Fusion Rings Research Articles

Building Vertex Algebras from Parts

Given a collection of modules of a vertex algebra parametrized by an abelian group, together with one dimensional spaces of composable intertwining operators, we assign a canonical element of the cohomology of an Eilenberg-Mac Lane space. This element describes the obstruction to locality, as the vanishing of this element is equivalent to the existence of a vertex algebra structure with multiplication given by our intertwining operators, and given existence, the structure is unique up to isomorphism. The homological obstruction reduces to an "evenness" problem that naturally vanishes for 2-divisible groups, so simple currents organized into odd order abelian groups always produce vertex algebras. Furthermore, in cases most relevant to conformal field theory (i.e., when we have well-behaved contragradients and tensor products), we obtain our spaces of intertwining operators naturally, and the evenness obstruction reduces to the question of whether the contragradient bilinear form on certain order 2 currents is symmetric or skew-symmetric. We show that if we are given a simple regular VOA and integral-weight modules parametrized by a group of even units in the fusion ring, then the direct sum admits the structure of a simple regular VOA, called the simple current extension, and this structure is unique up to isomorphism.

Wedge-direct sums of table algebras and applications to association schemes, I

In the remarkable paper (Blau, J. Algebr. 396:220–271, 2013) on fusion rings, Blau introduced a very important operation on two table algebras (the Blau-construction). It is very difficult to describe the structure of table algebras obtained by recursively applying the Blau-construction. In this paper, we define the wedge-direct sum for a sequence of table algebras. Using the wedge-direct sum of a sequence of table algebras, we are able to not only give a clear description of the structure of table algebras obtained by recursively applying the Blau-construction but also obtain the characterization and classification of a class of p-table algebras. The structure of the Bose-Mesner algebras of the wreath products and wedge products of association schemes can also be clearly described in terms of the wedge-direct sum. As an application, we get some new and known results about the wreath products and wedge products of association schemes.

Torsion-freeness for fusion rings and tensor C*-categories

Torsion-freeness for discrete quantum groups was introduced by R. Meyer in order to formulate a version of the Baum–Connes conjecture for discrete quantum groups. In this note, we introduce torsion-freeness for abstract fusion rings. We show that a discrete quantum group is torsion-free if its associated fusion ring is torsion-free. In the latter case, we say that the discrete quantum group is strongly torsion-free. As applications, we show that the discrete quantum group duals of the free unitary quantum groups are strongly torsion-free, and that torsion-freeness of discrete quantum groups is preserved under Cartesian and free products. We also discuss torsion-freeness in the more general setting of abstract rigid tensor C*-categories.

Ninjurin1 positively regulates osteoclast development by enhancing the survival of prefusion osteoclasts

Osteoclasts (OCs) are bone-resorbing cells that originate from hematopoietic stem cells and develop through the fusion of mononuclear myeloid precursors. Dysregulation of OC development causes bone disorders such as osteopetrosis, osteoporosis, and rheumatoid arthritis. Although the molecular mechanisms underlying osteoclastogenesis have been well established, the means by which OCs maintain their survival during OC development remain unknown. We found that Ninjurin1 (Ninj1) expression is dynamically regulated during osteoclastogenesis and that Ninj1−/− mice exhibit increased trabecular bone volume owing to impaired OC development. Ninj1 deficiency did not alter OC differentiation, transmigration, fusion, or actin ring formation but increased Caspase-9-dependent intrinsic apoptosis in prefusion OCs (preOCs). Overexpression of Ninj1 enhanced the survival of mouse macrophage/preOC RAW264.7 cells in osteoclastogenic culture, suggesting that Ninj1 is important for the survival of preOCs. Finally, analysis of publicly available microarray data sets revealed a potent correlation between high NINJ1 expression and destructive bone disorders in humans. Our data indicate that Ninj1 plays an important role in bone homeostasis by enhancing the survival of preOCs.

Modular categories of dimension $p^3m$ with $m$ square-free

We give a complete classification of modular categories of dimension $p^3m$ where $p$ is prime and $m$ is a square-free integer. When $p$ is odd, all such categories are pointed. For $p=2$ one encounters modular categories with the same fusion ring as orthogonal quantum groups at certain roots of unity, namely $SO(2m)_2$. As an immediate step we classify a more general class of so-called even metaplectic modular categories with the same fusion rules as $SO(2N)_2$ with $N$ odd.

