Classification Of Representations Research Articles

Morita equivalence and globalization for partial Hopf actions on nonunital algebras

In this work, we investigate partial actions of a Hopf algebra [Formula: see text] on nonunital algebras and the associated partial smash products, with the objective of providing a framework where one may obtain results for both [Formula: see text]-algebras with local units and [Formula: see text]-categories. We show that our partial actions correspond to nonunital algebras in the category of partial representations of [Formula: see text]. The central problem of existence of a globalization for a partial action is studied in detail, and we provide sufficient conditions for the existence (and uniqueness) of a minimal globalization for associative algebras in general. Extending previous results by Abadie, Dokuchaev, Exel and Simon, we define Morita equivalence for partial Hopf actions, and we show that if two symmetrical partial actions are Morita equivalent then their standard globalizations are also Morita equivalent. Particularizing to the case of a partial action on an algebra with local units, we obtain several strong results on equivalences of categories of modules of partial smash products of algebras and partial smash products of [Formula: see text]-categories.

Invariant Integrals on Coideals and Their Drinfeld Doubles

Abstract Let $A$ be a CQG Hopf *-algebra, that is, a Hopf *-algebra with a positive invariant state. Given a unital right coideal *-subalgebra $B$ of $A$, we provide conditions for the existence of a relatively invariant integral on the stabilizer coideal $B^{\perp }$ inside the dual discrete multiplier Hopf *-algebra of $A$. Given such a relatively invariant integral, we show how it can be extended to a relatively invariant integral on the Drinfeld double coideal. We moreover show that the representation category of the Drinfeld double coideal has a monoidal structure. As an application, we determine the relatively invariant integral for the Drinfeld double coideal *-algebra ${\mathcal{U}}_{q}(\mathfrak{s}\mathfrak{l}(2,{\mathbb{R}}))$ constructed from the Podleś spheres.

Twisted Whittaker category on affine flags and the category of representations of the mixed quantum group

Let $G$ be a reductive group, and let $\check {G}$ be its Langlands dual group. Arkhipov and Bezrukavnikov proved that the Whittaker category on the affine flags ${\operatorname {Fl}}_G$ is equivalent to the category of $\check {G}$ -equivariant quasi-coherent sheaves on the Springer resolution of the nilpotent cone. This paper proves this theorem in the quantum case. We show that the twisted Whittaker category on ${\operatorname {Fl}}_G$ and the category of representations of the mixed quantum group are equivalent. In particular, we prove that the quantum category $\mathsf {O}$ is equivalent to the twisted Whittaker category on ${\operatorname {Fl}}_G$ in the generic case. The strong version of our main theorem claims a motivic equivalence between the Whittaker category on ${\operatorname {Fl}}_G$ and a factorization module category, which holds in the de Rham setting, the Betti setting, and the $\ell$ -adic setting.

Finite dimensional Frobenius–Perron algebras

Motivated by the Frobenius–Perron (FP) theory of endofunctors of a [Formula: see text]-linear category, we introduce the notion of a FP algebra. This notion can be seen as a generalization of a fusion algebra, and is related to the PP theory of endofunctors of the categories of finite dimensional representations of Taft algebras. Our main result is to classify such algebras with small dimensions according to their FP dimension vectors. Moreover, we introduce the notion of a FP Hopf algebra which can be used to categorify a FP algebra via the Green ring of the Hopf algebra.

On the Product Functor on Inner forms of the General Linear Group Over A Non-Archimedean Local Field

AbstractLet $$G_n$$ G n be an inner form of a general linear group over a non-Archimedean local field. We fix an arbitrary irreducible representation $$\sigma $$ σ of $$G_n$$ G n . Building on the work of Lapid-Mínguez on the irreducibility of parabolic inductions, we show how to define a full subcategory of the category of smooth representations of some $$G_m$$ G m , on which the parabolic induction functor $$\tau \mapsto \tau \times \sigma $$ τ ↦ τ × σ is fully-faithful. A key ingredient of our proof for the fully-faithfulness is constructions of indecomposable representations of length 2. Such result for a special situation has been previously applied in proving the local non-tempered Gan-Gross-Prasad conjecture for non-Archimedean general linear groups. In this article, we apply the fully-faithful result to prove a certain big derivative arising from Jacquet functor satisfies the property that its socle is irreducible and has multiplicity one in the Jordan-Hölder sequence of the big derivative.

