Jürgen Fuchs, César Galindo, David Jaklitsch, Christoph Schweigert
We present a state sum construction that assigns a scalar to a skeleton in a closed oriented three-dimensional manifold. The input datum is the pivotal bicategory $mathbf{Mod}^{mathrm{sph}}(mathcal{A})$ of spherical module categories over a spherical fusion category $mathcal{A}$. The interplay of algebraic structures in this pivotal bicategory with moves of skeleta ensures that our state sum is independent of the skeleton on the manifold. We show that the bicategorical invariant recovers the value of the standard Turaev-Viro invariant associated to $mathcal{A}$, thereby proving the independence of the Turaev-Viro invariant under pivotal Morita equivalence without recurring to the Reshetikhin-Turaev construction. A key ingredient for the construction is the evaluation of graphs on the sphere with labels in $mathbf{Mod}^{mathrm{sph}}(mathcal{A})$ that we develop in this article. A central tool are Nakayama-twisted traces on pivotal bimodule categories which we study beyond semisimplicity.
我们提出了一种状态总和构造，它可以为封闭定向三维流形中的骨架分配一个标量。输入数据是球形融合类别 $mathcal{A}$ 上的球形模类的枢轴二分类 $mathbf{Mod}^{mathrm{sph}}(mathcal{A})$。这个关键二分类中的代数结构与骨架移动的相互作用，确保了我们的状态和与它们的骨架无关。我们证明，二分类不变量恢复了与 $mathcal{A}$ 相关联的标准图拉耶夫-维罗不变量的值，从而证明了图拉耶夫-维罗不变量在枢轴莫里塔等价性下的独立性，而无需重复雷谢提金-图拉耶夫的构造。该构造的一个关键要素是本文所发展的$mathbf{Mod}^{mathrm{sph}}(mathcal{A})$中标注的球面上图的评估。本文的核心工具是中山扭曲踪迹（Nakayama-twisted traces on pivotalbimodule categories），我们对其进行了超越半简单性的研究。
{"title":"A manifestly Morita-invariant construction of Turaev-Viro invariants","authors":"Jürgen Fuchs, César Galindo, David Jaklitsch, Christoph Schweigert","doi":"arxiv-2407.10018","DOIUrl":"https://doi.org/arxiv-2407.10018","url":null,"abstract":"We present a state sum construction that assigns a scalar to a skeleton in a\u0000closed oriented three-dimensional manifold. The input datum is the pivotal\u0000bicategory $mathbf{Mod}^{mathrm{sph}}(mathcal{A})$ of spherical module\u0000categories over a spherical fusion category $mathcal{A}$. The interplay of algebraic structures in this pivotal bicategory with moves\u0000of skeleta ensures that our state sum is independent of the skeleton on the\u0000manifold. We show that the bicategorical invariant recovers the value of the\u0000standard Turaev-Viro invariant associated to $mathcal{A}$, thereby proving the\u0000independence of the Turaev-Viro invariant under pivotal Morita equivalence\u0000without recurring to the Reshetikhin-Turaev construction. A key ingredient for the construction is the evaluation of graphs on the\u0000sphere with labels in $mathbf{Mod}^{mathrm{sph}}(mathcal{A})$ that we\u0000develop in this article. A central tool are Nakayama-twisted traces on pivotal\u0000bimodule categories which we study beyond semisimplicity.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"31 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141718013","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}
The skein module for a d-dimensional manifold is a vector space spanned by embedded framed graphs decorated by a category A with suitable extra structure depending on the dimension d, modulo local relations which hold inside d-balls. For a full subcategory S of A, an S-admissible skein module is defined analogously, except that local relations for a given ball may only be applied if outside the ball at least one edge is coloured in S. In this paper we prove that admissible skein modules in any dimension satisfy excision, namely that the skein module of a glued manifold is expressed as a coend over boundary values on the boundary components glued together. We furthermore relate skein modules for different choices of S, apply our result to cylinder categories, and recover the relation to modified traces.
