Group-like Structures Research Articles

Toric 2-group anomalies via cobordism

2-group symmetries arise in physics when a 0-form symmetry G[0] and a 1-form symmetry H[1] intertwine, forming a generalised group-like structure. Specialising to the case where both G[0] and H[1] are compact, connected, abelian groups (i.e. tori), we analyse anomalies in such ‘toric 2-group symmetries’ using the cobordism classification. As a warm up example, we use cobordism to study various ’t Hooft anomalies (and the phases to which they are dual) in Maxwell theory defined on non-spin manifolds. For our main example, we compute the 5th spin bordism group of B|\U0001d53e| where \U0001d53e is any 2-group whose 0-form and 1-form symmetry parts are both U(1), and |\U0001d53e| is the geometric realisation of the nerve of the 2-group \U0001d53e. By leveraging a variety of algebraic methods, we show that {varOmega}_5^{textrm{Spin}}left(Bleft|mathbbm{G}right|right)cong mathbb{Z}/m where m is the modulus of the Postnikov class for \U0001d53e, and we reproduce the expected physics result for anomalies in 2-group symmetries that appear in 4d QED. Moving down two dimensions, we recap that any (anomalous) U(1) global symmetry in 2d can be enhanced to a toric 2-group symmetry, before showing that its associated local anomaly reduces to at most an order 2 anomaly, when the theory is defined with a spin structure.

Fusion systems and localities – a dictionary

Linking systems were introduced to provide algebraic models for p-completed classifying spaces of fusion systems. Every linking system over a saturated fusion system F corresponds to a group-like structure called a locality. Given such a locality L, we prove that there is a one-to-one correspondence between the partial normal subgroups of L and the normal subsystems of the fusion system F. This is then used to obtain a kind of dictionary, which makes it possible to translate between various concepts in localities and corresponding concepts in fusion systems. As a byproduct, we obtain new proofs of many known theorems about fusion systems and also some new results. For example, we show in this paper that, in any saturated fusion system, there is a sensible notion of a product of normal subsystems.

The Galaxy Environment of Extremely Massive Quasars. I. An Overdensity of Hα Emitters at z = 1.47

Abstract We measure a strong excess in the galaxy number density around PG 1630+377, an extremely massive (M BH ≃ 109.7 M ⊙) quasar at z = 1.475, using near-infrared narrowband imaging. We identify 79 narrow H-band excess objects in a 525 arcmin2 area including the vicinity and surroundings of the quasar. These sources are likely Hα line emitting, star-forming galaxies at z ≈ 1.47. We detect a δ = 6.6 ± 2.7 overdensity of narrow H-band excess objects located at a projected distance ≈2.1 Mpc northeast of the quasar, which is the densest region in the target area. The overdensity is present in BzK color-selected galaxies, while a previously reported overdensity in the immediate vicinity of PG 1630+377 is not, and yet appears as a group-like structure. These megaparsec-scale environments are estimated to merge into a ≃1014.7 M ⊙ cluster at present. Our results support the view that extremely massive black holes form and grow in group-scale environments and later incorporate into a galaxy cluster.

Extensions of homomorphisms between localities

Abstract We show that the automorphism group of a linking system associated to a saturated fusion system $\mathcal {F}$ depends only on $\mathcal {F}$ as long as the object set of the linking system is $\mathrm {Aut}(\mathcal {F})$ -invariant. This was known to be true for linking systems in Oliver’s definition, but we demonstrate that the result holds also for linking systems in the considerably more general definition introduced previously by the author of this article. A similar result is proved for linking localities, which are group-like structures corresponding to linking systems. Our argument builds on a general lemma about the existence of an extension of a homomorphism between localities. This lemma is also used to reprove a theorem of Chermak showing that there is a natural bijection between the sets of partial normal subgroups of two possibly different linking localities over the same fusion system.

Gill Mucus and Gill Mucin O-glycosylation in Healthy and Amebic Gill Disease-Affected Atlantic Salmon.