On the Subregular J-Rings of Coxeter Systems

We recall Lusztig's construction of the asymptotic Hecke algebra $J$ of a Coxeter system $(W,S)$ via the Kazhdan--Lusztig basis of the corresponding Hecke algebra. The algebra $J$ has a direct summand $J_E$ for each two-sided Kazhdan--Lusztig cell of $W$, and we study the summand $J_C$ corresponding to a particular cell $C$ called the subregular cell. We develop a combinatorial method to compute $J_C$ without using the Kazhdan--Lusztig basis. As applications, we deduce some connections between $J_C$ and the Coxeter diagram of $W$, and we show that for certain Coxeter systems $J_C$ contains subalgebras that are free fusion rings in the sense of [Banica], thereby connecting the subalgebras to compact quantum groups arising from operator algebra theory.

Eigenvalues of rotations and braids in spherical fusion categories

We give formulae for the multiplicities of eigenvalues of generalized rotation operators in terms of generalized Frobenius–Schur indicators in a semisimple spherical tensor category C. In particular, this implies that the entire collection of rotation eigenvalues for a fusion category can be computed from the fusion rules and the traces of rotation at finitely many tensor powers. We also establish a rigidity property for FS indicators of fusion categories with a given fusion ring via Jones's theory of planar algebras. If C is also braided, these formulae yield the multiplicities of eigenvalues for a large class of braids in the associated braid group representations. When C is modular, this allows one to determine the eigenvalues and multiplicities of braids in terms of just the S and T matrices.

Irreducible ℤ+-modules of near-group fusion ring K(ℤ3, 3)

The near-group rings are an important class of fusion rings in the theory of tensor categories. In this paper, the irreducible ℤ+-modules over the near-group fusion ring K(ℤ3, 3) are explicitly classified. It turns out that there are only four inequivalent irreducible ℤ+-modules of rank 2 and two inequivalent irreducible ℤ+-modules of rank 4 over K(ℤ3, 3).

Topological automorphism groups of compact quantum groups

We study the topological structure of the automorphism groups of compact quantum groups showing that, in parallel to a classical result due to Iwasawa, the connected component of identity of the automorphism group and of the “inner” automorphism group coincide. For compact matrix quantum groups, which can be thought of as quantum analogues of compact Lie groups, we prove that the inner automorphism group is a compact Lie group and the outer automorphism group is discrete. Applications of this to the study of group actions on compact quantum groups are highlighted. We end by providing examples of compact matrix quantum groups with infinitely-generated fusion rings, in stark contrast with the classical situation. Along the way we study the invariant theory of finite group actions on free Laurent rings and show that the rings of invariants are, in general, not finitely generated.

Non-trivially graded self-dual fusion categories of rank 4

Let C be a self-dual spherical fusion categories of rank 4 with non-trivial grading. We complete the classification of Grothendieck ring K(C) of C; that is, we prove that K(C) ≅= Fib⊗ℤ[ℤ2], where Fib is the Fibonacci fusion ring and ℤ[ℤ2] is the group ring on ℤ2. In particular, if C is braided, then it is equivalent to Fib⊠ $$Vec_{{\mathbb{Z}_2}}^\omega $$ as fusion categories, where Fib is a Fibonacci category and $$Vec_{{\mathbb{Z}_2}}^\omega $$ is a rank 2 pointed fusion category.

Positivity and periodicity of Q-systems in the WZW fusion ring

We study properties of solutions of Q-systems in the WZW fusion ring obtained by the Kirillov–Reshetikhin modules. We make a conjecture about their positivity and periodicity and give a proof of it in some cases. We also construct a positive solution of the level k restricted Q-system of classical types in the fusion rings. As an application, we prove some conjectures of Kirillov and Kuniba–Nakanishi–Suzuki on the level k restricted Q-systems.

Rank 2 fusion rings are complete intersections

We give a non-constructive proof that fusion rings attached to a simple complex Lie algebra of rank 2 are complete intersections.

