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Drinfeld centers of fusion categories arising from generalized Haagerup subfactors 广义Haagerup子因子引起的融合范畴的Drinfeld中心
IF 1.1 2区 数学
Quantum Topology Pub Date : 2015-01-30 DOI: 10.4171/qt/167
Pinhas Grossman, Masaki Izumi
{"title":"Drinfeld centers of fusion categories arising from generalized Haagerup subfactors","authors":"Pinhas Grossman, Masaki Izumi","doi":"10.4171/qt/167","DOIUrl":"https://doi.org/10.4171/qt/167","url":null,"abstract":"We consider generalized Haagerup categories such that $1 oplus X$ admits a $Q$-system for every non-invertible simple object $X$. We show that in such a category, the group of order two invertible objects has size at most four. We describe the simple objects of the Drinfeld center and give partial formulas for the modular data. We compute the remaining corner of the modular data for several examples and make conjectures about the general case. We also consider several types of equivariantizations and de-equivariantizations of generalized Haagerup categories and describe their Drinfeld centers. \u0000In particular, we compute the modular data for the Drinfeld centers of a number of examples of fusion categories arising in the classification of small-index subfactors: the Asaeda-Haagerup subfactor; the $3^{Z_4} $ and $3^{Z_2 times Z_2} $ subfactors; the $2D2$ subfactor; and the $4442$ subfactor. \u0000The results suggest the possibility of several new infinite families of quadratic categories. A description and generalization of the modular data associated to these families in terms of pairs of metric groups is taken up in the accompanying paper cite{GI19_2}.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"94 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2015-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80723564","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 4
$L$-space surgeries on links $L$-链接上的空间手术
IF 1.1 2区 数学
Quantum Topology Pub Date : 2014-08-30 DOI: 10.4171/QT/96
Yajing Liu
{"title":"$L$-space surgeries on links","authors":"Yajing Liu","doi":"10.4171/QT/96","DOIUrl":"https://doi.org/10.4171/QT/96","url":null,"abstract":"An $L$-space link is a link in $S^3$ on which all large surgeries are $L$-spaces. In this paper, we initiate a general study of the definitions, properties, and examples of $L$-space links. In particular, we find many hyperbolic $L$-space links, including some chain links and two-bridge links; from them, we obtain many hyperbolic $L$-spaces by integral surgeries, including the Weeks manifold. We give bounds on the ranks of the link Floer homology of $L$-space links and on the coefficients in the multi-variable Alexander polynomials. We also describe the Floer homology of surgeries on any $L$-space link using the link surgery formula of Ozsv'{a}th and Manolescu. As applications, we compute the graded Heegaard Floer homology of surgeries on 2-component $L$-space links in terms of only the Alexander polynomial and the surgery framing, and give a fast algorithm to classify $L$-space surgeries among them.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"91 1","pages":"505-570"},"PeriodicalIF":1.1,"publicationDate":"2014-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90266242","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 22
A basis theorem for the affine oriented Brauer category and its cyclotomic quotients 仿射取向Brauer范畴及其分环商的一个基定理
IF 1.1 2区 数学
Quantum Topology Pub Date : 2014-04-25 DOI: 10.4171/QT/87
Jonathan Brundan, J. Comes, David Nash, Andrew Reynolds
{"title":"A basis theorem for the affine oriented Brauer category and its cyclotomic quotients","authors":"Jonathan Brundan, J. Comes, David Nash, Andrew Reynolds","doi":"10.4171/QT/87","DOIUrl":"https://doi.org/10.4171/QT/87","url":null,"abstract":"The affine oriented Brauer category is a monoidal category obtained from the oriented Brauer category (= the free symmetric monoidal category generated by a single object and its dual) by adjoining a polynomial generator subject to appropriate relations. In this article, we prove a basis theorem for the morphism spaces in this category, as well as for all of its cyclotomic quotients.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"38 1","pages":"75-112"},"PeriodicalIF":1.1,"publicationDate":"2014-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82423024","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 43
