Quotient Category Research Articles

Linkage and translation for tensor products of representations of simple algebraic groups and quantum groups

AbstractLet be either a simple linear algebraic group over an algebraically closed field of characteristic or a quantum group at an ‐th root of unity. We define a tensor ideal of singular ‐modules in the category of finite‐dimensional ‐modules and study the associated quotient category , called the regular quotient. For , the Coxeter number of , we establish a ‘linkage principle’ and a ‘translation principle’ for tensor products: Let be the essential image in of the principal block of . We first show that is closed under tensor products in . Then we prove that the monoidal structure of is governed to a large extent by the monoidal structure of . These results can be combined to give an external tensor product decomposition , where denotes the Verlinde category of .

Quadratic differentials as stability conditions: Collapsing subsurfaces

Abstract We introduce a new class of triangulated categories, which are Verdier quotients of three-Calabi–Yau categories from (decorated) marked surfaces, and show that its spaces of stability conditions can be identified with moduli spaces of framed quadratic differentials on Riemann surfaces with arbitrary order zeros and arbitrary higher order poles. A main tool in our proof is a comparison of two exchange graphs, obtained by tilting hearts in the quotient categories and by flipping mixed angulations associated with the quadratic differentials.

Risk assessment of Cr, As, and Sb to aquatic communities at the water-sediment interface in the lower Yellow River, China, integrated with the Diffusive Gradients in Thin‐films technique

The dynamic concentrations of labile metals/metalloids, measured by the Diffusive Gradients in Thin-films (DGT) technique, can be used for evaluating the toxicity risk of heavy metals to aquatic organisms in sediments. In this study, the different potential risks of the three sediment compartments, overlying water (W), sediment- water interface (SWI) and sediment (S) were independently assessed by studying the concentrations of Cr, As, and Sb in the sediment and water from the lower Yellow River. Total concentrations of these elements and bioavailable concentrations, using DGTs, were explored and probabilistic risk assessment (PRA) based on a Monte Carlo model was performed. The results show that the total heavy metal concentrations in sediment and overlying water were spatially variable due to influences of natural geography and sediment distribution. The trace elements mean concentrations using DGT were 1.68, 1.69 and 0.73 μg·L−1 for Cr, 0.61, 0.51 and 0.39 μg·L−1 for As, and 0.78, 1.16, and 0.41 μg·L−1 for Sb in W, SWI and S, respectively. Among them, the DGT-labile concentration of As displayed a large spatial variability, indicating a close association with anthropogenic activities. The risk quotient (RQ) categories for single heavy metal toxicity demonstrated to be moderate risk for Cr and almost no risk for As and Sb. The combined biotoxicity probability assessment of the three heavy metals showed that they would cause almost no harmful effects to benthic organisms in the whole ecosystem of the Yellow River.

Student’s Creative Thinking Process about Numeracy: A Case of Student’s Adversity Quotient (AQ)

The study aims to describe the creative thinking process of junior high school students with different Adversity Quotient (AQ) categories (quitter, camper, and climber) in solving numeracy problems. This qualitative research involves three subjects representing distinct AQ categories. Data collection includes the ARP questionnaire, numeracy problem-solving tests, and interviews, with analysis using the Pierce sign concept. Findings reveal camper and climber subjects effectively separate information in the ideation stage, while quitter and climber subjects associate knowledge concepts in generating idea. In preparing a plan for application, camper and climber subjects strategize based on information and knowledge, while in applying an idea, quitter subjects surrender, campers choose easier alternatives, and climbers persist with pre-planned strategies. Results suggest tailored motivation according to students' AQ categories can enhance teacher-student engagement.

FAITHFULNESS OF THE 2-BRAID GROUP VIA ZIGZAG ALGEBRA IN TYPE B

AbstractWe show that a certain category of bimodules over a finite-dimensional quiver algebra known as a type B zigzag algebra is a quotient category of the category of type B Soergel bimodules. This leads to an alternate proof of Rouquier’s conjecture on the faithfulness of the 2-braid groups for type B.

Quotients of exact categories by pseudo-cluster-tilting subcategories

We introduce the concept of a pseudo-cluster-tilting subcategory from the viewpoint of the fact that the quotient of an exact category by a cluster-tilting subcategory is an abelian category. We prove that the quotients in the case of pseudo-cluster-tilting are always semi-abelian. In addition, it is abelian if and only if some self-orthogonal conditions are satisfied. We revisit the abelian quotient category of conflations by splitting ones, and get that there exists a unique exact substructure such that it is a cluster quotient. Communicated by Julia Plavnik

A higher-dimensional Chevalley restriction theorem for orthogonal groups

We prove a higher-dimensional Chevalley restriction theorem for orthogonal groups, which was conjectured by Chen and Ngô for reductive groups. In characteristic p>2, we also prove a weaker statement. In characteristic 0, the theorem implies that the categorical quotient of a commuting scheme by the diagonal adjoint action of the group is integral and normal. As applications, we deduce some trace identities and a certain multiplicative property of the Pfaffian over an arbitrary commutative algebra.

