开来严格表述这些奇点。因此，我们仔细分析了仿射双代数关系在复平面上的解析结构，关注由于额外奇点可能带来的限制和扩展。作为首次尝试，我们以额外奇点作为Laurent多项式的展开点重新表述了仿射双代数关系，但存在一些不足。我们最终通过为有理情形和三角情形提出三点和四点仿射代数来克服剩余问题。通过基于Krichever-Novikov代数选择黎曼球面上具有多个穿刺点的亚纯函数的合适基，我们证明了多点仿射代数构成一致拟三角李双代数。
英文摘要
In this work, we revisit the analytical structure of the classical affine Lie bialgebra underlying the integrable structures of AdS/CFT worldsheet scattering and the one-dimensional Hubbard model. Earlier work showed that the algebraic relations implied two singularity locations in the evaluation parameter plane besides the common expansion points infinity and zero for Laurent polynomials forming the basis for loop and affine algebras. Our aim is to formulate these singularities rigorously using meromorphic functions rather than by expansion into geometric series. We thus carefully analyse the analytical structure of the affine bialgebra relations in the complex plane paying attention to potential restrictions and extensions due to the additional singularities. As a first attempt with some shortcomings, we reformulate the affine bialgebra relations with the extra singularities serving as the expansion points for Laurent polynomials. We finally overcome the remaining issues by proposing three-point and four-point affine algebras for the rational and trigonometric cases. With a suitable choice of basis for meromorphic functions on the Riemann sphere with several punctures based on Krichever-Novikov algebras, we show that the multi-point affine algebras form consistent quasi-triangular Lie bialgebras.
Comments49 pages