Amoebic gill disease (AGD) causes poor performance and death in salmonids. Mucins are mainly comprised by carbohydrates and are main components of the mucus covering the gill. Since glycans regulate pathogen binding and growth, glycosylation changes may affect susceptibility to primary and secondary infections. We investigated gill mucin O-glycosylation from Atlantic salmon with and without AGD using liquid chromatography–mass spectrometry. Gill mucin glycans were larger and more complex, diverse and fucosylated than skin mucins. Confocal microscopy revealed that fucosylated mucus coated sialylated mucus strands in ex vivo gill mucus. Terminal HexNAcs were more abundant among O-glycans from AGD-affected Atlantic salmon, whereas core 1 structures and structures with acidic moieties such as N-acetylneuraminic acid (NeuAc) and sulfate groups were less abundant compared to non-infected fish. The fucosylated and NeuAc-containing O-glycans were inversely proportional, with infected fish on the lower scale of NeuAc abundance and high on fucosylated structures. The fucosylated epitopes were of three types: Fuc-HexNAc-R, Gal-[Fuc-]HexNAc-R and HexNAc-[Fuc-]HexNAc-R. These blood group-like structures could be an avenue to diversify the glycan repertoire to limit infection in the exposed gills. Furthermore, care must be taken when using skin mucus as proxy for gill mucus, as gill mucins are distinctly different from skin mucins.

On some operators and dilations of frame generator and dual pair of frame generators of two structured unitary systems

ABSTRACT In this paper, we study the dilations of frame generators and pairs of dual frame generators of two special structured unitary systems for Hilbert spaces, i.e. the product unitary system and group-like unitary system. We first introduce certain operators which are associated with the frame generators of the unitary system and some properties of the operators are established. Then, as an application, we show that the canonical dual of a frame of group-like unitary systems remain the group-like structure. And we use these properties to study the dilations of the frame generators and dual pairs of frame generators of these two types of unitary systems.

Left Regular Representation of Gyrogroups

In this article, we examine a subspace L gyr ( G ) of the complex vector space, L ( G ) = { f : f is a function from G to C } , where G is a nonassociative group-like structure called a gyrogroup. The space L gyr ( G ) arises as a representation space for G associated with the left regular representation, consisting of complex-valued functions invariant under certain permutations of G. In the case when G is finite, we prove that dim ( L gyr ( G ) ) = 1 | γ ( G ) | ∑ ρ ∈ γ ( G ) | Fix ( ρ ) | , where γ ( G ) is the subgroup of Sym ( G ) generated by a class of permutations of G and Fix ( ρ ) = { a ∈ G : ρ ( a ) = a } .

Duality in Non-Abelian Algebra IV. Duality for groups and a universal isomorphism theorem

Abelian categories provide a self-dual axiomatic context for establishing homomorphism theorems (such as the isomorphism theorems and homological diagram lemmas) for abelian groups, and more generally, modules. In this paper we describe a self-dual context which allows one to establish the same theorems in the case of non-abelian group-like structures; the question of whether such a context can be found has been left open for seventy years. We also formulate and prove in our context a universal isomorphism theorem from which all other isomorphism theorems can be deduced.

Salamander lemma for non-abelian group-like structures

It is well known that the classical diagram lemmas of homological algebra for abelian groups can be generalized to non-abelian group-like structures, such as groups, rings, algebras, loops, etc. In this paper, we establish such a generalization of the “salamander lemma” due to G. M. Bergman, in a self-dual axiomatic context (developed originally by Z. Janelidze), which applies to all usual non-abelian group-like structures and also covers axiomatic contexts such as semi-abelian categories in the sense of G. Janelidze, L. Márki and W. Tholen and exact categories in the sense of M. Grandis.

Property (T) discrete quantum groups and subfactors with triangle presentations

Kazhdan's property (T) has been studied for several discrete group-like structures, including standard invariants of Jones' subfactors and discrete quantum groups. We prove a Żuk-type spectral gap criterion for property (T) in this setting. This allows us to construct a family of property (T) discrete quantum groups, as well as subfactor standard invariants, given by generators and a triangle presentation. These are the first examples of property (T) discrete quantum groups that are not directly related to a discrete group with property (T).

