arXiv - MATH - Quantum Algebra最新文献_第3页

Central Hopf Monads and Braided Commutative Algebras 中心霍普夫单元与编织交换代数

arXiv - MATH - Quantum Algebra Pub Date : 2024-09-03 DOI: arxiv-2409.01918

Noelia Bortolussi, Adriana Mejía Castaño, Martín Mombelli

{"title":"Central Hopf Monads and Braided Commutative Algebras","authors":"Noelia Bortolussi, Adriana Mejía Castaño, Martín Mombelli","doi":"arxiv-2409.01918","DOIUrl":"https://doi.org/arxiv-2409.01918","url":null,"abstract":"Let $ V$ be a braided tensor category and $ C$ a tensor category equipped\u0000with a braided tensor functor $G:Vto Z(C)$. For any exact indecomposable\u0000$C$-module category $M$, we explicitly construct a right adjoint of the action\u0000functor $rho:Z^V(C)to C^*_{M}$ afforded by $M$. Here $Z^V(C)$ is the\u0000M\"uger's centralizer of the subcategory $G(V)$ inside the center $Z^V(C)$,\u0000also known as the relative center. The construction is parallel to the one\u0000presented by K. Shimizu, but using instead the relative coend end. This\u0000adjunction turns out to be monadic, thus inducing Hopf monads $T_{V}: Cto C$,\u0000such that there is a monoidal equivalence of categories $ C_{T_{V}}simeq\u0000Z^V(C).$ If $bar{rho}: C^*_{ M}to Z^V(C)$ is the right adjoint of $rho,$\u0000then $bar{rho}(Id_{M})$ is the braided commutative algebra constructed in [R.\u0000Laugwitz and C. Walton. Braided commutative algebras over quantized enveloping\u0000algebras, Transform. Groups 26(3) (2021), 957--993]. As a consequence of our\u0000construction of these algebras, in terms of the right adjoint to $rho$, we can\u0000provide a recipe to compute them when $C=Rep(H# T)$ is the category of\u0000finite-dimensional representations of a finite-dimensional Hopf algebra $H# T$\u0000obtained by bosonization, and choosing an arbitrary $Rep(H# T)$-module\u0000category $M$. We show an explicit example in the case of Taft algebras.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142225241","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Plethysm Stability of Schur's $Q$-functions 舒尔 Q$ 函数的 Plethysm 稳定性

arXiv - MATH - Quantum Algebra Pub Date : 2024-09-02 DOI: arxiv-2409.01479

John Graf, Naihuan Jing

引用次数: 0

Generalised 6j symbols over the category of $G$-graded vector spaces $G$等级向量空间类别上的广义6j符号

arXiv - MATH - Quantum Algebra Pub Date : 2024-08-31 DOI: arxiv-2409.09055

Fabio Lischka

{"title":"Generalised 6j symbols over the category of $G$-graded vector spaces","authors":"Fabio Lischka","doi":"arxiv-2409.09055","DOIUrl":"https://doi.org/arxiv-2409.09055","url":null,"abstract":"Any choice of a spherical fusion category defines an invariant of oriented\u0000closed 3-manifolds, which is computed by choosing a triangulation of the\u0000manifold and considering a state sum model that assigns a 6j symbol to every\u0000tetrahedron in this triangulation. This approach has been generalized to\u0000oriented closed 3-manifolds with defect data by Meusburger. In a recent paper,\u0000she constructed a family of invariants for such manifolds parametrised by the\u0000choice of certain spherical fusion categories, bimodule categories, finite\u0000bimodule functors and module natural transformations. Meusburger defined\u0000generalised 6j symbols for these objects, and introduces a state sum model that\u0000assigns a generalised 6j symbol to every tetrahedron in the triangulation of a\u0000manifold with defect data, where the type of 6j symbol used depends on what\u0000defect data occur within the tetrahedron. The present work provides non-trivial\u0000examples of suitable bimodule categories, bimodule functors and module natural\u0000transformation, all over categories of $G$-graded vector spaces. Our main\u0000result is the description of module functors in terms of matrices, which allows\u0000us to classify these functors when $G$ is a finite cyclic group. Furthermore,\u0000we calculate the generalised 6j symbols for categories of $G$-graded vector\u0000spaces, (bi-)module categories over such categories and (bi-)module functors.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"54 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253468","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Miraculous cancellations and the quantum Frobenius for $SL_3$ skein modules SL_3$绺裂模块的神奇抵消和量子弗罗本尼斯

arXiv - MATH - Quantum Algebra Pub Date : 2024-08-31 DOI: arxiv-2409.00351

Vijay Higgins

引用次数: 0

Higher categories of push-pull spans, II: Matrix factorizations 推拉跨度的更高类别，II：矩阵因式分解

arXiv - MATH - Quantum Algebra Pub Date : 2024-08-30 DOI: arxiv-2409.00219

Lorenzo Riva

引用次数: 0

On $mathbb{Z}/2mathbb{Z}$ permutation gauging 关于 $mathbb{Z}/2mathbb{Z}$ 周期测量

arXiv - MATH - Quantum Algebra Pub Date : 2024-08-30 DOI: arxiv-2408.17195

Zhengwei Liu, Yuze Ruan