Classification Of Representations Research Articles

Use of Artificial Intelligence in the Classification of Elementary Oral Lesions from Clinical Images.

Artificial intelligence has generated a significant impact in the health field. The aim of this study was to perform the training and validation of a convolutional neural network (CNN)-based model to automatically classify six clinical representation categories of oral lesion images. The CNN model was developed with the objective of automatically classifying the images into six categories of elementary lesions: (1) papule/nodule; (2) macule/spot; (3) vesicle/bullous; (4) erosion; (5) ulcer and (6) plaque. We selected four architectures and using our dataset we decided to test the following architectures: ResNet-50, VGG16, InceptionV3 and Xception. We used the confusion matrix as the main metric for the CNN evaluation and discussion. A total of 5069 images of oral mucosa lesions were used. The oral elementary lesions classification reached the best result using an architecture based on InceptionV3. After hyperparameter optimization, we reached more than 71% correct predictions in all six lesion classes. The classification achieved an average accuracy of 95.09% in our dataset. We reported the development of an artificial intelligence model for the automated classification of elementary lesions from oral clinical images, achieving satisfactory performance. Future directions include the study of including trained layers to establish patterns of characteristics that determine benign, potentially malignant and malignant lesions.

Indecomposable objects in Khovanov–Sazdanovic's generalizations of Deligne's interpolation categories

Khovanov and Sazdanovic recently introduced symmetric monoidal categories parameterized by rational functions and given by quotients of categories of two-dimensional cobordisms. These categoriesgeneralizeDeligne'sinterpolation categories of representations of symmetric groups. In this paper, we classify indecomposable objects and identify the associated graded Grothendieck rings of Khovanov–Sazdanovic's categories through sums of representation categories over crossed products of polynomial rings over a general field. To obtain these results, we introduce associated graded categories for Krull–Schmidt categories with filtrations as a categorification of the associated graded Grothendieck ring, and study field extensions and Galois descent for Krull–Schmidt categories.

From octonions to composition superalgebras via tensor categories

The nontrivial unital composition superalgebras, of dimension 3 and 6, which exist only in characteristic 3, are obtained from the split Cayley algebra and its order 3 automorphisms, by means of the process of semisimplification of the symmetric tensor category of representations of the cyclic group of order 3. Connections with the extended Freudenthal magic square in characteristic 3, that contains some exceptional Lie superalgebras specific of this characteristic are discussed too. In the process, precise recipes to go from (nonassociative) algebras in this tensor category to the corresponding superalgebras are given.

The indecomposable objects in the center of Deligne's category Re̲pSt$\protect\underline{{\rm Re}}\!\operatorname{p}S_t$

AbstractWe classify the indecomposable objects in the monoidal center of Deligne's interpolation category by viewing as a model‐theoretic limit in rank and characteristic. We further prove that the center of is semisimple if and only if is not a non‐negative integer. In addition, we identify the associated graded Grothendieck ring of this monoidal center with that of the graded sum of the centers of representation categories of finite symmetric groups with an induction product. We prove analogous statements for the abelian envelope.

Homotopy theory of net representations

The homotopy theory of representations of nets of algebras over a (small) category with values in a closed symmetric monoidal model category is developed. We illustrate how each morphism of nets of algebras determines a change-of-net Quillen adjunction between the model categories of net representations, which is furthermore, a Quillen equivalence when the morphism is a weak equivalence. These techniques are applied in the context of homotopy algebraic quantum field theory with values in cochain complexes. In particular, an explicit construction is presented that produces constant net representations for Maxwell [Formula: see text]-forms on a fixed oriented and time-oriented globally hyperbolic Lorentzian manifold.

Careers of Algerian Women Managers: The Psychological Drivers of Discrimination in Professional Careers

Today, the texts governing the Algerian economy claim equality between men and women in the labour market: no training, no profession, and no position of responsibility is legally closed to women. Algerian law prohibits all discrimination in hiring and career development. The imbalance has even been reversed in one essential respect: women are entering the university field more than men and are more successful. However, they account for only 19% (ONS, 2020) of the working population and are still only marginally present in positions of responsibility in the public and private sectors. However, this progressive and constant access of women to so-called ‘male’ bastions has led to the emergence of a professional category, namely female managers. The reason that led us to choose this research object is the desire to understand and apprehend the professional pathway of this category of women (executives), to attempt, through professional and family representations and perceptions, to understand the identity construction of women occupying executive positions. In this perspective, empirical research built on a qualitative approach, based on semi-directive interviews with 20 women executives working in a public paramilitary institution, appeared relevant. This research aimed to understand the career path through the professional experiences of women managers insofar as it highlighted professional trajectories interacting with family life and different professional rhythms in terms of career. More generally, how women managers in this institution articulate their private and professional life and place them in an organisational context. In this sense, studying the practices and social representations of women managers means understanding the codes, values, and ideologies that women’s work occupies in Algerian society as a whole and this paramilitary institution.

