Classification Of Representations Research Articles

The Jordan-Hölder property of torsion-free classes in a quiver of type a

ABSTRACT We investigate the torsion-free classes satisfying the Jordan-Hölder property in the representation category of a quiver of type A. Explicitly, we use reflection functors and c-sortable elements to give a combinatorial criterion for determining such torsion-free classes.

Spectral Networks and Stability Conditions for Fukaya Categories with Coefficients

Given a holomorphic family of Bridgeland stability conditions over a surface, we define a notion of spectral network which is an object in a Fukaya category of the surface with coefficients in a triangulated DG-category. These spectral networks are analogs of special Lagrangian submanifolds, combining a graph with additional algebraic data, and conjecturally correspond to semistable objects of a suitable stability condition on the Fukaya category with coefficients. They are closely related to the spectral networks of Gaiotto–Moore–Neitzke. One novelty of our approach is that we establish a general uniqueness results for spectral network representatives. We also verify the conjecture in the case when the surface is disk with six marked points on the boundary and the coefficients category is the derived category of representations of an A2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A_2$$\\end{document} quiver. This example is related, via homological mirror symmetry, to the stacky quotient of an elliptic curve by the cyclic group of order six.

Generalised Gabriel-Roiter measure and thin representations

For Dynkin and Euclidean quivers, it is shown that Gabriel-Roiter measures of thin representations equal the induced chain length functions on the corresponding system of subquivers. This allows a combinatorial procedure to find GR filtrations of thin representations, showing that GR measures of thin representations are field-independent. It is proved that an indecomposable filtration of a thin representation is a GR filtration for a suitable choice of a length function on the category of finite-dimensional representations.

Equivariant 𝒟-modules on 2×2×𝓃 hypermatrices

We study D \mathcal {D} -modules and related invariants on the space of 2 × 2 × n 2\times 2\times n hypermatrices for n ≥ 3 n\geq 3 , which has finitely many orbits under the action of G = G L 2 ( C ) × G L 2 ( C ) × G L n ( C ) G=GL_2(\mathbb {C}) \times GL_2(\mathbb {C}) \times GL_n(\mathbb {C}) . We describe the category of coherent G G -equivariant D \mathcal {D} -modules as the category of representations of a quiver with relations. We classify the simple equivariant D \mathcal {D} -modules, determine their characteristic cycles and find special representations that appear in their G G -structures. We determine the explicit D \mathcal {D} -module structure of the local cohomology groups with supports given by orbit closures. As a consequence, we calculate the Lyubeznik numbers and intersection cohomology groups of the orbit closures. All but one of the orbit closures have rational singularities: we use local cohomology to prove that the one exception is neither normal nor Cohen–Macaulay. While our results display special behavior in the cases n = 3 n=3 and n = 4 n=4 , they are completely uniform for n ≥ 5 n\geq 5 .

Modular tensor categories arising from central extensions and related applications

A modular tensor category is a non-degenerate ribbon finite tensor category and a ribbon factorizable Hopf algebra is a Hopf algebra whose finite-dimensional representations form a modular tensor category. In this paper, we provide a method of constructing ribbon factorizable Hopf algebras using central extensions. We then apply this method to n-rank Taft algebras, which are considered finite-dimensional quantum groups associated with abelian Lie algebras (see Section 2 for the definition), and obtain a family of non-semisimple ribbon factorizable Hopf algebras Eq, thus producing non-semisimple modular tensor categories using their representation categories. And we provide a prime decomposition of Rep(Eq) (the representation category of Eq). By further studying the simplicity of Eq (whether it is a simple Hopf algebra or not), we conclude that(1)there exists a twist J of uq(sl2⊕3) such that uq(sl2⊕3)J is a simple Hopf algebra,(2)there is no relation between the simplicity of a Hopf algebra H and the primality of Rep(H),(3)there are many ribbon factorizable Hopf algebras that are distinct from some known ones, i.e., not isomorphic to any tensor products of trivial Hopf algebras (group algebras or their dual), Drinfeld doubles, and small quantum groups.