Fourier transform for quantum D-modules via the punctured torus mapping class group 通过穿孔环面映射类群的量子d模的傅里叶变换
IF 1.1 2区 数学
Quantum Topology Pub Date : 2014-03-07 DOI: 10.4171/QT/92
Adrien Brochier, D. Jordan
{"title":"Fourier transform for quantum D-modules via the punctured torus mapping class group","authors":"Adrien Brochier, D. Jordan","doi":"10.4171/QT/92","DOIUrl":"https://doi.org/10.4171/QT/92","url":null,"abstract":"We construct a certain cross product of two copies of the braided dual $tilde H$ of a quasitriangular Hopf algebra $H$, which we call the elliptic double $E_H$, and which we use to construct representations of the punctured elliptic braid group extending the well-known representations of the planar braid group attached to $H$. We show that the elliptic double is the universal source of such representations. We recover the representations of the punctured torus braid group obtained in arXiv:0805.2766, and hence construct a homomorphism to the Heisenberg double $D_H$, which is an isomorphism if $H$ is factorizable. \u0000The universal property of $E_H$ endows it with an action by algebra automorphisms of the mapping class group $widetilde{SL_2(mathbb{Z})}$ of the punctured torus. One such automorphism we call the quantum Fourier transform; we show that when $H=U_q(mathfrak{g})$, the quantum Fourier transform degenerates to the classical Fourier transform on $D(mathfrak{g})$ as $qto 1$.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"6 1","pages":"361-379"},"PeriodicalIF":1.1,"publicationDate":"2014-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80406597","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 13
A categorification of quantum $mathfrak{sl}_3$ projectors and the $mathfrak{sl}_3$ Reshetikhin–Turaev invariant of tangles 量子$mathfrak{sl}_3$投影的分类和缠结$mathfrak{sl}_3$ Reshetikhin-Turaev不变量
IF 1.1 2区 数学
Quantum Topology Pub Date : 2014-03-03 DOI: 10.4171/QT/46
David E. V. Rose
{"title":"A categorification of quantum $mathfrak{sl}_3$ projectors and the $mathfrak{sl}_3$ Reshetikhin–Turaev invariant of tangles","authors":"David E. V. Rose","doi":"10.4171/QT/46","DOIUrl":"https://doi.org/10.4171/QT/46","url":null,"abstract":"","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"1 1","pages":"1-59"},"PeriodicalIF":1.1,"publicationDate":"2014-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75582578","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 9
|ZKup|=|ZHenn|2 for lens spaces 镜头空间|ZKup|=|ZHenn|2
IF 1.1 2区 数学
Quantum Topology Pub Date : 2013-11-15 DOI: 10.4171/QT/44
Liang-Chu Chang, Zhenghan Wang
{"title":"|ZKup|=|ZHenn|2 for lens spaces","authors":"Liang-Chu Chang, Zhenghan Wang","doi":"10.4171/QT/44","DOIUrl":"https://doi.org/10.4171/QT/44","url":null,"abstract":"","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"98 1","pages":"411-445"},"PeriodicalIF":1.1,"publicationDate":"2013-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78375041","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
An odd categorification of $U_q (mathfrak{sl}_2)$ $U_q (mathfrak{sl}_2)$的奇分类
IF 1.1 2区 数学
Quantum Topology Pub Date : 2013-07-30 DOI: 10.4171/QT/78
Alexander P. Ellis, Aaron D. Lauda
{"title":"An odd categorification of $U_q (mathfrak{sl}_2)$","authors":"Alexander P. Ellis, Aaron D. Lauda","doi":"10.4171/QT/78","DOIUrl":"https://doi.org/10.4171/QT/78","url":null,"abstract":"We define a 2-category that categorifies the covering Kac-Moody algebra for sl(2) introduced by Clark and Wang. This categorification forms the structure of a super-2-category as formulated by Kang, Kashiwara, and Oh. The super-2-category structure introduces a (Z x Z_2)-grading giving its Grothendieck group the structure of a free module over the group algebra of Z x Z_2. By specializing the Z_2-action to +1 or to -1, the construction specializes to an \"odd\" categorification of sl(2) and to a supercategorification of osp(1|2), respectively.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"18 1","pages":"329-433"},"PeriodicalIF":1.1,"publicationDate":"2013-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84467243","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 15