d 维流形的绺裂模块是一个向量空间，它由内嵌的框架图所跨越，框架图由一个类别 A 装饰，类别 A 具有适当的额外结构化，取决于维数 d，并模数化了在 d 球内部成立的局部关系。对于 A 的全子类 S，S-admissible skein 模块的定义与此类似，只是给定球的局部关系只有在球外至少有一条边在 S 中着色的情况下才适用。本文证明了任意维度的 admissible skein 模块满足苛刻条件，即粘合流形的 skein 模块表示为粘合在一起的边界成分上的边界值。我们进一步将不同 S 选择下的矢量模块联系起来，将我们的结果应用于圆柱范畴，并恢复了与修正迹线的关系。
{"title":"Excision for Spaces of Admissible Skeins","authors":"Ingo Runkel, Christoph Schweigert, Ying Hong Tham","doi":"arxiv-2407.09302","DOIUrl":"https://doi.org/arxiv-2407.09302","url":null,"abstract":"The skein module for a d-dimensional manifold is a vector space spanned by\u0000embedded framed graphs decorated by a category A with suitable extra structure\u0000depending on the dimension d, modulo local relations which hold inside d-balls.\u0000For a full subcategory S of A, an S-admissible skein module is defined\u0000analogously, except that local relations for a given ball may only be applied\u0000if outside the ball at least one edge is coloured in S. In this paper we prove that admissible skein modules in any dimension satisfy\u0000excision, namely that the skein module of a glued manifold is expressed as a\u0000coend over boundary values on the boundary components glued together. We\u0000furthermore relate skein modules for different choices of S, apply our result\u0000to cylinder categories, and recover the relation to modified traces.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141718015","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}
We investigate the invertible and non-invertible symmetries of topological finite group gauge theories in general spacetime dimensions, where the gauge group can be Abelian or non-Abelian. We focus in particular on the 0-form symmetry. The gapped domain walls that generate these symmetries are specified by boundary conditions for the gauge fields on either side of the wall. We investigate the fusion rules of these symmetries and their action on other topological defects including the Wilson lines, magnetic fluxes, and gapped boundaries. We illustrate these constructions with various novel examples, including non-invertible electric-magnetic duality symmetry in 3+1d $mathbb{Z}_2$ gauge theory, and non-invertible analogs of electric-magnetic duality symmetry in non-Abelian finite group gauge theories. In particular, we discover topological domain walls that obey Fibonacci fusion rules in 2+1d gauge theory with dihedral gauge group of order 8. We also generalize the Cheshire string defect to analogous defects of general codimensions and gauge groups and show that they form a closed fusion algebra.
{"title":"Non-invertible symmetries in finite group gauge theory","authors":"Clay Cordova, Davi B. Costa, Po-Shen Hsin","doi":"arxiv-2407.07964","DOIUrl":"https://doi.org/arxiv-2407.07964","url":null,"abstract":"We investigate the invertible and non-invertible symmetries of topological\u0000finite group gauge theories in general spacetime dimensions, where the gauge\u0000group can be Abelian or non-Abelian. We focus in particular on the 0-form\u0000symmetry. The gapped domain walls that generate these symmetries are specified\u0000by boundary conditions for the gauge fields on either side of the wall. We\u0000investigate the fusion rules of these symmetries and their action on other\u0000topological defects including the Wilson lines, magnetic fluxes, and gapped\u0000boundaries. We illustrate these constructions with various novel examples,\u0000including non-invertible electric-magnetic duality symmetry in 3+1d\u0000$mathbb{Z}_2$ gauge theory, and non-invertible analogs of electric-magnetic\u0000duality symmetry in non-Abelian finite group gauge theories. In particular, we\u0000discover topological domain walls that obey Fibonacci fusion rules in 2+1d\u0000gauge theory with dihedral gauge group of order 8. We also generalize the\u0000Cheshire string defect to analogous defects of general codimensions and gauge\u0000groups and show that they form a closed fusion algebra.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"81 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141612959","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}
I propose a notation for biracks that includes from the begining the knowledege of the associated (or underlying, or derived) rack structure. Motivated by results of Rump in the involutive case, this notation allows to generalize some results from involutive case to the non necessarily involutive solutions, and also to view some twisting constructions and its relation to the underlying rack structure in a more transparent way. Two applications are given.