Stated SL(n$n$)‐skein modules and algebras

AbstractWe develop a theory of stated SL()‐skein modules, , of 3‐manifolds marked with intervals in their boundaries. These skein modules, generalizing stated SL(2)‐modules of the first author, stated SL(3)‐modules of Higgins', and SU(n)‐skein modules of the second author, consist of linear combinations of framed, oriented graphs, called ‐webs, with ends in , considered up to skein relations of the ‐Reshetikhin–Turaev functor on tangles, involving coupons representing the anti‐symmetrizer and its dual. We prove the Splitting Theorem asserting that cutting of a marked 3‐manifold along a disk resulting in a 3‐manifold yields a homomorphism for all . That result allows to analyze the skein modules of 3‐manifolds through the skein modules of their pieces. The theory of stated skein modules is particularly rich for thickened surfaces , in whose case, is an algebra, denoted by . One of the main results of this paper asserts that the skein algebra of the ideal bigon is isomorphic with and it provides simple geometric interpretations of the product, coproduct, counit, the antipode, and the cobraided structure on . (In particular, the coproduct is given by a splitting homomorphism.) We show that for surfaces with boundary every splitting homomorphism is injective and that is a free module with a basis induced from the Kashiwara–Lusztig canonical bases. Additionally, we show that a splitting of a thickened bigon near a marking defines a right ‐comodule structure on , or dually, a left ‐module structure. Furthermore, we show that the skein algebra of surfaces glued along two sides of a triangle is isomorphic with the braided tensor product of Majid. These results allow for geometric interpretation of further concepts in the theory of quantum groups, for example, of the braided products and of Majid's transmutation operation. Building upon the above results, we prove that the factorization homology with coefficients in the category of representations of is equivalent to the category of left modules over for surfaces with . We also establish isomorphisms of our skein algebras with the quantum moduli spaces of Alekseev–Schomerus and with the internal algebras of the skein categories for these surfaces and .

Development of coparenting during the transition to parenthood: Integrating parents' representations with observed interactions.

The transition to parenthood is a significant period of transformation and adjustment for all members of a new family, reshaping relational dynamics that often endure throughout the initial years of a child's life. This qualitative and longitudinal study aimed to explore coparenting representations held by both partners in a parental couple, along with observations of their family interactions, at three distinct points in time, with the aim to better understand the interaction between these representational and behavioral observations. In-depth interviews were conducted with 17 heterosexual, first-time parents in Santiago de Chile, during pregnancy and at two different times during the first year of their child's life. Family interactions were assessed using the Lausanne Trilogue Play task (LTP) on all occasions. Three main coparenting representation categories emerged: traditional, ambiguous, and co-responsible. On observation, approximately half of the couples showed cooperative coparenting interactions, while the other half had conflictual interactions. Couples generally followed stable trajectories over time. Cooperative couples demonstrated co-responsibility and open dialogue, while conflictual couples tended to follow traditional gender roles with tacit communication styles. This study highlights the importance of promoting co-responsibility and dialogue for fostering cooperative relational dynamics during the critical transition to parenthood.

Homotopy Quotients and Comodules of Supercommutative Hopf Algebras

We study model structures on the category of comodules of a supercommutative Hopf algebra A over fields of characteristic 0. Given a graded Hopf algebra quotient A→B\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A \\rightarrow B$$\\end{document} satisfying some finiteness conditions, the Frobenius tensor category D\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {D}}$$\\end{document} of graded B-comodules with its stable model structure induces a monoidal model structure on C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {C}}$$\\end{document}. We consider the corresponding homotopy quotient γ:C→HoC\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma : {\\mathcal {C}} \\rightarrow Ho {\\mathcal {C}}$$\\end{document} and the induced quotient T→HoT\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {T}} \\rightarrow Ho {\\mathcal {T}}$$\\end{document} for the tensor category T\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {T}}$$\\end{document} of finite dimensional A-comodules. Under some mild conditions we prove vanishing and finiteness theorems for morphisms in HoT\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Ho {\\mathcal {T}}$$\\end{document}. We apply these results in the Rep(GL(m|n))-case and study its homotopy category HoT\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Ho {\\mathcal {T}}$$\\end{document} associated to the parabolic subgroup of upper triangular block matrices. We construct cofibrant replacements and show that the quotient of HoT\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Ho{\\mathcal {T}}$$\\end{document} by the negligible morphisms is again the representation category of a supergroup scheme.

On neutral Tannakian subcategories of loop quiver representations

We explored the categories of (twisted) representations of a loop quiver. These representation categories have two choices of tensor structures: Kronecker tensor and Simpson tensor. By studying the rigidity properties, we have provided several examples of (semi-) Tannakian categories using the category of (twisted) representations of a loop quiver for both tensors.We have introduced the concept of essentially finite loop quiver bundles based on the work of Nori, Borne, and Vistoli. As an application, we have given some examples of (semi-) Tannakian categories of equivariant bundles and Hitchin pairs. Additionally, we have defined the notion of H-nflat twisted loop quiver bundles and have established Tannakian category structures for certain classes of varieties.

Completeness of Induced Cotorsion Pairs in Representation Categories of Rooted Quivers

Completeness of Induced Cotorsion Pairs in Representation Categories of Rooted Quivers

A comparison of endomorphism algebras

Let F be a non-archimedean local field and G be a connected reductive group over F. For a Bernstein block in the category of smooth complex representations of G(F), we have two kinds of progenerators: the compactly induced representation indKG(F)(ρ) of a type (K,ρ), and the parabolically induced representation IPG(ΠM) of a progenerator ΠM of a Bernstein block for a Levi subgroup M of G. In this paper, we construct an explicit isomorphism of these two progenerators. Moreover, we compare the description of the endomorphism algebra EndG(F)(indKG(F)(ρ)) for a depth-zero type (K,ρ) in [20] with the description of the endomorphism algebra EndG(F)(IPG(ΠM)) in [33], that are described in terms of affine Hecke algebras.