Measuring and Evaluating Convergence Processes Across a Series of Group Discussions

As groups develop, members’ knowledge and expectations regarding the task and the group tend to converge. Such convergence allows members to anticipate and coordinate their own and others’ actions, facilitating productive group work. Using zero-history laboratory groups, this study analyzes the presence and trajectory of cognitive convergence as groups worked on a series of three similar tasks. We focus on two types of convergence: anticipatory expectations for future work, and reflective assessments regarding previous group discussions. Results indicate immediate convergence for reflective cognitions but delayed convergence for anticipatory cognitions. Associations among the different types of convergence also vary at the group- and individual-levels of analysis. The discussion addresses measurement implications of both types of convergence in regard to emergence of group-like cognitive structures.

Möbius’s functional equation and Schur’s lemma with applications to the complex unit disk

Mobius addition is defined on the complex open unit disk by $$\begin{aligned} a\oplus _M b = \dfrac{a+b}{1+\bar{a}b} \end{aligned}$$ and Mobius’s exponential equation takes the form $$L(a\oplus _M b) = L(a)L(b)$$ , where L is a complex-valued function defined on the complex unit disk. In the present article, we indicate how Mobius’s exponential equation is connected to Cauchy’s exponential equation. Mobius’s exponential equation arises when one determines the irreducible linear representations of the unit disk equipped with Mobius addition, considered as a nonassociative group-like structure. This suggests studying Schur’s lemma in a more general setting.

A Note on the Abelianization Functor

It is well known that the abelianization of a group G can be computed as the cokernel of the diagonal morphism (1G, 1G): G → G × G in the category of groups. We generalize this to arbitrary regular subtractive categories, among which are the category of groups, the category of topological groups, and the categories of other group-like structures. We also establish that an abelian category is the same as a regular subtractive category in which every monomorphism is a kernel of some morphism.

Duality in non-abelian algebra II. From Isbell bicategories to Grandis exact categories

This paper continues the study of self-dual axioms on forms, i.e. faithful amnestic functors, motivated by properties of subobject forms in non-abelian algebra (which in many cases are Grothendieck bifibrations), and in particular, by properties of forms of substructures of group-like structures. In this paper we explore axiomatic origins of this kind, of a hierarchy of contexts introduced by M. Grandis for his projective approach to non-abelian homological algebra. This reveals new links between those contexts and the theory of factorization systems. Among other things, we show that a Grandis exact category is the same as an Isbell bicategory whose form (fibration) of projections is isomorphic to the form (opfibration) of injections.

On a quasi-relativistic formula in polarization theory.

In a pure operatorial (nonmatrix) Pauli algebraic approach, this Letter shows that the Poincaré vector of the light transmitted by a dichroic device can be expressed as function of the Poincaré vectors of the incoming light and of the device by a composition law of the same kind as the composition law of the noncolinear relativistic velocities. This is, in fact, a general law of composition for three-dimensional (3D) vectors remaining in the Poincaré ball (where they have a group-like structure). The differences between this problem and that of the composition law of two dichroic devices are pointed out and justified.

A novel structure-aware sparse learning algorithm for brain imaging genetics.

Brain imaging genetics is an emergent research field where the association between genetic variations such as single nucleotide polymorphisms (SNPs) and neuroimaging quantitative traits (QTs) is evaluated. Sparse canonical correlation analysis (SCCA) is a bi-multivariate analysis method that has the potential to reveal complex multi-SNP-multi-QT associations. Most existing SCCA algorithms are designed using the soft threshold strategy, which assumes that the features in the data are independent from each other. This independence assumption usually does not hold in imaging genetic data, and thus inevitably limits the capability of yielding optimal solutions. We propose a novel structure-aware SCCA (denoted as S2CCA) algorithm to not only eliminate the independence assumption for the input data, but also incorporate group-like structure in the model. Empirical comparison with a widely used SCCA implementation, on both simulated and real imaging genetic data, demonstrated that S2CCA could yield improved prediction performance and biologically meaningful findings.