Graph Representation and Prototype Learning for webly supervised fine-grained image recognition

Webly supervised fine-grained image recognition (FGIR) learns to distinguish sub-ordinate categories based on webly-retrieved data, which can dramatically alleviate the dependency on manually annotated labels. This is quite a challenging task due to the heavy noise labels and the inherent dilemma of small inter-class variance and large intra-class variance. Current webly supervised algorithms learn holistic category prototypes to help correct noisy labels but ignore local features that can distinguish different sub-ordinate categories. In this work, we propose a Graph Representation and Prototype Learning (GRPL) framework to automatically mine discriminative local regions and their interactions with holistic image to learn instance graph representation both category graph prototype to help correct noisy labels and retrieve out-of-distribution (OOD) samples. Specifically, an attention-focused module is designed to extract the discriminative regions and then build a structured graph to correlate them with the holistic image for each instance and an identical graph to model holistic-local correlations for each category. Next, we apply two stacked graph convolution networks to explore holistic-local interaction within each graph and across two graphs to learn graph representation for each instance and graph prototype for each category. Finally, the similarities between the instance-level and prototype-level graph representation are learned to help correct noisy labels and exclude OOD samples. Extensive experiments conducted on several datasets show the proposed approach achieves superior performance compared with current leading algorithms.

Realization of ıquantum groups via Δ-Hall algebras

For an essentially small hereditary abelian category A, we associate an algebra HΔ(A), called the Δ-Hall algebra of A. The basis of HΔ(A) is the isomorphism classes of objects in A, and the Δ-Hall numbers calculate certain three-cycles of exact sequences in A. We show that the Δ-Hall algebra HΔ(A) is isomorphic to the 1-periodic derived Hall algebra associated to A. By taking suitable extension and twisting, we can obtain the ıHall algebra and the semi-derived Hall algebra related to A respectively.When applied to the nilpotent representation category A=repnil(kQ) for an arbitrary quiver Q without loops, the (resp. extended) Δ-Hall algebra provides a realization of the (resp. universal) ıquantum group associated to Q.

(Co)homology of partial smash products

Given a cocommutative Hopf algebra H over a commutative ring K and a symmetric partial action of H on a K-algebra A, we obtain a first quadrant Grothendieck spectral sequence converging to the Hochschild homology of the smash product A#H, involving the Hochschild homology of A and the partial homology of H. An analogous third quadrant cohomological spectral sequence is also obtained. The definition of the partial (co)homology of H under consideration is based on the category of the partial representations of H. A specific partial representation of H on a subalgebra B of the partial “Hopf” algebra Hpar is involved in the definition and we construct a projective resolution of B.

Enriched string-net models and their excitations

Boundaries of Walker-Wang models have been used to construct commuting projector models which realize chiral unitary modular tensor categories (UMTCs) as boundary excitations. Given a UMTC A representing the Witt class of an anomaly, the article \cite{MR4640433} gave a commuting projector model associated to an A-enriched unitary fusion category X on a 2D boundary of the 3D Walker-Wang model associated to A. That article claimed that the boundary excitations were given by the enriched center/Müger centralizer ZA(X) of A in Z(X).In this article, we give a rigorous treatment of this 2D boundary model, and we verify this assertion using topological quantum field theory (TQFT) techniques, including skein modules and a certain semisimple algebra whose representation category describes boundary excitations. We also use TQFT techniques to show the 3D bulk point excitations of the Walker-Wang bulk are given by the Müger center Z2(A), and we construct bulk-to-boundary hopping operators Z2(A)→ZA(X) reflecting how the UMTC of boundary excitations ZA(X) is symmetric-braided enriched in Z2(A).This article also includes a self-contained comprehensive review of the Levin-Wen string net model from a unitary tensor category viewpoint, as opposed to the skeletal 6j symbol viewpoint.

Entanglement-symmetries of covariant channels

Let G and G′ be monoidally equivalent compact quantum groups, and let H be a Hopf-Galois object realising a monoidal equivalence between these groups' representation categories. This monoidal equivalence induces an equivalence Chan(G) → Chan(G′), where Chan(G) is the category whose objects are finite-dimensional C∗-algebras with an action of G and whose morphisms are covariant channels. We show that, if the Hopf-Galois object H has a finite-dimensional *-representation, then channels related by this equivalence can simulate each other using a finite-dimensional entangled resource. We use this result to calculate the entanglement-assisted capacities of certain quantum channels.