{"title":"Biracks: a notational proposal and applications","authors":"Marco A. Farinati","doi":"arxiv-2407.07650","DOIUrl":"https://doi.org/arxiv-2407.07650","url":null,"abstract":"I propose a notation for biracks that includes from the begining the\u0000knowledege of the associated (or underlying, or derived) rack structure.\u0000Motivated by results of Rump in the involutive case, this notation allows to\u0000generalize some results from involutive case to the non necessarily involutive\u0000solutions, and also to view some twisting constructions and its relation to the\u0000underlying rack structure in a more transparent way. Two applications are\u0000given.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141585523","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}
In this article, we construct matrices associated to Pachner $frac{n-1}{2}$-$frac{n-1}{2}$ moves for odd $n$ and matrices associated to Pachner $(frac{n}{2}-1)$-$frac{n}{2}$ moves for even $n$. The entries of these matrices are rational functions of formal variables in a field. We prove that these matrices satisfy the $n$-gon equation for any $n$.
{"title":"A matrix solution to any polygon equation","authors":"Zheyan Wan","doi":"arxiv-2407.07131","DOIUrl":"https://doi.org/arxiv-2407.07131","url":null,"abstract":"In this article, we construct matrices associated to Pachner\u0000$frac{n-1}{2}$-$frac{n-1}{2}$ moves for odd $n$ and matrices associated to\u0000Pachner $(frac{n}{2}-1)$-$frac{n}{2}$ moves for even $n$. The entries of\u0000these matrices are rational functions of formal variables in a field. We prove\u0000that these matrices satisfy the $n$-gon equation for any $n$.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"125 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141585524","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}
For every differential graded Lie algebra $mathfrak{g}$ one can define two different group actions on the Maurer-Cartan elements: the ubiquitous gauge action and the action of $mathrm{Lie}_infty$-isotopies of $mathfrak{g}$, which we call the ambient action. In this note, we explain how the assertion of gauge triviality of a homologically trivial ambient action relates to the calculus of dendriform, Zinbiel, and Rota-Baxter algebras, and to Eulerian idempotents. In particular, we exhibit new relationships between these algebraic structures and the operad of rational functions defined by Loday.
对于每一个微分级联代数 $mathfrak{g}$，我们都可以在毛勒-卡尔坦元素上定义两种不同的群作用：无处不在的量规作用和 $mathrm{Lie}_infty$-isotopies of $mathfrak{g}$的作用，我们称之为环境作用。在本注释中，我们将解释同源琐碎环境作用的几何琐碎性断言是如何与树枝形、津比尔和罗塔-巴克斯特代数以及欧拉幂等的计算相关联的。特别是，我们展示了这些代数结构与洛代定义的有理函数操作数之间的新关系。
{"title":"Hidden structures behind ambient symmetries of the Maurer-Cartan equation","authors":"Vladimir Dotsenko, Sergey Shadrin","doi":"arxiv-2407.06589","DOIUrl":"https://doi.org/arxiv-2407.06589","url":null,"abstract":"For every differential graded Lie algebra $mathfrak{g}$ one can define two\u0000different group actions on the Maurer-Cartan elements: the ubiquitous gauge\u0000action and the action of $mathrm{Lie}_infty$-isotopies of $mathfrak{g}$,\u0000which we call the ambient action. In this note, we explain how the assertion of\u0000gauge triviality of a homologically trivial ambient action relates to the\u0000calculus of dendriform, Zinbiel, and Rota-Baxter algebras, and to Eulerian\u0000idempotents. In particular, we exhibit new relationships between these\u0000algebraic structures and the operad of rational functions defined by Loday.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"29 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141568761","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}
A finite-dimensional Hopf algebra is called quasi-split if it is Morita equivalent to a split abelian extension of Hopf algebras. Combining results of Schauenburg and Negron, it is shown that every quasi-split finite-dimensional Hopf algebra satisfies the finite generation cohomology conjecture of Etingof and Ostrik. This is applied to a family of pointed Hopf algebras in odd characteristic introduced by Angiono, Heckenberger and the first author, proving that they satisfy the aforementioned conjecture.
