{"title":"Factorization Problems on Rational Loop Groups, and the Poisson Geometry of Yang-Baxter Maps","authors":"Luen-Chau Li","doi":"10.1007/s11040-022-09419-4","DOIUrl":null,"url":null,"abstract":"<div><p>The study of set-theoretic solutions of the Yang-Baxter equation, also known as Yang-Baxter maps, is historically a meeting ground for various areas of mathematics and mathematical physics. In this work, we study factorization problems on rational loop groups, which give rise to Yang-Baxter maps on various geometrical objects. We also study the symplectic and Poisson geometry of these Yang-Baxter maps, which we show to be integrable maps in the sense of having natural collections of Poisson commuting integrals. In a special case, the factorization problems we consider are associated with the <i>N</i>-soliton collision process in the <i>n</i>-Manakov system, and in this context we show that the polarization scattering map is a symplectomorphism.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2022-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Physics, Analysis and Geometry","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s11040-022-09419-4","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 2
Abstract
The study of set-theoretic solutions of the Yang-Baxter equation, also known as Yang-Baxter maps, is historically a meeting ground for various areas of mathematics and mathematical physics. In this work, we study factorization problems on rational loop groups, which give rise to Yang-Baxter maps on various geometrical objects. We also study the symplectic and Poisson geometry of these Yang-Baxter maps, which we show to be integrable maps in the sense of having natural collections of Poisson commuting integrals. In a special case, the factorization problems we consider are associated with the N-soliton collision process in the n-Manakov system, and in this context we show that the polarization scattering map is a symplectomorphism.
期刊介绍：
MPAG is a peer-reviewed journal organized in sections. Each section is editorially independent and provides a high forum for research articles in the respective areas.
The entire editorial board commits itself to combine the requirements of an accurate and fast refereeing process.
The section on Probability and Statistical Physics focuses on probabilistic models and spatial stochastic processes arising in statistical physics. Examples include: interacting particle systems, non-equilibrium statistical mechanics, integrable probability, random graphs and percolation, critical phenomena and conformal theories. Applications of probability theory and statistical physics to other areas of mathematics, such as analysis (stochastic pde''s), random geometry, combinatorial aspects are also addressed.
The section on Quantum Theory publishes research papers on developments in geometry, probability and analysis that are relevant to quantum theory. Topics that are covered in this section include: classical and algebraic quantum field theories, deformation and geometric quantisation, index theory, Lie algebras and Hopf algebras, non-commutative geometry, spectral theory for quantum systems, disordered quantum systems (Anderson localization, quantum diffusion), many-body quantum physics with applications to condensed matter theory, partial differential equations emerging from quantum theory, quantum lattice systems, topological phases of matter, equilibrium and non-equilibrium quantum statistical mechanics, multiscale analysis, rigorous renormalisation group.
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