Ribbon tensor structure on the full representation categories of the singlet vertex algebras

We show that the category of finite-length generalized modules for the singlet vertex algebra M(p), p∈Z>1, is equal to the category OM(p) of C1-cofinite M(p)-modules, and that this category admits the vertex algebraic braided tensor category structure of Huang-Lepowsky-Zhang. Since OM(p) includes the uncountably many typical M(p)-modules, which are simple M(p)-module structures on Heisenberg Fock modules, our results substantially extend our previous work on tensor categories of atypical M(p)-modules. We also introduce a tensor subcategory OM(p)T, graded by an algebraic torus T, which has enough projectives and is conjecturally tensor equivalent to the category of finite-dimensional weight modules for the unrolled restricted quantum group of sl2 at a 2pth root of unity. We compute all tensor products involving simple and projective M(p)-modules, and we prove that both tensor categories OM(p) and OM(p)T are rigid and thus also ribbon. As an application, we use vertex operator algebra extension theory to show that the representation categories of all finite cyclic orbifolds of the triplet vertex algebras W(p) are non-semisimple modular tensor categories, and we confirm a conjecture of Adamović-Lin-Milas on the classification of simple modules for these finite cyclic orbifolds.

On Minimal Tilting Complexes in Highest Weight Categories

We explain the construction of minimal tilting complexes for objects of highest weight categories and we study in detail the minimal tilting complexes for standard objects and simple objects. For certain categories of representations of complex simple Lie algebras, affine Kac-Moody algebras and quantum groups at roots of unity, we relate the multiplicities of indecomposable tilting objects appearing in the terms of these complexes to the coefficients of Kazhdan-Lusztig polynomials.

Monoidal abelian envelopes and a conjecture of Benson and Etingof

We give several criteria to decide whether a given tensor category is the abelian envelope of a fixed symmetric monoidal category. As a main result we prove that the category of finite-dimensional representations of a semisimple simply connected algebraic group is the abelian envelope of the category of tilting modules. Benson and Etingof conjectured that a certain limit of finite symmetric tensor categories is tensor equivalent to the finite dimensional representations of $SL_2$ in characteristic $2$. We use our results on the abelian envelopes to prove this conjecture and its variants for any prime $p$.

The parity of Lusztig's restriction functor and Green's formula

Our investigation in the present paper is based on three important results. (1) In [14], Ringel introduced Hall algebra for representations of a quiver over finite fields and proved the elements corresponding to simple representations satisfy the quantum Serre relation. This gives a realization of the nilpotent part of quantum group if the quiver is of finite type. (2) In [6], Green found a homological formula for the representation category of the quiver and equipped Ringel's Hall algebra with a comultiplication. The generic form of the composition subalgebra of Hall algebra generated by simple representations realizes the nilpotent part of quantum group of any type. (3) In [11], Lusztig defined induction and restriction functors for the perverse sheaves on the variety of representations of the quiver which occur in the direct images of constant sheaves on flag varieties, and he found a formula between his induction and restriction functors which gives the comultiplication as algebra homomorphism for quantum group. In the present paper, we prove the formula holds for all semisimple complexes with Weil structure. This establishes the categorification of Green's formula.

Exotic t-structures for two-block Springer fibres

We study the category of representations of s l m + 2 n \mathfrak {sl}_{m+2n} in positive characteristic, with p p -character given by a nilpotent with Jordan type ( m + n , n ) (m+n,n) . Recent work of Bezrukavnikov-Mirkovic [Ann. of Math. (2) 178 (2013), pp. 835–919] implies that this representation category is equivalent to D m , n 0 \mathcal {D}_{m,n}^0 , the heart of the exotic t-structure on the derived category of coherent sheaves on a Springer fibre for that nilpotent. Using work of Cautis and Kamnitzer [Duke Math. J. 142 (2008), pp. 511–588], we construct functors indexed by affine tangles, between these categories D m , n \mathcal {D}_{m,n} (i.e. for different values of n n ). This allows us to describe the irreducible objects in D m , n 0 \mathcal {D}_{m,n}^0 and enumerate them by crossingless ( m , m + 2 n ) (m,m+2n) matchings. We compute the E x t \mathrm {Ext} spaces between the irreducible objects, and conjecture that the resulting Ext algebra is an annular variant of Khovanov’s arc algebra. In subsequent work, we use these results to give combinatorial dimension formulae for the irreducible representations. These results may be viewed as a positive characteristic analogue of results about two-block parabolic category O \mathcal {O} due to Lascoux-Schutzenberger [Astérisque, vol. 87, Soc. Math. France, Paris, 1981, pp. 249–266], Bernstein-Frenkel-Khovanov [Selecta Math. (N.S.) 5 (1999), pp. 199–241], Brundan-Stroppel [Represent. Theory 15 (2011), pp. 170–243], et al.