On the SL(2,ℂ) quantum 6j-symbols and their relation to the hyperbolic volume SL(2，)量子6j符号及其与双曲体积的关系
IF 1.1 2区 数学
Quantum Topology Pub Date : 2013-07-02 DOI: 10.4171/QT/41
F. Costantino, J. Murakami
{"title":"On the SL(2,ℂ) quantum 6j-symbols and their relation to the hyperbolic volume","authors":"F. Costantino, J. Murakami","doi":"10.4171/QT/41","DOIUrl":"https://doi.org/10.4171/QT/41","url":null,"abstract":"We generalize the colored Alexander invariant of knots to an invariant of graphs and we construct a face model for this invariant by using the corresponding 6j -symbols, which come from the non-integral representations of the quantum group Uq.sl2/. We call it the SL.2; C/-quantum 6j -symbols, and show their relation to the hyperbolic volume of a truncated tetrahedron. Mathematics Subject Classification (2010). Primary 46L37; Secondary 46L54, 82B99.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"12 1","pages":"303-351"},"PeriodicalIF":1.1,"publicationDate":"2013-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77160897","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 20
Cyclic extensions of fusion categories via the Brauer-Picard groupoid 基于Brauer-Picard群的融合范畴的循环扩展
IF 1.1 2区 数学
Quantum Topology Pub Date : 2012-11-27 DOI: 10.4171/QT/64
Pinhas Grossman, D. Jordan, Noah Snyder
{"title":"Cyclic extensions of fusion categories via the Brauer-Picard groupoid","authors":"Pinhas Grossman, D. Jordan, Noah Snyder","doi":"10.4171/QT/64","DOIUrl":"https://doi.org/10.4171/QT/64","url":null,"abstract":"We construct a long exact sequence computing the obstruction space, pi_1(BrPic(C_0)), to G-graded extensions of a fusion category C_0. The other terms in the sequence can be computed directly from the fusion ring of C_0. We apply our result to several examples coming from small index subfactors, thereby constructing several new fusion categories as G-extensions. The most striking of these is a Z/2Z-extension of one of the Asaeda-Haagerup fusion categories, which is one of only two known 3-supertransitive fusion categories outside the ADE series. \u0000In another direction, we show that our long exact sequence appears in exactly the way one expects: it is part of a long exact sequence of homotopy groups associated to a naturally occuring fibration. This motivates our constructions, and gives another example of the increasing interplay between fusion categories and algebraic topology.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"7 1","pages":"313-331"},"PeriodicalIF":1.1,"publicationDate":"2012-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79906901","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 19
Orbifold completion of defect bicategories 缺陷分类的轨道完成
IF 1.1 2区 数学
Quantum Topology Pub Date : 2012-10-23 DOI: 10.4171/QT/76
Nils Carqueville, I. Runkel
{"title":"Orbifold completion of defect bicategories","authors":"Nils Carqueville, I. Runkel","doi":"10.4171/QT/76","DOIUrl":"https://doi.org/10.4171/QT/76","url":null,"abstract":"Orbifolds of two-dimensional quantum field theories have a natural formulation in terms of defects or domain walls. This perspective allows for a rich generalisation of the orbifolding procedure, which we study in detail for the case of topological field theories. Namely, a TFT with defects gives rise to a pivotal bicategory of \"worldsheet phases\" and defects between them. We develop a general framework which takes such a bicategory B as input and returns its \"orbifold completion\" B_orb. The completion satisfies the natural properties B subset B_orb and (B_orb)_orb = B_orb, and it gives rise to various new equivalences and nondegeneracy results. When applied to TFTs, the objects in B_orb correspond to generalised orbifolds of the theories in B. In the example of Landau-Ginzburg models we recover and unify conventional equivariant matrix factorisations, prove when and how (generalised) orbifolds again produce open/closed TFTs, and give nontrivial examples of new orbifold equivalences.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"141 1","pages":"203-279"},"PeriodicalIF":1.1,"publicationDate":"2012-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80297423","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 113
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