Jesse Liptrap
Fusion rules generalize groups by allowing multivalued multiplication. Groups are fusion rules of simple current index 1. We classify nilpotent (in the sense of Gelaki and Nikshych) fusion rules of simple current index 2, and characterize the associated fusion categories.
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We introduce formulae of Frobenius-Schur indicators of simple objects of Tambara-Yamagami categories. By using techniques of the Fourier transform on finite abelian groups, we study some arithmetic properties of indicators.
Braidings for Non-Split Tambara-Yamagami Categories over the Reals
研究实数上非分裂Tambara-Yamagami范畴的辫子结构，并分类其辫子等价类，同时证明分裂实数和复数Tambara-Yamagami范畴的新结果。
Tambara-Yamagami Categories over the Reals: The Non-Split Case
该研究将Tambara-Yamagami的融合规则范畴化结论推广到实数域上非分裂单对象（实、复或四元数对象）的情形，得到了类似的范畴化结果。
The Tambara-Yamagami calculus and finite semigroups
本文研究将 Tambara-Yamagami 范畴中的有限群替换为有限半群后，其图解演算的保留与发展，提出并构建了相应的演算体系。
NIM-representations of Tambara-Yamagami generalizations
本文研究了Tambara-Yamagimi泛化的NIM-表示法，分类了其不可约表示并检测了相关代数对象。
We prove that if two Tambara-Yamagami categories TY(A,χ,ν) and TY(A',χ',ν') give rise to the same state sum invariants of 3-manifolds and the order of one of the groups A, A' is odd, then ν=ν' and there is a group isomorphism A\approx A' carrying χto χ'. The proof is based on an explicit computation of the state sum invariants for the lens spaces of type (k,1).
We prove that the higher Frobenius-Schur indicators, introduced by Ng and Schauenburg, give a strong enough invariant to distinguish between any two Tambara-Yamagami fusion categories. Our proofs are based on computation of the higher indicators as quadratic Gauss sums for certain quadratic forms on finite abelian groups and relies on the classification of quadratic forms on finite abelian groups, due to Wall. As a corollary to our work, we show that the state-sum invariants of a Tambara-Yamagami category determine the category as long as we restrict to Tambara-Yamagami categories coming from …
Modular $\mathbb{Z}_2$-Crossed Tambara-Yamagami-like Categories for Even Groups
本文构建了非退化braided Z₂交叉张量范畴，扩展了Tambara-Yamagami范畴，允许Γ群有偶阶且非平凡部分包含多个单对象。
Continuous Tambara-Yamagami tensor categories
该研究将Tambara-Yamagami张量范畴构造从有限阿贝尔群推广到局部紧阿贝尔群，给出其分类，并证明满足对应融合规则的$\mathrm{W}^*$-张量范畴的结合子自动连续。
We show that all isomorphism classes of even rank Tambara-Yamagami categories arise as $\mathbb{Z}_2$-twisted representations of conformal nets. As a consequence, we show that their Drinfel'd centers are realized by (generalized) orbifolds of conformal nets associated with (self-dual) lattices. The quantum double subfactors of even rank Tambara-Yamagami categories are Bisch-Haagerup subfactors and we describe their (dual) principal graphs. For every abelian group of odd order the Drinfel'd centers of the associated Tambara-Yamagami categories give a fusion ring generalizing the Verlinde ring $…
We determine the structure of Hopf algebra extensions of a group algebra by the cyclic group of order 2. We study the corepresentation theory of such Hopf algebras, which provide a generalization, at the Hopf algebra level, of the so called Tambara-Yamagami fusion categories. As a byproduct, we show that every semisimple Hopf algebra of dimension $<36$ is necessarily group-theoretical; thus 36 is the smallest possible dimension where a non group-theoretical example occurs.