Early visual cortices reveal interrelated item and category representations in aging.

Neural dedifferentiation, the finding that neural representations tend to be less distinct in older adults compared with younger adults, has been associated with age-related declines in memory performance. Most studies assessing the relation between memory and neural dedifferentiation have evaluated how age impacts the distinctiveness of neural representations for different visual categories (e.g., scenes and objects). However, how age impacts the quality of neural representations at the level of individual items is still an open question. Here, we present data from an age-comparative fMRI study that aimed to understand how the distinctiveness of neural representations for individual stimuli differs between younger and older adults and relates to memory outcomes. Pattern similarity searchlight analyses yielded indicators of neural dedifferentiation at the level of individual items as well as at the category level in posterior occipital cortices. We asked whether age differences in neural distinctiveness at each representational level were associated with inter- and/or intraindividual variability in memory performance. While age-related dedifferentiation at both the item and category level related to between-person differences in memory, neural distinctiveness at the category level also tracked within-person variability in memory performance. Concurrently, neural distinctiveness at the item level was strongly associated with neural distinctiveness at the category level both within and across participants, elucidating a potential representational mechanism linking item- and category-level distinctiveness. In sum, we provide evidence that age-related neural dedifferentiation co-exists across multiple representational levels and is related to memory performance.Significance Statement Age-related memory decline has been associated with neural dedifferentiation, the finding that older adults have less distinctive neural representations than younger adults. This has been mostly shown for category information, while evidence for age differences in the specificity of item representations is meager. We used pattern similarity searchlight analyses to find indicators of neural dedifferentiation at both levels of representation (category and item) and linked distinctiveness to memory performance. Both item- and category-level dedifferentiation in the calcarine cortex were related to interindividual differences in memory performance, while category-level distinctiveness further tracked intraindividual variability. Crucially, neural distinctiveness was strongly tied between the item and category levels, indicating that intersecting representational properties of posterior occipital cortices reflect both individual exemplars and categories.

Quantum current and holographic categorical symmetry

We establish the formulation for quantum current. Given a symmetry group G, let \mathcal{C}𝒞:=Rep G be its representation category. Physically, symmetry charges are objects of \mathcal{C}𝒞 and symmetric operators are morphisms in \mathcal{C}𝒞. The addition of charges is given by the tensor product of representations. For any symmetric operator O crossing two subsystems, the exact symmetry charge transported by O can be extracted. The quantum current is defined as symmetric operators that can transport symmetry charges over an arbitrary long distance. A quantum current exactly corresponds to an object in the Drinfeld center Z_1(\mathcal{C})Z1(𝒞). The condition for quantum currents to be is also specified, which corresponds to condensation of anyons in one higher dimension. To express the local conservation, the internal hom must be used to compute the charge difference, and the framework of enriched category is inevitable. To illustrate these ideas, we develop a rigorous scheme of renormalization in one-dimensional lattice systems and analyse the fixed-point models. It is proved that in the fixed-point models, superconducting quantum currents form a Lagrangian algebra in Z_1(\mathcal{C})Z1(𝒞) and the boundary-bulk correspondence is verified in the enriched setting. Overall, the quantum current provides a natural physical interpretation to the holographic categorical symmetry.

On homological properties of the category of [formula omitted]-representations over a linear quiver of type [formula omitted

Let Q be a quiver of type An with linear orientation and rep(Q,F1) the category of representations of Q over the virtual field F1. It is proved that rep(Q,F1) has global dimension 2 whenever n≥3 and it is hereditary if n≤2. As a consequence, the Euler form 〈L,M〉=∑i=0∞dim⁡Exti(L,M) is well-defined. However, it does not descend to the Grothendieck group of rep(Q,F1). This yields negative answers to questions raised by Szczesny (2012) [4].

Equivariant Fusion Subcategories

AbstractWe provide a parameterization of all fusion subcategories of the equivariantization by a group action on a fusion category. As applications, we classify the Hopf subalgebras of a family of semisimple Hopf algebras of Kac-Paljutkin type and recover Naidu-Nikshych-Witherspoon classification of the fusion subcategories of the representation category of a twisted quantum double of a finite group.