{"title":"On the finite generation of the cohomology of abelian extensions of Hopf algebras","authors":"Nicolás Andruskiewitsch, Sonia Natale","doi":"arxiv-2407.05881","DOIUrl":"https://doi.org/arxiv-2407.05881","url":null,"abstract":"A finite-dimensional Hopf algebra is called quasi-split if it is Morita\u0000equivalent to a split abelian extension of Hopf algebras. Combining results of\u0000Schauenburg and Negron, it is shown that every quasi-split finite-dimensional\u0000Hopf algebra satisfies the finite generation cohomology conjecture of Etingof\u0000and Ostrik. This is applied to a family of pointed Hopf algebras in odd\u0000characteristic introduced by Angiono, Heckenberger and the first author,\u0000proving that they satisfy the aforementioned conjecture.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"30 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141577399","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}
We study the cohomology of forested graph complexes with ordered and unordered hairs whose cohomology computes the cohomology of a family of groups $Gamma_{g,r}$ that generalize the (outer) automorphism group of free groups. We give examples and a recipe for constructing additional differentials on these complexes. These differentials can be used to construct spectral sequences that start with the cohomology of the standard complexes. We focus on one such sequence that relates cohomology classes of graphs with different numbers of hairs and compute its limit.
{"title":"Differentials on Forested and Hairy Graph Complexes with Dishonest Hairs","authors":"Nicolas Grunder","doi":"arxiv-2407.05326","DOIUrl":"https://doi.org/arxiv-2407.05326","url":null,"abstract":"We study the cohomology of forested graph complexes with ordered and\u0000unordered hairs whose cohomology computes the cohomology of a family of groups\u0000$Gamma_{g,r}$ that generalize the (outer) automorphism group of free groups.\u0000We give examples and a recipe for constructing additional differentials on\u0000these complexes. These differentials can be used to construct spectral\u0000sequences that start with the cohomology of the standard complexes. We focus on\u0000one such sequence that relates cohomology classes of graphs with different\u0000numbers of hairs and compute its limit.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"66 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141568762","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}
We show that the Inozemtsev spin chain is integrable. The conserved quantities (commuting Hamiltonians) are constructed using elliptic Dunkl operators. We also suggest a generalisation.
{"title":"Integrability of the Inozemtsev spin chain","authors":"Oleg Chalykh","doi":"arxiv-2407.03276","DOIUrl":"https://doi.org/arxiv-2407.03276","url":null,"abstract":"We show that the Inozemtsev spin chain is integrable. The conserved\u0000quantities (commuting Hamiltonians) are constructed using elliptic Dunkl\u0000operators. We also suggest a generalisation.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"92 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141549791","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}
It is well-known that the cohomology of symmetric quandles generates robust cocycle invariants for unoriented classical and surface links. Expanding on the recently introduced module-theoretic generalized cohomology for symmetric quandles, we derive a four-term exact sequence that relates 1-cocycles, second cohomology, and a specific group of automorphisms associated with the extensions of symmetric quandles. This exact sequence shows that the obstruction to lifting and extending automorphisms is found in the second symmetric quandle cohomology. Additionally, some general aspects of dynamical cocycles and extensions are discussed.
{"title":"Automorphisms, cohomology and extensions of symmetric quandles","authors":"Biswadeep Karmakar, Deepanshi Saraf, Mahender Singh","doi":"arxiv-2407.02971","DOIUrl":"https://doi.org/arxiv-2407.02971","url":null,"abstract":"It is well-known that the cohomology of symmetric quandles generates robust\u0000cocycle invariants for unoriented classical and surface links. Expanding on the\u0000recently introduced module-theoretic generalized cohomology for symmetric\u0000quandles, we derive a four-term exact sequence that relates 1-cocycles, second\u0000cohomology, and a specific group of automorphisms associated with the\u0000extensions of symmetric quandles. This exact sequence shows that the\u0000obstruction to lifting and extending automorphisms is found in the second\u0000symmetric quandle cohomology. Additionally, some general aspects of dynamical\u0000cocycles and extensions are discussed.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"64 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141549788","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：