This paper is part of a sequence interpreting quantities of conformal field theories K-theoretically. Here we give geometric constructions of the associated module categories (modular invariants, nimreps, etc). In particular, we give a KK-theory interpretation of all modular invariants for the loop groups of tori, as well as most known modular invariants of loop groups. In addition, we find unexpectedly that the Tambara-Yamagami fusion category has an elegant description as bundles over a groupoid, and use that to interpret its module categories as KK-elements. We establish reconstruction for …
Fiber 2-Functors and Tambara-Yamagami Fusion 2-Categories
本文定义并刻画群论融合 2-范畴，分析其结构、融合规则与分次，并分类带 fiber 2-函子及 Tambara-Yamagami 缺陷的融合 2-范畴。
An algorithm for Tambara-Yamagami quantum invariants of 3-manifolds, parameterized by the first Betti number
本文研究由 Tambara-Yamagami 范畴导出的三维流形状态和不变量的计算复杂性，证明其 #P 难性，并提出以第一贝蒂数为参数的固定参数可解算法，揭示困难性源于经典三维流形拓扑。
We establish some relations between the orders of simple objects in a fusion category and the structure of its universal grading group. We consider fusion categories which have a faithful simple object and show that its universal grading group must be cyclic. As for the converse, we prove that a braided nilpotent fusion category with cyclic universal grading group always has a faithful simple object. We study the universal grading of fusion categories with generalized Tambara-Yamagami fusion rules. As an application, we classify modular categories in this class and describe the modularizations…
Let C be a fusion category faithfully graded by a finite group G and let D be the trivial component of this grading. The center Z(C) of C is shown to be canonically equivalent to a G-equivariantization of the relative center Z_D(C). We use this result to obtain a criterion for C to be group-theoretical and apply it to Tambara-Yamagami fusion categories. We also find several new series of modular categories by analyzing the centers of Tambara-Yamagami categories. Finally, we prove a general result about existence of zeroes in S-matrices of weakly integral modular categories.
For an abelian group $ A $, we study a close connection between braided crossed $ A $-categories with a trivialization of the $ A $-action and $ A $-graded braided tensor categories. Additionally, we prove that the obstruction to the existence of a trivialization of a categorical group action $T$ on a monoidal category $\mathcal{C}$ is given by an element $O(T)\in H^2(G,\operatorname{Aut}_\otimes(\operatorname{Id}_{\mathcal{C}}))$. In the case that $O(T)=0$, the set of obstructions form a torsor over $\operatorname{Hom}(G,\operatorname{Aut}_\otimes(\operatorname{Id}_{\mathcal{C}}))$, where $\o…
We advance the classification of fusion categories in two directions. Firstly, we completely classify integral fusion categories -- and consequently, semi-simple Hopf algebras -- of dimension $pq^2$, where $p$ and $q$ are distinct primes. This case is especially interesting because it is the simplest class of dimensions where not all integral fusion categories are group-theoretical. Secondly, we classify a certain family of $\ZZ/3\ZZ$-graded fusion categories, which are generalizations of the $\ZZ/2\ZZ$-graded Tambara-Yamagami categories. Our proofs are based on the recently developed theory o…
Let C be a fusion category which is an extension of a fusion category D by a finite group G. We classify module categories over C in terms of module categories over D and the extension data (c,M,a) of C. We also describe functor categories over C (and in particular the dual categories of C). We use this in order to classify module categories over the Tambara Yamagami fusion categories, and their duals.
We describe the structure of a generalized near-group fusion category and present an example of this class of fusion categories which arises from the extension of a Fibonacci category. We then classify slightly degenerate generalized near-group fusion categories. We also prove a structure result for braided generalized Tambara-Yamagami fusion categories.
In this paper, we first show for a slightly degenerate pre-modular fusion category $\mathcal{C}$ that squares of dimensions of simple objects divide half of the dimension of $\mathcal{C}$, and that slightly degenerate fusion categories of FP-dimensions $2p^nd$ and $4p^nd$ are nilpotent, where $p$ is an odd prime and $d$ is an odd square-free integer. Then we classify slightly degenerate generalized Tambara-Yamagami fusion categories and weakly integral slightly degenerate fusion categories of particular dimensions.
We classify integral modular categories of dimension pq^4 and p^2q^2 where p and q are distinct primes. We show that such categories are always group-theoretical except for categories of dimension 4q^2. In these cases there are well-known examples of non-group-theoretical categories, coming from centers of Tambara-Yamagami categories and quantum groups. We show that a non-group-theoretical integral modular category of dimension 4q^2 is equivalent to either one of these well-known examples or is of dimension 36 and is twist-equivalent to fusion categories arising from a certain quantum group.
