Tetsuya Ito
We give a classification of Wada-type representations of the braid groups, and solutions of a variant of the set-theoretical Yang-Baxter equation adapted to the free-product group structure. As a consequence, we prove Wada's conjecture: There are only seven types of Wada-type representations up to certain symmetries.
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A group-theoretical method, via Wada's representations, is presented to distinguish Kishino's virtual knot from the unknot. Biquandles are constructed for any group using Wada's braid group representations. Cocycle invariants for these biquandles are studied. These invariants are applied to show the non-existence of Alexander numberings and checkerboard colorings.
We prove that for a fixed braid index there are only finitely many possible shapes of the annular Rasmussen $d_t$ invariant of braid closures. Applying the same perspective to the knot Floer invariant $Υ_K(t)$, we show that for a fixed concordance genus of $K$ there are only finitely many possibilities for $Υ_K(t)$. Focusing on the case of 3-braids, we compute the Rasmussen $s$ invariant and the annular Rasmussen $d_t$ invariant of all 3-braid closures. As a corollary, we show that the vanishing/non-vanishing of the $ψ$ invariant is entirely determined by the $s$ invariant and the self-linking…
A braid representation is a monoidal functor from the braid category $\mathsf{B}$, for example given by a solution to the constant Yang-Baxter equation. Given a monoidal category $\mathsf{C}$ with $ob(\mathsf{C})=\mathbb{N}$, a rank-$N$ charge-conserving representation (or spin-chain representation) is a strict monoidal functor $F$ from $\mathsf{C}$ to the category $\mathrm{Match}^N$ of rank-$N$ charge-conserving matrices that is natural in the sense that $F(1)=1$}. In this work we construct all spin-chain braid representations, and classify up to suitable notions of isomorphism.
For a finite group $G$, Turaev introduced the notion of a braided $G$-crossed fusion category. The classification of braided $G$-crossed extensions of braided fusion categories was studied by Etingof, Nikshych and Ostrik in terms of certain group cohomological data. In this paper we will define the notion of a $G$-crossed Frobenius $\star$-algebra and give a classification of (strict) $G$-crossed extensions of a commutative Frobenius $\star$-algebra $R$ equipped with a given action of $G$, in terms of the second group cohomology $H^2(G,R^\times)$. Now suppose that $\mathcal{B}$ is a non-degene…
Here a loop braid representation is a monoidal functor $\mathsf{F}$ from the loop braid category $\mathsf{L}$ to a suitable target category, and is $N$-charge-conserving if that target is the category $\mathsf{Match}^N$ of charge-conserving matrices (specifically $\mathsf{Match}^N$ is the same rank-$N$ charge-conserving monoidal subcategory of the monoidal category $\mathsf{Mat}$ used to classify braid representations in arXiv:2112.04533) with $\mathsf{F}$ strict, and surjective on $\mathbb{N}$, the object monoid. We classify and construct all such representations. In particular we prove that …
The Witt group of nondegenerate braided fusion categories $\mathcal{W}$ contains a subgroup $\mathcal{W}_\text{un}$ consisting of Witt equivalence classes of pseudo-unitary nondegenerate braided fusion categories. For each finite-dimensional simple Lie algebra $\mathfrak{g}$ and positive integer $k$ there exists a pseudo-unitary category $\mathcal{C}(\mathfrak{g},k)$ consisting of highest weight integerable $\hat{g}$-modules of level $k$ where $\hat{\mathfrak{g}}$ is the corresponding affine Lie algebra. Relations between the classes $[\mathcal{C}(\mathfrak{sl}_2,k)]$, $k\geq1$ have been compl…
Low-dimensional indecomposable representations of the braid group $B_3$
本研究完成了特征0代数闭域上2维与3维辫群$B_3$可约但不可分解表示的等价类完整分类，并举例说明其应用价值。
Using various tools from representation theory and group theory, but without using hard classification theorems such as the classification of finite simple groups, we show that the Jones representations of braid groups are dense in the complex Zariski topology when the parameter $t$ is not a root of unity. As first established by Freedman, Larsen, and Wang, we the same result when t is a non-lattice root of unity, other than one initial case when t has order 10. We also compute the real Zariski closure of these representations. When such a representation is indiscrete in the analytic topology,…
Categorification and Dynamics in Generalised Braid Groups
该博士论文受三角范畴与曲面的关联启发，通过范畴化Burau表示将秩2广义辫群实现为三角范畴自等价群，借助范畴熵研究其动力学，用类似映射类群分类的方法完成分类并给出可计算质量增长的算法。
Classification of certain weakly integral fusion categories
本文证明若干维数形式的弱整辫化融合范畴具有弱群论性，并结合维数84、90等结果，完成Frobenius-Perron维数小于120的弱整融合范畴按Morita等价的分类。
We classify all fusion categories for a given set of fusion rules with three simple object types. If a conjecture of Ostrik is true, our classification completes the classification of fusion categories with three simple object types. To facilitate the discussion we describe a convenient, concrete and useful variation of graphical calculus for fusion categories, discuss pivotality and sphericity in this framework, and give a short and elementary re-proof of the fact that the quadruple dual functor is naturally isomorphic to the identity.
