Bisch and Jones suggested the skein theoretic classification of planar algebras and investigated the ones generated by 2-boxes with the second author. In this paper, we consider 3-box generators and classify subfactor planar algebras generated by a non-trivial 3-box satisfying a relation proposed by Thurston. The subfactor planar algebras in the classification are either $E^6$ or the ones from representations of quantum $SU(N)$. We introduce a new method to determine positivity of planar algebras and new techniques to reduce the complexity of computations.
{"title":"Classification of Thurston relation subfactor planar algebras","authors":"Corey Jones, Zhengwei Liu, Yunxiang Ren","doi":"10.4171/qt/126","DOIUrl":"https://doi.org/10.4171/qt/126","url":null,"abstract":"Bisch and Jones suggested the skein theoretic classification of planar algebras and investigated the ones generated by 2-boxes with the second author. In this paper, we consider 3-box generators and classify subfactor planar algebras generated by a non-trivial 3-box satisfying a relation proposed by Thurston. The subfactor planar algebras in the classification are either $E^6$ or the ones from representations of quantum $SU(N)$. We introduce a new method to determine positivity of planar algebras and new techniques to reduce the complexity of computations.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2016-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86797957","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}
We categorify a coideal subalgebra of the quantum group of $mathfrak{sl}_{2r+1}$ by introducing a $2$-category a la Khovanov-Lauda-Rouquier, and show that self-dual indecomposable $1$-morphisms categorify the canonical basis of this algebra. This allows us to define a categorical action of this coideal algebra on the categories of modules over cohomology rings of partial flag varieties and on the BGG category $mathcal{O}$ of type B/C.
{"title":"Categorification of quantum symmetric pairs I","authors":"Huanchen Bao, P. Shan, Weiqiang Wang, Ben Webster","doi":"10.4171/QT/117","DOIUrl":"https://doi.org/10.4171/QT/117","url":null,"abstract":"We categorify a coideal subalgebra of the quantum group of $mathfrak{sl}_{2r+1}$ by introducing a $2$-category a la Khovanov-Lauda-Rouquier, and show that self-dual indecomposable $1$-morphisms categorify the canonical basis of this algebra. This allows us to define a categorical action of this coideal algebra on the categories of modules over cohomology rings of partial flag varieties and on the BGG category $mathcal{O}$ of type B/C.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2016-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83211314","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}
{"title":"Skein and cluster algebras of marked surfaces","authors":"G. Muller","doi":"10.4171/QT/79","DOIUrl":"https://doi.org/10.4171/QT/79","url":null,"abstract":"","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89011928","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}
We give a presentation of the asymptotic expansion of the Kashaev invariant of the 52 knot. As the volume conjecture states, the leading term of the expansion presents the hyperbolic volume and the Chern-Simons invariant of the complement of the 52 knot. Further, we obtain a method to compute the full Poincare asymptotics to all orders of the Kashaev invariant of the 52 knot.
{"title":"On the asymptotic expansion of the Kashaev invariant of the $5_2$ knot","authors":"T. Ohtsuki","doi":"10.4171/QT/83","DOIUrl":"https://doi.org/10.4171/QT/83","url":null,"abstract":"We give a presentation of the asymptotic expansion of the Kashaev invariant of the 52 knot. As the volume conjecture states, the leading term of the expansion presents the hyperbolic volume and the Chern-Simons invariant of the complement of the 52 knot. Further, we obtain a method to compute the full Poincare asymptotics to all orders of the Kashaev invariant of the 52 knot.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86420733","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}
{"title":"The symplectic properties of the PGL($n,mathbb C$)-gluing equations","authors":"S. Garoufalidis, C. Zickert","doi":"10.4171/QT/80","DOIUrl":"https://doi.org/10.4171/QT/80","url":null,"abstract":"","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75380976","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}
In this article we describe an algebraic framework which can be used in three related but different contexts: string topology, symplectic field theory, and Lagrangian Floer theory of higher genus. It turns out that the relevant algebraic structure for all three contexts is a homotopy version of involutive bi-Lie algebras, which we call IBL$_infty$-algebras.
