Related papers: The Extended Haagerup fusion categories
We classify $E_2$ condensable algebras in a modular tensor category $\mathcal{C}$ up to 2-Morita equivalence. From a physical perspective, this is equivalent to providing a criterion for when different $E_2$ condensable algebras result in…
We prove that two dual operator algebras are weak$^*$ Morita equivalent if and only if they have equivalent categories of dual operator modules via completely contractive functors which are also weak$^*$-continuous on appropriate morphism…
We provide a complete classification of the class of unital graph $C^*$-algebras - prominently containing the full family of Cuntz-Krieger algebras - showing that Morita equivalence in this case is determined by ordered, filtered…
We show that several Morita equivalence classes of tame algebras do not occur as blocks of finite groups. This refines classifications by Erdmann of classes of blocks with dihedral, semidihedral, and generalised quaternion defect groups. In…
Three exceptional modular invariants of SU(4) exist at levels 4, 6 and 8. They can be obtained from appropriate conformal embeddings and the corresponding graphs have self-fusion. From these embeddings, or from their associated modular…
We discuss the equivalence between the categories of certain ribbon graphs and subgroups of the modular group $\Gamma$ and use it to construct exponentially large families of not Hurwitz equivalent simple braid monodromy factorizations of…
This work is a detailed version of arXiv:0704.0195 [math.QA]. We introduce a new notion of the core of a braided fusion category. It allows to separate the part of a braided fusion category that does not come from finite groups. We also…
Although it has been a well-known fact, for more than two decades, that category theory is needed for the study of topological orders, it is still a non-trivial challenge for students and working physicists to master the abstract language…
The main goal of this paper is to classify $\ast$-module categories for the $SO(3)_{2m}$ modular tensor category. This is done by classifying $SO(3)_{2m}$ nimrep graphs and cell systems, and in the process we also classify the $SO(3)$…
2-Theories are a canonical way of describing categories with extra structure. 2-theory-morphisms are used when discussing how one structure can be replaced with another structure. This is central to categorical coherence theory. We place a…
We develop a homotopy theory for additive categories endowed with endofunctors, analogous to the concept of a model structure. We use it to construct the homotopy theory of a Hovey triple (which consists of two compatible complete cotorsion…
Compact quantum groups can be studied by investigating their co-representation categories in analogy to the Schur-Weyl/Tannaka-Krein approach. For the special class of (unitary) "easy" quantum groups these categories arise from a…
We associate to a projective $n$-dimensional toric variety $X_{\Delta}$ a pair of co-commutative (but generally non-commutative) Hopf algebras $H^{\alpha}_X, H^{T}_X$. These arise as Hall algebras of certain categories $\Coh^{\alpha}(X),…
We compute the modular data (that is, the $S$ and $T$ matrices) for the centre of the extended Haagerup subfactor. The full structure (i.e. the associativity data, also known as 6-$j$ symbols or $F$ matrices) still appears to be…
In this work, we propose to study noncommutative geometry using the language of categories of sheaves of algebras with polynomial identities and their properties, introducing new (graded) noncommutative geometries. These include, for…
In this paper we find the fusion rules of the free wreath products $\widehat{\Gamma}\wr_*S_N^+$ for any (discrete) group $\Gamma$. To do this we describe the spaces of intertwiners between basic corepresentations which allows us to identify…
We study the birational geometry of the moduli spaces of hyperelliptic curves with marked points. We show that these moduli spaces have non $\mathbb{Q}$-factorial singularities. We complete the Kodaira classification by proving that these…
Given a unitary fusion category, one can define the Hilbert space of a so-called ``anyonic spin-chain'' and nearest neighbor Hamiltonians providing a real-time evolution. There is considerable evidence that suitable scaling limits of such…
We complete the foundational architecture of Algebraic Phase Theory by developing a categorical and $2$-categorical framework for algebraic phases. Building on the structural notions introduced in Papers~I-III, we define phase morphisms,…
We prove some results on the structure of certain classes of integral fusion categories and semisimple Hopf algebras under restrictions on the set of its irreducible degrees.