A near-group fusion category is a fusion category C where all but 1 simple objects are invertible. Examples of these include the Tambara-Yamagami categories and the even sectors of the E6 and affine-D5 subfactors, though there are infinitely many others. We classify the near-group fusion categories, and compute their doubles and the modular data relevant to conformal field theory. Among other things, we explicitly construct over 40 new finite depth subfactors, with Jones index ranging from around 6.85 to around 14.93. We expect all of these doubles to be realised by rational conformal field th…
This is a study of weakly integral braided fusion categories with elementary fusion rules to determine which possess nondegenerately braided extensions of theoretically minimal dimension, or equivalently in this case, which satisfy the minimal modular extension conjecture. We classify near-group braided fusion categories satisfying the minimal modular extension conjecture; the remaining Tambara-Yamagami braided fusion categories provide arbitrarily large families of braided fusion categories with identical fusion rules violating the minimal modular extension conjecture. These examples generali…
We introduce Manifold tensor categories, which make precise the notion of a tensor category with a manifold of simple objects. A basic example is the category of vector spaces graded by a Lie group. Unlike classic tensor category theory, our setup keeps track of the smooth (and topological) structure of the manifold of simple objects. We set down the necessary definitions for Manifold tensor categories and a generalisation we term Orbifold tensor categories. We also construct a number of examples, most notably two families of examples we call Interpolated Tambara-Yamagami categories and Interp…
Given an action of a finite group G on a fusion category C we give a criterion for the category of G-equivariant objects in C to be group-theoretical, i.e., to be categorically Morita equivalent to a category of group-graded vector spaces. We use this criterion to answer affirmatively the question about existence of non group-theoretical semisimple Hopf algebras asked by P. Etingof, V. Ostrik, and the author in math/0203060. Namely, we show that certain Z/2Z-equivariantizations of fusion categories constructed by D. Tambara and S. Yamagami are equivalent to representation categories of non gro…
On bicrossed product of fusion categories and exact factorizations
本文推广群的匹配对概念至融合环与融合范畴，定义其双交叉积并构造精确分解，证明融合环精确分解可表为双交叉积，刻画融合范畴精确分解的伴随子范畴与泛分次群，还给出新例子的融合规则及结合约束。
Classifying Module Categories for Generalized Temperley-Lieb-Jones *-2-Categories
本文提出通过加权双向公平和平衡图对广义Temperley-Lieb-Jones *-2-范畴的单位ary模块进行分类。
In this paper we provide a complete classification of fusion categories of Frobenius-Perron (FP) dimension pq, where p<q are distinct primes, thus giving a categorical generalization of math.QA/9801129. As a corollary we also obtain the classification of semisimple quasi-Hopf algebras of dimension pq. A concise formulation of our main result is: Let C be a fusion category over the complex numbers of FP dimension pq. Then either p=2 and C is a Tambara-Yamagami category of dimension 2q, or C is group-theoretical in the sense of math.QA/0203060 (which easily yields the full classification). As a …
We give a pedagogical survey of those aspects of the abstract representation theory of quantum groups which are related to the Tannaka-Krein reconstruction problem. We show that every concrete semisimple tensor *-category with conjugates is equivalent to the category of finite dimensional non-degenerate *-representations of a discrete algebraic quantum group. Working in the self-dual framework of algebraic quantum groups, we then relate this to earlier results of S. L. Woronowicz and S. Yamagami. We establish the relation between braidings and R-matrices in this context. Our approach emphasize…
Braidings on topological operators, anomaly of higher-form symmetries and the SymTFT
该研究利用可编织拓扑算子对应 SymTFT 对称性的观点，提出绕过六边形方程计算融合范畴编织的算法，并刻画多类融合 2-范畴可编织的充要条件。
We apply the yoga of classical homotopy theory to classification problems of G-extensions of fusion and braided fusion categories, where G is a finite group. Namely, we reduce such problems to classification (up to homotopy) of maps from BG to classifiying spaces of certain higher groupoids. In particular, to every fusion category C we attach the 3-groupoid BrPic(C) of invertible C-bimodule categories, called the Brauer-Picard groupoid of C, such that equivalence classes of G-extensions of C are in bijection with homotopy classes of maps from BG to the classifying space of BrPic(C). This gives…
On Low-Rank Multiplicity-Free Fusion Categories
本文提出算法获取秩至7的所有无重数pivotal融合范畴的显式F符号、R符号和pivotal系数，并探讨了任意子模型与图辫模型。
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