{"title":"Homological algebra related to surfaces with boundary","authors":"K. Cieliebak, K. Fukaya, J. Latschev","doi":"10.4171/qt/144","DOIUrl":"https://doi.org/10.4171/qt/144","url":null,"abstract":"In this article we describe an algebraic framework which can be used in three related but different contexts: string topology, symplectic field theory, and Lagrangian Floer theory of higher genus. It turns out that the relevant algebraic structure for all three contexts is a homotopy version of involutive bi-Lie algebras, which we call IBL$_infty$-algebras.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2015-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86437072","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}
This is the third article in the series begun with [BonWon3, BonWon4], devoted to finite-dimensional representations of the Kauffman bracket skein algebra of an oriented surface $S$. In [BonWon3] we associated a classical shadow to an irreducible representation $rho$ of the skein algebra, which is a character $r_rho in mathcal R_{mathrm{SL}_2(mathbb C)}(S)$ represented by a group homomorphism $pi_1(S) to mathrm{SL}_2(mathbb C)$. The main result of the current article is that, when the surface $S$ is closed, every character $rin mathcal R_{mathrm{SL}_2(mathbb C)}(S)$ occurs as the classical shadow of an irreducible representation of the Kauffman bracket skein algebra. We also prove that the construction used in our proof is natural, and associates to each group homomorphism $rcolon pi_1(S) to mathrm{SL}_2(mathbb C)$ a representation of the skein algebra $mathcal S^A(S)$ that is uniquely determined up to isomorphism.
本文是从[BonWon3, BonWon4]开始的系列文章中的第三篇，专门讨论定向曲面$S$的Kauffman括号串代数的有限维表示。在[BonWon3]中，我们将经典阴影与交织代数的不可约表示$rho$联系起来，这是一个由群同态$pi_1(S) to mathrm{SL}_2(mathbb C)$表示的字符$r_rho in mathcal R_{mathrm{SL}_2(mathbb C)}(S)$。当前文章的主要结果是，当表面$S$是封闭的，每个字符$rin mathcal R_{mathrm{SL}_2(mathbb C)}(S)$都作为Kauffman括号串代数的不可约表示的经典阴影出现。我们还证明了在我们的证明中使用的构造是自然的，并且将每个群同态$rcolon pi_1(S) to mathrm{SL}_2(mathbb C)$关联到一个到同态为止唯一确定的串代数$mathcal S^A(S)$的表示。
{"title":"Representations of the Kauffman bracket skein algebra III: closed surfaces and naturality","authors":"F. Bonahon, H. Wong","doi":"10.4171/QT/125","DOIUrl":"https://doi.org/10.4171/QT/125","url":null,"abstract":"This is the third article in the series begun with [BonWon3, BonWon4], devoted to finite-dimensional representations of the Kauffman bracket skein algebra of an oriented surface $S$. In [BonWon3] we associated a classical shadow to an irreducible representation $rho$ of the skein algebra, which is a character $r_rho in mathcal R_{mathrm{SL}_2(mathbb C)}(S)$ represented by a group homomorphism $pi_1(S) to mathrm{SL}_2(mathbb C)$. The main result of the current article is that, when the surface $S$ is closed, every character $rin mathcal R_{mathrm{SL}_2(mathbb C)}(S)$ occurs as the classical shadow of an irreducible representation of the Kauffman bracket skein algebra. We also prove that the construction used in our proof is natural, and associates to each group homomorphism $rcolon pi_1(S) to mathrm{SL}_2(mathbb C)$ a representation of the skein algebra $mathcal S^A(S)$ that is uniquely determined up to isomorphism.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2015-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81681110","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}
In this paper we introduce an extension of the hat Heegaard Floer TQFT which allows cobordisms with disconnected ends. Our construction goes by way of sutured Floer homology, and uses some elementary results from contact geometry. We provide some model computations, which allow us to realize the $H_1(Y;mathbb{Z})/text{Tors}$ action and the first order term, $partial_1$, of the differential of $CF^infty$ as cobordism maps. As an application we prove a conjectured formula for the action of $pi_1(Y,p)$ on $hat{HF}(Y,p)$. We provide enough model computations to completely determine the new cobordism maps without the use of any contact geometric constructions.