Quantum cluster characters of Hall algebras revisited

Let Q be a finite acyclic valued quiver. We define a bialgebra structure and an integration map on the Hall algebra associated to the morphism category of projective representations of Q. As an application, we recover the surjective homomorphism defined in [12], which realizes the principal coefficient quantum cluster algebra \(\mathcal {A}\,_q(Q)\) as a sub-quotient of the Hall algebra of morphisms. Moreover, we also recover the quantum Caldero–Chapoton formula, as well as some multiplication formulas between quantum Caldero–Chapoton characters.

Dyck path categories and its relationships with cluster algebras

Dyck path categories are introduced as a combinatorial model of the category of representations of quivers of Dynkin-type [Formula: see text]. In particular, it is proved that there is a bijection between some Dyck paths and perfect matchings of some snake graphs. The approach allows us to give formulas for cluster variables in cluster algebras of Dynkin-type [Formula: see text] in terms of Dyck paths.

Harish-Chandra Pairs in the Verlinde Category in Positive Characteristic

Abstract In this article, we develop the theory of commutative and cocommutative Hopf algerbas in the Verlinde category and prove that the category of affine group schemes of finite type in the Verlinde category is equivalent to the category of Harish-Chandra pairs in the Verlinde category. Subsequently, we extend this equivalence to an equivalence between corresponding representation categories.

Coherent categorification of quantum loop algebras: The SL(2) case

Abstract We construct an equivalence of graded Abelian categories from a category of representations of the quiver-Hecke algebra of type A 1 ( 1 ) {A_{1}^{(1)}} to the category of equivariant perverse coherent sheaves on the nilpotent cone of type A. We prove that this equivalence is weakly monoidal. This gives a representation-theoretic categorification of the preprojective K-theoretic Hall algebra considered by Schiffmann and Vasserot. Using this categorification, we compare the monoidal categorification of the quantum open unipotent cells of type A 1 ( 1 ) {A_{1}^{(1)}} given by Kang, Kashiwara, Kim, Oh and Park in terms of quiver-Hecke algebras with the one given by Cautis and Williams in terms of equivariant perverse coherent sheaves on the affine Grassmannians.

Certain complex representations of [formula omitted

We introduce the representation category C(G) for a connected reductive algebraic group G which is defined over a finite field Fq of q elements. We show that this category has many good properties for G=SL2(F¯q). In particular, it is an abelian category and a highest weight category. Moreover, we classify the simple objects in C(G) for G=SL2(F¯q).

On Frobenius-Perron dimension

We propose a notion of Frobenius-Perron dimension for certain free Z \mathbb {Z} -modules of infinite rank and compute it for the Z \mathbb {Z} -modules of finite dimensional complex representations of unitary groups with nonnegative dominant weights. The definition of Frobenius-Perron dimension that we are introducing naturally generalizes the well-known Frobenius-Perron dimension on the category of finite dimensional complex representations of a finite group.

Adjoint Functors on the Representation Category of OI

Adjoint Functors on the Representation Category of OI

A covariant Stinespring theorem

We prove a finite-dimensional covariant Stinespring theorem for compact quantum groups. Let G be a compact quantum group, and let T≔Rep(G) be the rigid C*-tensor category of finite-dimensional continuous unitary representations of G. Let Mod(T) be the rigid C*-2-category of cofinite semisimple finitely decomposable T-module categories. We show that finite-dimensional G-C*-algebras can be identified with equivalence classes of 1-morphisms out of the object T in Mod(T). For 1-morphisms X:T→M1, Y:T→M2, we show that covariant completely positive maps between the corresponding G-C*-algebras can be “dilated” to isometries τ: X → Y ⊗ E, where E:M2→M1 is some “environment” 1-morphism. Dilations are unique up to partial isometry on the environment; in particular, the dilation minimizing the quantum dimension of the environment is unique up to a unitary. When G is a compact group, this recovers previous covariant Stinespring-type theorems.

Categorical Koszul duality

In this paper we establish Koszul duality between dg categories and a class of curved coalgebras, generalizing the corresponding result for dg algebras and conilpotent curved coalgebras. We show that the normalized chain complex functor transforms the Quillen equivalence between quasicategories and simplicial categories into this Koszul duality. This allows us to give a conceptual interpretation of the dg nerve of a dg category and its adjoint. As an application, we prove that the category of representations of a quasicategory K is equivalent to the coderived category of comodules over C⁎(K), the chain coalgebra of K. A corollary of this is a characterization of the category of constructible dg sheaves on a stratified space as the coderived category of a certain dg coalgebra.