The Double Semion State in Infinite Volume

AbstractWe describe in a simple setting how to extract a braided tensor category from a collection of superselection sectors of a two-dimensional quantum spin system, corresponding to abelian anyons. We extract from this category its fusion ring as well as its F and R-symbols. We then construct the double semion state in infinite volume and extract the braided tensor category describing its semion, anti-semion, and bound state excitations. We verify that this category is equivalent to the representation category of the twisted quantum double $$\mathcal {D}^{\phi }(\mathbb {Z}_2)$$ D ϕ ( Z 2 ) .

Relative tilting theory in abelian categories I: Auslander–Buchweitz–Reiten approximations theory in subcategories and cotorsion-like pairs

In this paper, we introduce a special kind of relative (co)resolutions associated with a pair of classes of objects in an abelian category C.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {C}.$$\\end{document} We will see that, by studying this relative (co)resolutions, we get a possible generalization of a part of the Auslander–Buchweitz approximation theory that is useful for developing n-X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {X}$$\\end{document}-tilting theory in Argudin Monroy and Mendoza Hernández (relative tilting theory in abelian categories II: n-X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {X}$$\\end{document} tilting theory. arXiv:2112.14873, 2021). With this goal, new concepts as X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {X}$$\\end{document}-complete and X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {X}$$\\end{document}-hereditary pairs are introduced as a generalization of complete and hereditary cotorsion pairs. These pairs appear in a natural way in the study of the category of representations of a quiver in an abelian category (Argudin Monroy and Mendoza Hernández in categories of quiver representations and relative cotorsion pairs. arXiv:2311.12774v1, 2023). Our main results will include an existence theorem for relative approximations, among other results related with closure properties of relative (co)resolution classes and relative homological dimensions which are essential in the development of n-X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {X}$$\\end{document}-tilting theory in Argudin Monroy and Mendoza Hernández (2021).

РЕПРЕЗЕНТАЦІЇ ІГРОВОГО МОДЕЛЮВАННЯ ЖИТТЄДІЯЛЬНОСТІ ЛЮДИНИ В УМОВАХ НЕВИЗНАЧЕНОСТІ

Purpose. Features of modernity that correspond to the features of a world dominated by volatility, uncertainty, complexity and ambiguity, threats of using such features as means of geopolitical influence, conditions of a full-scale russian-ukrainian war, the necessity restoration for a safe environment throughout the territory of Ukraine requires effective proposals from domestic psychology to ensure human life in the given conditions. A well-founded system-forming position for this increasingly appears as a game as a universal socio-cultural phenomenon and an integral factor of human self-realization, and game modeling as a way and method of human life activity in uncertainty. The purpose of the article is a theoretical study of the psychological regularities of game representation in the aspects of modeling human life, substantiation of psychological content, functional essence, classification of representations as an attributive factor of game modeling of human life in uncertainty conditions. Methods. The research uses methods of analysis and systematization for psychological and interdisciplinary approaches in the study of representation and modeling of human life activities, theoretical reconstruction and classification of the psychological content and functional essence for the game modeling representation of human life activities in current living conditions. Results. Based on the results of the research, the categorical and conceptual content and functional essence for game modeling representations of human life activity in uncertainty conditions were specified. The attribution of representation in the game modeling aspects is updated and substantiated. The definition of "game modeling representations of human life activity in uncertainty conditions" and the corresponding classification are proposed. Scientific ideas about representation, game and game modeling are deepened. It was established that the game modeling representations of human life activity in uncertainty conditions is a universal, pragmatic, attributive phenomenon; non-specific mental formation; the way of life of a person and a social group, which occurs simultaneously in two complementary dimensions: a) the intrasubjective space of a person as an interiorizational mental representation and b) in the external activity dimension as a deed, activity representation of exteriorization. Conclusions. The results of the study allow us to draw conclusions about the relevance of continuing psychological explorations for game modeling representations of human life activity in uncertainty conditions. The clarifying regularities tasks of influence for representations on psychological well-being and productivity of individual and group activities are considered promising.

Quantum $\operatorname{SL}(2)$ and logarithmic vertex operator algebras at $(p,1)$-central charge

We provide a ribbon tensor equivalence between the representation category of small quantum \operatorname{SL}(2) , at parameter q=e^{\pi i/p} , and the representation category of the triplet vertex operator algebra at integral parameter p>1 . We provide similar quantum group equivalences for representation categories associated to the Virasoro and singlet vertex operator algebras at central charge c=1-6(p-1)^{2}/p . These results resolve a number of fundamental conjectures coming from studies of logarithmic CFTs in type A_{1} .

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.

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.