{"title":"A graph TQFT for hat Heegaard Floer homology","authors":"Ian Zemke","doi":"10.4171/qt/154","DOIUrl":"https://doi.org/10.4171/qt/154","url":null,"abstract":"In this paper we introduce an extension of the hat Heegaard Floer TQFT which allows cobordisms with disconnected ends. Our construction goes by way of sutured Floer homology, and uses some elementary results from contact geometry. We provide some model computations, which allow us to realize the $H_1(Y;mathbb{Z})/text{Tors}$ action and the first order term, $partial_1$, of the differential of $CF^infty$ as cobordism maps. As an application we prove a conjectured formula for the action of $pi_1(Y,p)$ on $hat{HF}(Y,p)$. We provide enough model computations to completely determine the new cobordism maps without the use of any contact geometric constructions.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2015-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80287401","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}
We consider the asymptotics of the Turaev-Viro and the Reshetikhin-Turaev invariants of a hyperbolic $3$-manifold, evaluated at the root of unity $exp({2pisqrt{-1}}/{r})$ instead of the standard $exp({pisqrt{-1}}/{r})$. We present evidence that, as $r$ tends to $infty$, these invariants grow exponentially with growth rates respectively given by the hyperbolic and the complex volume of the manifold. This reveals an asymptotic behavior that is different from that of Witten's Asymptotic Expansion Conjecture, which predicts polynomial growth of these invariants when evaluated at the standard root of unity. This new phenomenon suggests that the Reshetikhin-Turaev invariants may have a geometric interpretation other than the original one via $SU(2)$ Chern-Simons gauge theory.
{"title":"Volume conjectures for the Reshetikhin–Turaev and the Turaev–Viro invariants","authors":"Qingtao Chen, Tian Yang","doi":"10.4171/QT/111","DOIUrl":"https://doi.org/10.4171/QT/111","url":null,"abstract":"We consider the asymptotics of the Turaev-Viro and the Reshetikhin-Turaev invariants of a hyperbolic $3$-manifold, evaluated at the root of unity $exp({2pisqrt{-1}}/{r})$ instead of the standard $exp({pisqrt{-1}}/{r})$. We present evidence that, as $r$ tends to $infty$, these invariants grow exponentially with growth rates respectively given by the hyperbolic and the complex volume of the manifold. This reveals an asymptotic behavior that is different from that of Witten's Asymptotic Expansion Conjecture, which predicts polynomial growth of these invariants when evaluated at the standard root of unity. This new phenomenon suggests that the Reshetikhin-Turaev invariants may have a geometric interpretation other than the original one via $SU(2)$ Chern-Simons gauge theory.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2015-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89281660","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}
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. In 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. The 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}.
我们考虑广义haagup范畴，使得$1 oplus X$对于每一个不可逆的简单物体$X$都承认一个$Q$ -系统。我们证明了在这样一个范畴中，二阶可逆对象群的大小最多为4。我们描述了德林菲尔德中心的简单对象，并给出了模数据的部分公式。我们计算了几个例子的模块化数据的剩余角，并对一般情况进行了推测。我们还考虑了广义Haagerup范畴的几种类型的等变化和去等变化，并描述了它们的Drinfeld中心。特别地，我们计算了在小指数子因子分类中产生的融合类别的一些例子的Drinfeld中心的模数据:Asaeda-Haagerup子因子;$3^{Z_4} $和$3^{Z_2 times Z_2} $子因子;$2D2$子因子;还有$4442$子因子。结果提示了几个新的二次类无限族的可能性。在随附的论文cite{GI19_2}中，描述和概括了与这些族相关的模数据对度量群。
{"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":null,"pages":null},"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}
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