Path algebras are a convenient way of describing decompositions of tensor powers of an object in a tensor category. If the category is braided, one obtains representations of the braid groups $B_n$ for all $n\in \N$. We say that such representations are rigid if they are determined by the path algebra and the representations of $B_2$. We show that besides the known classical cases also the braid representations for the path algebra for the 7-dimensional representation of $G_2$ satisfies the rigidity condition, provided $B_3$ generates $\End(V^{\otimes 3})$. We obtain a complete classification …
On Higher Representation Theory via Categories of type Charge-Conserving--with--Glue
本文通过电荷守恒类别提出一种新的方法，用于研究更高表示论中的非半单性问题，并应用于braid表示的分类和分析。
We give two proofs of a level-rank duality for braided fusion categories obtained from quantum groups of type $C$ at roots of unity. The first proof uses conformal embeddings, while the second uses a classification of braided fusion categories associated with quantum groups of type $C$ at roots of unity. In addition we give a similar result for non-unitary braided fusion categories quantum groups of types $B$ and $C$ at odd roots of unity.
We give a complete classification of simple representations of the braid group B_3 with dimension $\leq 5$ over any algebraically closed f ield. In particular, we prove that a simple d-dimensional representation $ρ: B_3 \to GL(V)$ is determined up to isomorphism by the eigenvalues $λ_1, λ_2, ..., λ_d$ of the image of the generators for d=2,3 and a choice of a $δ=\sqrt{\det ρ(σ_1)}$ for d=4 or a choice of $δ=\sqrt[5]{\det ρ(σ_1)}$ for d=5. We also s howed that such representations exist whenever the eigenvalues and $δ$ are not roots of certain polynomials $Q_{ij}^{(d)}$, which are explicitly gi…
Morita invariants of quasitriangular coideal subalgebras
本文通过BC和D型辫群的表示，研究准三角余子代数表示范畴的莫里塔不变量，并给出K矩阵的分类结果。
A Categorical Perspective on Braid Representations
本文从范畴论视角研究辫子表示，通过引入函子范畴的幺半结构、子对象与商对象等工具，揭示了电荷守恒 Yang-Baxter 算子分类的范畴机制，并证明了足够自由条件下幺半函子可严格化。
We construct a [(n+1)/2]+1 parameters family of irreducible representations of the Braid group B_3 in arbitrary dimension n\in N, using a q-deformation of the Pascal triangle. This construction extends in particular results by S.P.Humphries [8], who constructed representations of the braid group B_3 in arbitrary dimension using the classical Pascal triangle. E.Ferrand [7] obtained an equivalent representation of B_3 by considering two special operators in the space C^n[X]. Slightly more general representations were given by I.Tuba and H.Wenzl [11]. They involve [(n+1)/2] parameters (and also u…
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.
Ng and Schauenburg proved that the kernel of a $(2+1)$-dimensional topological quantum field theory representation of $\mathrm{SL}(2, \mathbb{Z})$ is a congruence subgroup. Motivated by their result, we explore when the kernel of an irreducible representation of the braid group $B_3$ with finite image enjoys a congruence subgroup property. In particular, we show that in dimensions two and three, when the projective order of the image of the braid generator $σ_1$ is between 2 and 5 the kernel projects onto a congruence subgroup of $\mathrm{PSL}(2,\mathbb{Z})$ and compute its level. However, we …
We complete the classification of Hopf algebras whose infinitesimal braiding is a principal Yetter-Drinfeld realization of a braided vector space of Cartan type $G_2$ over a cosemisimple Hopf algebra. We develop a general formula for a class of liftings in which the quantum Serre relations hold. We give a detailed explanation of the procedure for finding the relations, based on the recent work of Andruskiewitsch, Angiono and Rossi Bertone.