Modular representations in type A with a two-row nilpotent central character

We study the category of representations of slm+2n over a field of characteristic p with p≫0, whose p-character is a nilpotent whose Jordan type is the two-row partition (m+n,n). In a previous paper with Anno, we used Bezrukavnikov-Mirkovic-Rumynin's theory of positive characteristic localization and exotic t-structures to give a geometric parametrization of the simples using annular crossingless matchings. Building on this, here we give combinatorial dimension formulae for the simple objects, and compute the Jordan-Hölder multiplicities of the simples inside the baby Vermas. We use Cautis-Kamnitzer's geometric categorification of the tangle calculus to study the images of the simple objects under the BMR equivalence. Our results generalize Jantzen's formulae in the subregular nilpotent case (i.e. when n=1), and may be viewed as a positive characteristic analogue of the combinatorial description for Kazhdan-Lusztig polynomials of Grassmannian permutations due to Lascoux and Schutzenberger.

Extensions of the category of comodules of the Taft algebra

We construct a family of non-equivalent pairwise extensions of the category of comodules of the Taft algebra, which are equivalent to representation categories of non-triangular quasi-Hopf algebras.

What is the Problem of Gender Inequality Represented to be in Inter-National Development Policy in Burkina Faso?

This article contributes to critical policy analysis scholarship from a post-structuralism perspective. Employing the ‘What’s the problem represented to be’ (WPR) framework, a Foucault-influenced post-structural approach, we investigate what is the problem of Gender Inequality (GI) represented to be in development in Burkina Faso. Based on systematic analysis of selected (inter)national development policy documents and in-depth stakeholder interviews, our results show two main categories of problem representations: a) local culture/ informal structures that strengthen and are strengthened by patriarchy, and b) women’s weak agency that undermine their effective participation. These problem representations are framed from two different but overlapping standpoints: rights and development. Furthermore, the informal structures are presented as the source of the problem of GI while formal structures are portrayed as the solution. The underlying assumptions ignore the gendered impacts of history, colonial legacies, the interconnectedness and often-conflicting state policies and globalisation. Consequently, the problem of GI is depoliticised, rendered local, technical, and static. This deflects responsibility in solving the problem, limits local agency and the exploration of effective cultural and bottom-up policy responses. Alternatively, GI could be represented as a problem of structural unequal power relations – rather than a simplistic blame of local culture.

Separated monic correspondence of cotorsion pairs and semi-Gorenstein-projective modules

Given a finite dimensional algebra A over a field k, and a finite acyclic quiver Q, let Λ=A⊗kkQ/I, where kQ is the path algebra of Q over k and I is a monomial ideal. This paper is devoted to studying the separated monic representations over the bounded quiver (Q,I) over a subcategory X of A-mod, denoted by smon(Q,I,X). We show that (X,Y) is a (complete) hereditary cotorsion pair in A-mod if and only if (smon(Q,I,X),rep(Q,I,Y)) is a (complete) hereditary cotorsion pair in Λ-mod. We also show that A is left weakly Gorenstein if and only if so is Λ. Provided that kQ/I is non-semisimple, the category ▪ of semi-Gorenstein-projective Λ-modules coincides with the category of separated monic representations ▪ if and only if A is left weakly Gorenstein.

Wightman Fields for Two-Dimensional Conformal Field Theories with Pointed Representation Category

Two-dimensional full conformal field theories have been studied in various mathematical frameworks, from algebraic, operator-algebraic to categorical. In this work, we focus our attention on theories with chiral components having pointed braided tensor representation subcategories, namely having automorphisms whose equivalence classes necessarily form an abelian group. For such theories, we exhibit the explicit Hilbert space structure and construct primary fields as Wightman fields for the two-dimensional full theory. Given a finite collection of chiral components with automorphism categories with trivial total braiding, we also construct a local extension of their tensor product as a chiral component. We clarify the relations with the Longo–Rehren construction, and illustrate these results with concrete examples including the U(1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extrm{U}}(1)$$\\end{document}-current.

A Lie conformal superalgebra and duality of representations for E(4,4)

We construct a duality functor in the category of continuous representations of the Lie superalgebra E(4,4), the only exceptional simple linearly compact Lie superalgebra, for which it wasn't known. This is achieved by constructing a Lie conformal superalgebra of type (4,4) for which E(4,4) is the annihilation algebra. Along the way we obtain an explicit realization of E(4,4) by vector fields on a (4|4)-dimensional supermanifold.