{"title":"Yangian Deformations of $$\\mathcal {S}$$ -Commutative Quantum Vertex Algebras and Bethe Subalgebras","authors":"","doi":"10.1007/s00031-023-09837-w","DOIUrl":null,"url":null,"abstract":"<h3>Abstract</h3> <p>We construct a new class of quantum vertex algebras associated with the normalized Yang <em>R</em>-matrix. They are obtained as Yangian deformations of certain <span> <span>\\(\\mathcal {S}\\)</span> </span>-commutative quantum vertex algebras, and their <span> <span>\\(\\mathcal {S}\\)</span> </span>-locality takes the form of a single <em>RTT</em>-relation. We establish some preliminary results on their representation theory and then further investigate their braiding map. In particular, we show that its fixed points are closely related with Bethe subalgebras in the Yangian quantization of the Poisson algebra <span> <span>\\(\\mathcal {O}(\\mathfrak {gl}_N((z^{-1})))\\)</span> </span>, which were recently introduced by Krylov and Rybnikov. Finally, we extend this construction of commutative families to the case of trigonometric <em>R</em>-matrix of type <em>A</em>.</p>","PeriodicalId":49423,"journal":{"name":"Transformation Groups","volume":"36 1","pages":""},"PeriodicalIF":0.4000,"publicationDate":"2024-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Transformation Groups","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00031-023-09837-w","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We construct a new class of quantum vertex algebras associated with the normalized Yang R-matrix. They are obtained as Yangian deformations of certain \(\mathcal {S}\)-commutative quantum vertex algebras, and their \(\mathcal {S}\)-locality takes the form of a single RTT-relation. We establish some preliminary results on their representation theory and then further investigate their braiding map. In particular, we show that its fixed points are closely related with Bethe subalgebras in the Yangian quantization of the Poisson algebra \(\mathcal {O}(\mathfrak {gl}_N((z^{-1})))\), which were recently introduced by Krylov and Rybnikov. Finally, we extend this construction of commutative families to the case of trigonometric R-matrix of type A.
期刊介绍：
Transformation Groups will only accept research articles containing new results, complete Proofs, and an abstract. Topics include: Lie groups and Lie algebras; Lie transformation groups and holomorphic transformation groups; Algebraic groups; Invariant theory; Geometry and topology of homogeneous spaces; Discrete subgroups of Lie groups; Quantum groups and enveloping algebras; Group aspects of conformal field theory; Kac-Moody groups and algebras; Lie supergroups and superalgebras.
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