Let $C$ be a modular category of Frobenius-Perron dimension $dq^n$, where $q$ is a prime number and $d$ is a square-free integer. We show that if $q>2$ then $C$ is integral and nilpotent. In particular, $C$ is group-theoretical. In the general case, we describe the structure of $C$ in terms of equivariantizations of group-crossed braided fusion categories.
In [1] we have constructed a [n+1/2]+1 parameters family of irreducible representations of the Braid group B_3 in arbitrary dimension using a $q-$deformation of the Pascal triangle. This construction extends in particular results by S.P. Humphries (2000), who constructed representations of the braid group B_3 in arbitrary dimension using the classical Pascal triangle. E. Ferrand (2000) obtained an equivalent representation of B_3 by considering two special operators in the space ${\mathbb C}^n[X].$ Slightly more general representations were given by I. Tuba and H. Wenzl (2001). They involve [n…
Over fields of arbitrary characteristic we classify all braid-indecomposable tuples of at least two absolutely simple Yetter-Drinfeld modules over non-abelian groups such that the group is generated by the support of the tuple and the Nichols algebra of the tuple is finite-dimensional. Such tuples are classified in terms of analogs of Dynkin diagrams which encode much information about the Yetter-Drinfeld modules. We also compute the dimensions of these finite-dimensional Nichols algebras. Our proof uses the Weyl groupoid of a tuple of simple Yetter-Drinfeld modules.
We investigate invertible elements and gradings in braided tensor categories. This leads us to the definition of theta-, product-, subgrading and orbitcategories in order to construct new families of BTC's from given ones. We use the representation theory of Hecke algebras in order to relate the fusionring of a BTC generated by an object $X$ with a two component decomposition of its tensorsquare to the fusionring of quantum groups of type $A$ at roots of unity. We find the condition of `local isomorphie' on a special fusionring morphism implying that a BTC is obtained from the above constructi…
For transcendental values of q the quantum tangent spaces of all left-covariant first order differential calculi of dimension less than four on the quantum group $\SLq 2$ are given. All such differential calculi $Γ$ are determined and investigated for which the left-invariant differential one-forms $ω(u^1_2)$, $ω(u^2_1)$ and $ω(u^1_1-u^2_2)$ generate $Γ$ as a bimodule and the universal higher order differential calculus has the same dimension as in the classical case. Important properties (cohomology spaces, *-structures, braidings, generalized Lie brackets) of these calculi are examined as we…
Unitary Yang--Baxter Operators: Towards a Classification
该研究针对酉杨-巴克斯特算子分类缺失的问题，提出三类生成源的猜想性分类，通过计算搜索得到特殊解，回答了Lechner的存在性问题并验证了Rowell-Wang局域化猜想的开放案例。
Let g be a complex, simple Lie algebra with Cartan subalgebra h and Weyl group W. We construct a one-parameter family of flat connections D on h with values in any finite-dimensional h-module V and simple poles on the root hyperplanes. The corresponding monodromy representation of the braid group B of type g is a deformation of the action of (a finite extension of) W on V. The residues of D are the Casimirs of the sl(2)-subalgebras of g corresponding to its roots. The irreducibility of a subspace U of V under these implies that, for generic values of the parameter, the braid group B acts irred…
Simple $B_4$ representations associated to cyclotomic Hecke algebras
本文在生成元可对角化的非半单情形下，确定了复反射群$G_{25}$对应的分圆Hecke代数结构，分类了辫群$B_4$满足三次多项式的简单表示，并应用于$G_2$型辫子张量范畴的分类。
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…
We classify all groups G and all pairs (V,W) of absolutely simple Yetter-Drinfeld modules over G such that the support of the direct sum of V and W generates G, the square of the braiding between V and W is not the identity, and the Nichols algebra of the direct sum of V and W admits a finite root system. As a byproduct, we determine the dimensions of such Nichols algebras, and several new families of finite-dimensional Nichols algebras are obtained. Our main tool is the Weyl groupoid of pairs of absolutely simple Yetter-Drinfeld modules over groups.
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