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A manifestly Morita-invariant construction of Turaev-Viro invariants 图拉耶夫-维罗不变式的明显莫里塔不变构造
arXiv - MATH - Quantum Algebra Pub Date : 2024-07-13 DOI: arxiv-2407.10018
Jürgen Fuchs, César Galindo, David Jaklitsch, Christoph Schweigert
{"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}
引用次数: 0
Excision for Spaces of Admissible Skeins 对可接受的骨架空间进行切除
arXiv - MATH - Quantum Algebra Pub Date : 2024-07-12 DOI: arxiv-2407.09302
Ingo Runkel, Christoph Schweigert, Ying Hong Tham
{"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}
引用次数: 0
Non-invertible symmetries in finite group gauge theory 有限群规理论中的不可逆对称性
arXiv - MATH - Quantum Algebra Pub Date : 2024-07-10 DOI: arxiv-2407.07964
Clay Cordova, Davi B. Costa, Po-Shen Hsin
{"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}
引用次数: 0
Biracks: a notational proposal and applications Biracks：符号建议和应用
arXiv - MATH - Quantum Algebra Pub Date : 2024-07-10 DOI: arxiv-2407.07650
Marco A. Farinati
{"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}
引用次数: 0
A matrix solution to any polygon equation 任何多边形方程的矩阵解
arXiv - MATH - Quantum Algebra Pub Date : 2024-07-09 DOI: arxiv-2407.07131
Zheyan Wan
{"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}
引用次数: 0
Hidden structures behind ambient symmetries of the Maurer-Cartan equation 毛勒-卡尔坦方程环境对称性背后的隐藏结构
arXiv - MATH - Quantum Algebra Pub Date : 2024-07-09 DOI: arxiv-2407.06589
Vladimir Dotsenko, Sergey Shadrin
{"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}
引用次数: 0
On the finite generation of the cohomology of abelian extensions of Hopf algebras 论霍普夫代数无性扩展同调的有限生成
arXiv - MATH - Quantum Algebra Pub Date : 2024-07-08 DOI: arxiv-2407.05881
Nicolás Andruskiewitsch, Sonia Natale
{"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}
引用次数: 0
Differentials on Forested and Hairy Graph Complexes with Dishonest Hairs 有不诚实毛发的森林图和毛发图复合物上的差分
arXiv - MATH - Quantum Algebra Pub Date : 2024-07-07 DOI: arxiv-2407.05326
Nicolas Grunder
{"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}
引用次数: 0
Integrability of the Inozemtsev spin chain 伊诺泽姆采夫自旋链的积分性
arXiv - MATH - Quantum Algebra Pub Date : 2024-07-03 DOI: arxiv-2407.03276
Oleg Chalykh
{"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}
引用次数: 0
Automorphisms, cohomology and extensions of symmetric quandles 对称广域的自形、同调与扩展
arXiv - MATH - Quantum Algebra Pub Date : 2024-07-03 DOI: arxiv-2407.02971
Biswadeep Karmakar, Deepanshi Saraf, Mahender Singh
{"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}
引用次数: 0
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