An axiomatic survey of diagram lemmas for non-abelian group-like structures

It is well known that diagram lemmas for abelian groups (and more generally in abelian categories) used in algebraic topology, can be suitably extended to “non-abelian” structures such as groups, rings, loops, etc. Moreover, they are equivalent to properties which arise in the axiomatic study of these structures. For the five lemma this is well known, and in the present paper we establish this for the snake lemma and the 3×3 lemma, which, when suitably formulated, turn out to be equivalent to each other for all (pointed) algebraic structures, and also in general categories of a special type. In particular, we show that among varieties of universal algebras, they are satisfied precisely in the so-called (pointed) ideal determined varieties.

First Results from a Photometric Survey of Strong Gravitational Lens Environments

Many strong gravitational lenses lie in complex environments, such as poor groups of galaxies, that significantly bias conclusions from lens analyses. We are undertaking a photometric survey of all known galaxy-mass strong lenses to characterize their environments and include them in careful lens modeling, and to build a large, uniform sample of galaxy groups at intermediate redshifts for evolutionary studies. In this paper we present wide-field photometry of the environments of twelve lens systems with 0.24 < z_lens < 0.5. Using a red-sequence identifying technique, we find that eight of the twelve lenses lie in groups, and that ten group-like structures are projected along the line of sight towards seven of these lenses. Follow-up spectroscopy of a subset of these fields confirms these results. For lenses in groups, the group centroid position is consistent with the direction of the external tidal shear required by lens models. Lens galaxies are not all super-L_* ellipticals; the median lens luminosity is < L_*, and the distribution of lens luminosities extends 3 magnitudes below L_* (in agreement with theoretical models). Only two of the lenses in groups are the brightest group galaxy, in qualitative agreement with theoretical predictions. As in the local Universe, the highest velocity-dispersion groups contain a brightest member spatially coincident with the group centroid, whereas lower-dispersion groups tend to have an offset brightest group galaxy. This suggests that higher-dispersion groups are more dynamically relaxed than lower-dispersion groups and that at least some evolved groups exist by z ~ 0.5.

Gyrogroups and the decomposition of groups into twisted subgroups and subgroups

Gyrogroups are generalized groups modelled on the Einstein groupoid of all relativistically admissible velocities with their Einstein’s velocity addition as a binary operation. Einstein’s gyrogroup fails to form a group since it is nonassociative. The breakdown of associativity in the Einstein addition does not result in loss of mathematical regularity owing to the presence of the relativistic effect known as the Thomas precession which, by abstraction, becomes an automorphism called the Thomas gyration. The Thomas gyration turns out to be the missing link that gives rise to analogies shared by gyrogroups and groups. In particular, it gives rise to the gyroassociative and the gyrocommuttive laws that Einstein’s addition possesses, in full analogy with the associative and the commutative laws that vector addition possesses in a vector space. The existence of striking analogies shared by gyrogroups and groups implies the existence of a general theory which underlies the theories of groups and gyrogroups and unifies them with respect to their central features. Accordingly, our goal is to construct finite and infinite gyrogroups, both gyrocommutative and non-gyrocommutaive, in order to demonstrate that gyrogroups abound in group theory of which they form an integral part. A gyrogroup is a grouplike structure that is defined in [7] along with a weaker structure called a left gyrogroup. We show that any given group can be turned into a left gyrogroup which is, in turn, a gyrogroup if and only if the given group is central by a 2-Engel group. The importance of left gyrogroups stems from the facts that (i) any gyrotransversal groupoid is a left gyrogroup, and (ii) any left gyrogroup is a twisted subgroup in a specified group. Twisted subgroups are subsets of groups, introduced by Aschbacher [1], which under general conditions are near subgroups. The concept of near subgroup of a finite group was introduced by Feder and Vardi [6 ]a s at ool to study problems in computational complexity involving the class NP .

Umbral Calculus and Cancellative Semigroup Algebras

We describe some connections between three different fields: combinatorics (umbral calculus), functional analysis (linear functionals and operators) and harmonic analysis (convolutions on group-like structures). Systematic usage of cancellative semigroups, their convolution algebras, and tokens between them provides a common language for description of objects from these three fields.