Generalized orbifold construction for conformal nets

Let [Formula: see text] be a conformal net. We give the notion of a proper action of a finite hypergroup [Formula: see text] acting by vacuum preserving unital completely positive (so-called stochastic) maps on [Formula: see text] which generalizes the proper action of a finite group [Formula: see text]. Taking the fixed point under such an action gives a finite index subnet [Formula: see text] of [Formula: see text], which generalizes the [Formula: see text]-orbifold net. Conversely, we show that if [Formula: see text] is a finite inclusion of conformal nets, then [Formula: see text] is a generalized orbifold [Formula: see text] of the conformal net [Formula: see text] by a unique finite hypergroup [Formula: see text]. There is a Galois correspondence between intermediate nets [Formula: see text] and subhypergroups [Formula: see text] given by [Formula: see text]. In this case, the fixed point of [Formula: see text] is the generalized orbifold by the hypergroup of double cosets [Formula: see text].If [Formula: see text] is a finite index inclusion of completely rational nets, we show that the inclusion [Formula: see text] is conjugate to an intermediate subfactor of a Longo–Rehren inclusion. This implies that if [Formula: see text] is a holomorphic net, and [Formula: see text] acts properly on [Formula: see text], then there is a unitary fusion category [Formula: see text] which is a categorification of [Formula: see text] and [Formula: see text] is braided equivalent to the Drinfel’d center [Formula: see text]. More generally, if [Formula: see text] is a completely rational conformal net and [Formula: see text] acts properly on [Formula: see text], then there is a unitary fusion category [Formula: see text] extending [Formula: see text], such that [Formula: see text] is given by the double cosets of the fusion ring of [Formula: see text] by the Verlinde fusion ring of [Formula: see text] and [Formula: see text] is braided equivalent to the Müger centralizer of [Formula: see text] in the Drinfel’d center [Formula: see text].

The center of the extended Haagerup subfactor has 22 simple objects

We explain a technique for discovering the number of simple objects in [Formula: see text], the center of a fusion category [Formula: see text], as well as the combinatorial data of the induction and restriction functors at the level of Grothendieck rings. The only input is the fusion ring [Formula: see text] and the dimension function [Formula: see text]. In particular, we apply this to deduce that the center of the extended Haagerup subfactor has 22 simple objects, along with their decompositions as objects in either of the fusion categories associated to the subfactor. This information has been used subsequently in [T. Gannon and S. Morrison, Modular data for the extended Haagerup subfactor (2016), arXiv:1606.07165 .] to compute the full modular data. This is the published version of arXiv:1404.3955 .

On realization of fusion rings from generalized Cartan matrices

The Casimir element of a fusion ring (R,B) gives rise to the so called Casimir matrix C of (R,B). This enables us to construct a generalized Cartan matrix D − C in the sense of Kac for a suitable diagonal matrix D. In this paper, we study some elementary properties of the Casimir matrix C and use them to realize certain fusion rings from the generalized Cartan matrix D−C of finite (resp. affine) type. It turns out that there exists a fusion ring with D−C being of finite (resp. affine) type if and only if D−C has only the form A 2 (resp. A 1 (1) ). We also realize all fusion rings with D−C being a particular generalized Cartan matrix of indefinite type.

A Comparison of Accuracy of Image- versus Hardware-based Tracking Technologies in 3D Fusion in Aortic Endografting

Fusion of three-dimensional (3D) computed tomography and intraoperative two-dimensional imaging in endovascular surgery relies on manual rigid co-registration of bony landmarks and tracking of hardware to provide a 3D overlay (hardware-based tracking, HWT). An alternative technique (image-based tracking, IMT) uses image recognition to register and place the fusion mask. We present preliminary experience with an agnostic fusion technology that uses IMT, with the aim of comparing the accuracy of overlay for this technology with HWT. Data were collected prospectively for 12 patients. All devices were deployed using both IMT and HWT fusion assistance concurrently. Postoperative analysis of both systems was performed by three blinded expert observers, from selected time-points during the procedures, using the displacement of fusion rings, the overlay of vascular markings and the true ostia of renal arteries. The Mean overlay error and the deviation from mean error was derived using image analysis software. Comparison of the mean overlay error was made between IMT and HWT. The validity of the point-picking technique was assessed. IMT was successful in all of the first 12 cases, whereas technical learning curve challenges thwarted HWT in four cases. When independent operators assessed the degree of accuracy of the overlay, the median error for IMT was 3.9mm (IQR 2.89-6.24, max 9.5) versus 8.64mm (IQR 6.1-16.8, max 24.5) for HWT (p=.001). Variance per observer was 0.69mm(2) and 95% limit of agreement ±1.63. In this preliminary study, the error of magnitude of displacement from the "true anatomy" during image overlay in IMT was less than for HWT. This confirms that ongoing manual re-registration, as recommended by the manufacturer, should be performed for HWT systems to maintain accuracy. The error in position of the fusion markers for IMT was consistent, thus may be considered predictable.

