Tensor K-Matrices for Quantum Symmetric Pairs

IF 2.2 1区 物理与天体物理 Q1 PHYSICS, MATHEMATICAL
Andrea Appel, Bart Vlaar
{"title":"Tensor K-Matrices for Quantum Symmetric Pairs","authors":"Andrea Appel,&nbsp;Bart Vlaar","doi":"10.1007/s00220-025-05241-5","DOIUrl":null,"url":null,"abstract":"<div><p>Let <span>\\({{\\mathfrak {g}}}\\)</span> be a symmetrizable Kac–Moody algebra, <span>\\(U_q({{\\mathfrak {g}}})\\)</span> its quantum group, and <span>\\(U_q({\\mathfrak {k}})\\subset U_q({{\\mathfrak {g}}})\\)</span> a quantum symmetric pair subalgebra determined by a Lie algebra automorphism <span>\\(\\theta \\)</span>. We introduce a category <span>\\(\\mathcal {W}_{\\theta }\\)</span> of <i>weight</i> <span>\\(U_q({\\mathfrak {k}})\\)</span>-modules, which is acted on by the category of weight <span>\\(U_q({{\\mathfrak {g}}})\\)</span>-modules via tensor products. We construct a universal tensor K-matrix <span>\\({{\\mathbb {K}}} \\)</span> (that is, a solution of a reflection equation) in a completion of <span>\\(U_q({\\mathfrak {k}})\\otimes U_q({{\\mathfrak {g}}})\\)</span>. This yields a natural operator on any tensor product <span>\\(M\\otimes V\\)</span>, where <span>\\(M\\in \\mathcal {W}_{\\theta }\\)</span> and <span>\\(V\\in {{\\mathcal {O}}}_\\theta \\)</span>, <i>i.e.</i>, <i>V</i> is a <span>\\(U_q({{\\mathfrak {g}}})\\)</span>-module in category <span>\\({{\\mathcal {O}}}\\)</span> satisfying an integrability property determined by <span>\\(\\theta \\)</span>. Canonically, <span>\\(\\mathcal {W}_{\\theta }\\)</span> is equipped with a structure of a bimodule category over <span>\\({{\\mathcal {O}}}_\\theta \\)</span> and the action of <span>\\({{\\mathbb {K}}} \\)</span> is encoded by a new categorical structure, which we call a <i>boundary</i> structure on <span>\\(\\mathcal {W}_{\\theta }\\)</span>. This generalizes a result of Kolb which describes a braided module structure on finite-dimensional <span>\\(U_q({\\mathfrak {k}})\\)</span>-modules when <span>\\({{\\mathfrak {g}}}\\)</span> is finite-dimensional. We also consider our construction in the case of the category <span>\\({{\\mathcal {C}}}\\)</span> of finite-dimensional modules of a quantum affine algebra, providing the most comprehensive universal framework to date for large families of solutions of parameter-dependent reflection equations. In this case the tensor K-matrix gives rise to a formal Laurent series with a well-defined action on tensor products of any module in <span>\\(\\mathcal {W}_{\\theta }\\)</span> and any module in <span>\\({{\\mathcal {C}}}\\)</span>. This series can be normalized to an operator-valued rational function, which we call trigonometric tensor K-matrix, if both factors in the tensor product are in <span>\\({{\\mathcal {C}}}\\)</span>.</p></div>","PeriodicalId":522,"journal":{"name":"Communications in Mathematical Physics","volume":"406 5","pages":""},"PeriodicalIF":2.2000,"publicationDate":"2025-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00220-025-05241-5.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Communications in Mathematical Physics","FirstCategoryId":"101","ListUrlMain":"https://link.springer.com/article/10.1007/s00220-025-05241-5","RegionNum":1,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
引用次数: 0

Abstract

Let \({{\mathfrak {g}}}\) be a symmetrizable Kac–Moody algebra, \(U_q({{\mathfrak {g}}})\) its quantum group, and \(U_q({\mathfrak {k}})\subset U_q({{\mathfrak {g}}})\) a quantum symmetric pair subalgebra determined by a Lie algebra automorphism \(\theta \). We introduce a category \(\mathcal {W}_{\theta }\) of weight \(U_q({\mathfrak {k}})\)-modules, which is acted on by the category of weight \(U_q({{\mathfrak {g}}})\)-modules via tensor products. We construct a universal tensor K-matrix \({{\mathbb {K}}} \) (that is, a solution of a reflection equation) in a completion of \(U_q({\mathfrak {k}})\otimes U_q({{\mathfrak {g}}})\). This yields a natural operator on any tensor product \(M\otimes V\), where \(M\in \mathcal {W}_{\theta }\) and \(V\in {{\mathcal {O}}}_\theta \), i.e., V is a \(U_q({{\mathfrak {g}}})\)-module in category \({{\mathcal {O}}}\) satisfying an integrability property determined by \(\theta \). Canonically, \(\mathcal {W}_{\theta }\) is equipped with a structure of a bimodule category over \({{\mathcal {O}}}_\theta \) and the action of \({{\mathbb {K}}} \) is encoded by a new categorical structure, which we call a boundary structure on \(\mathcal {W}_{\theta }\). This generalizes a result of Kolb which describes a braided module structure on finite-dimensional \(U_q({\mathfrak {k}})\)-modules when \({{\mathfrak {g}}}\) is finite-dimensional. We also consider our construction in the case of the category \({{\mathcal {C}}}\) of finite-dimensional modules of a quantum affine algebra, providing the most comprehensive universal framework to date for large families of solutions of parameter-dependent reflection equations. In this case the tensor K-matrix gives rise to a formal Laurent series with a well-defined action on tensor products of any module in \(\mathcal {W}_{\theta }\) and any module in \({{\mathcal {C}}}\). This series can be normalized to an operator-valued rational function, which we call trigonometric tensor K-matrix, if both factors in the tensor product are in \({{\mathcal {C}}}\).

求助全文
约1分钟内获得全文 求助全文
来源期刊
Communications in Mathematical Physics
Communications in Mathematical Physics 物理-物理:数学物理
CiteScore
4.70
自引率
8.30%
发文量
226
审稿时长
3-6 weeks
期刊介绍: The mission of Communications in Mathematical Physics is to offer a high forum for works which are motivated by the vision and the challenges of modern physics and which at the same time meet the highest mathematical standards.
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信