On Regularised Quantum Dimensions of the Singlet Vertex Operator Algebra and False Theta Functions

We study a family of non-C2-cofinite vertex operator algebras, called the singlet vertex operator algebras, and connect several important concepts in the theory of vertex operator algebras, quantum modular forms, and modular tensor categories. More precisely, starting from explicit formulae for characters of modules over the singlet vertex operator algebra, which can be expressed in terms of false theta functions and their derivatives, we first deform these characters by using a complex parameter ϵϵ. We then apply modular transformation properties of regularisedpartial theta functions to study asymptotic behaviour of regularised characters of irreducible modules and compute their regularised quantum dimensions. We also give a purely geometric description of the regularisation parameter as a uniformisation parameter of the fusion variety coming from atypical blocks. It turns out that the quantum dimensions behave very differently depending on the sign of the real part of ϵϵ. The map from the space of characters equipped with the Verlinde product to the space of regularised quantum dimensions turns out to be a genuine ring isomorphism for positive real part of ϵϵ, while for sufficiently negative real part of ϵϵ its surjective image gives the fusion ring of a rational vertex operator algebra. The category of modules of this rational vertex operator algebra should be viewed as obtained through the process of a semi-simplification procedure widely used in the theory of quantum groups. Interestingly, the modular tensor category structure constants of this vertex operator algebra can be also detected from vector-valued quantum modular forms formed by distinguished atypical characters.

Generalized zeta function representation of groups and 2-dimensional topological Yang-Mills theory: The example of GL(2, 𝔽q) and PGL(2, 𝔽q)

We recall the relation between zeta function representation of groups and two-dimensional topological Yang-Mills theory through Mednikh formula. We prove various generalisations of Mednikh formulas and define generalization of zeta function representations of groups. We compute some of these functions in the case of the finite group GL(2, 𝔽q) and PGL(2, 𝔽q). We recall the table characters of these groups for any q, compute the Frobenius-Schur indicator of their irreducible representations, and give the explicit structure of their fusion rings.

Green rings of pointed rank one Hopf algebras of non-nilpotent type

In this paper, we continue our study of the Green rings of finite dimensional pointed Hopf algebras of rank one initiated in [22], but focus on those Hopf algebras of non-nilpotent type. Let H be a finite dimensional pointed rank one Hopf algebra of non-nilpotent type. We first determine all non-isomorphic indecomposable H-modules and describe the Clebsch–Gordan formulas for them. We then study the structures of both the Green ring r(H) and the Grothendieck ring G0(H) of H and establish the precise relation between the two rings. We use the Cartan map of H to study the Jacobson radical and the idempotents of r(H). It turns out that the Jacobson radical of r(H) is exactly the kernel of the Cartan map, a principal ideal of r(H), and r(H) has no non-trivial idempotents. Besides, we show that the stable Green ring of H is a transitive fusion ring. This enables us to calculate Frobenius–Perron dimensions of objects in the stable category of H. Finally, as an example, we present both the Green ring and the Grothendieck ring of the Radford Hopf algebra in terms of generators and relations.

Membrane Repair: Mechanisms and Pathophysiology.

Eukaryotic cells have been confronted throughout their evolution with potentially lethal plasma membrane injuries, including those caused by osmotic stress, by infection from bacterial toxins and parasites, and by mechanical and ischemic stress. The wounded cell can survive if a rapid repair response is mounted that restores boundary integrity. Calcium has been identified as the key trigger to activate an effective membrane repair response that utilizes exocytosis and endocytosis to repair a membrane tear, or remove a membrane pore. We here review what is known about the cellular and molecular mechanisms of membrane repair, with particular emphasis on the relevance of repair as it relates to disease pathologies. Collective evidence reveals membrane repair employs primitive yet robust molecular machinery, such as vesicle fusion and contractile rings, processes evolutionarily honed for simplicity and success. Yet to be fully understood is whether core membrane repair machinery exists in all cells, or whether evolutionary adaptation has resulted in multiple compensatory repair pathways that specialize in different tissues and cells within our body.