ERROR ANALYSIS OF PROSPECTIVE MATHEMATICS TEACHER IN SOLVING ADDITIONAL AND MULTIPLICATION RULES PROBLEM IN COMBINATORICS BASED ON ADVERSITY QUOTIENT

The background of this research is that there are still many prospective teacher students who still experience difficulties in solving addition and multiplication rules in combinatorics. This research is a qualitative descriptive research that aims to find out the mistakes of prospective teacher students in solving addition and multiplication rules in combinatorics based on adversity quitient. The research subjects were student teacher candidates in the 3rd semester of Mathematics Education, Faculty of Teaching and Education, Sriwijaya University. There are stages for this research including 1) preparation stage, 2) implementation stage, 3) final stage. Data collection was in the form of questionnaires and written tests. The results of the study were that student teacher candidates who had a Quitter type adversity quotient had 0 student teacher candidates with a percentage of 0%, Camper type adversity quotient had 58 student teacher candidates with a percentage of 82.86%, Climber type adversity quotient had 12 student teacher candidates with a percentage of 17 .14%. As well as prospective teacher students in the Adversity quotient category of the Climber type, they made 4 mistakes according to Newman's procedure, namely Comprehension Error, Transformation Error, Process Skill Error and Encoding Error. Student teacher candidates in the adversity quotient category of Camperr type made 5 mistakes according to Newman's procedure, namely Reading Errors, Comprehension Errors, Transformation Errors, Process Skill Errors and Encoding Errors.

Pre-(n + 2)-angulated categories

In this article, we introduce the notion of a pre-(n+2)-angulated category as a higher dimensional analogue of a pre-triangulated category defined by Beligiannis-Reiten. We first show that the idempotent completion of a pre-(n+2)-angulated category admits a unique pre-(n+2)-angulated structure. Let (C,E,s) be an n-exangulated category and X be a strongly functorially finite subcategory of C. We then show that the quotient category C/X is a pre-(n+2)-angulated category. These results allow to construct several examples of pre-(n+2)-angulated categories. Moreover, we also give a necessary and sufficient condition for the quotient C/X to be an (n+2)-angulated category.

The effect of integrating society, science, environment, technology, and collaborative mind mapping (ISSETCM2) model on mathematical literacy in terms of adversity quotient

The implementation of this study aims to ensure the effectiveness of the Integrating Society, Science, Environment, Technology, and Collaborative Mind Mapping (ISSETCM2) learning model on mathematical literacy skills in terms of adversity quotient (AQ). This research design is quasi-experimental, with the population being all eighth-grade students in the academic year 2021/2022. The research sample was selected by cluster random. The average value of the mathematical literacy ability of the experimental class students is 72.03, and that of the control class is 50.34. Based on data interpretation and analysis of variance (ANOVA) of two-path unequal cells with a significant value of . The value of Sig is 0.001, so Sig . Based on the data, there are differences in the results of mathematical literacy skills between the learning model (ISSETCM2) and direct instruction learning models. Also, there are differences in adversity quotient categories (climbers, campers, and quitters) on mathematical literacy abilities. There is no interaction between learning model factors and adversity quotient on the results of students' mathematical literacy abilities. From these data, it can be concluded that there is an effect of the ISSETCM2 learning model on mathematical literacy skills in terms of the adversity quotient of class VIII. The research results show that each student has a different adversity quotient, which can affect students' mathematical reasoning abilities. For this reason, students need to know the adversity quotient to maximize their abilities.

Stable categories and triangulated equivalences

Extriangulated categories are a generalization of triangular categories and exact categories, and many important conclusions can be unified under this framework. In this paper, the localized quotient induced by the model structure is realized by the ideal quotient in the extriangulated category, i.e., the triangulated equivalence between the two quotient categories is established. At the same time, this result is also applied to the Frobenius exact category, the homotopy category of a ring $R$, and the stable category of Gorenstein objects.

Localization, monoid sets and K-theory

We develop the K-theory of sets with an action of a pointed monoid (or monoid scheme), analogous to the K-theory of modules over a ring (or scheme). In order to form localization sequences, we construct the quotient category of a nice regular category by a Serre subcategory.

K‐flat complexes and derived categories

Let R $R$ be a ring with identity. Inspired by recent work in Emmanouil, Preprint, 2021, we show that the derived category of R $R$ is equivalent to the chain homotopy category of all K-flat complexes with pure-injective components. This is implicitly related to a recollement we exhibit. It expresses D p u r ( R ) $\mathcal {D}_{pur}(R)$ , the pure derived category of R $R$ , as an attachment of the usual derived category D ( R ) $\mathcal {D}(R)$ with Emmanouil's quotient category D K − f l a t ( R ) : = K ( R ) / K − F l a t $\mathcal {D}_{\textnormal {K-flat}}(R):=K(R)/K\textnormal {-Flat}$ , which we call here the K-flat derived category. It follows that this Verdier quotient is a compactly generated triangulated category. We obtain our results by using methods of cotorsion pairs to construct (cofibrantly generated) monoidal abelian model structures on the exact category of chain complexes along with the degreewise pure exact structure. In fact, most of our model structures are obtained as corollaries of a general method which associates an abelian model structure to any class of so-called C $\mathcal {C}$ -acyclic complexes, where C $\mathcal {C}$ is any given class of chain complexes. Finally, we also give a new characterization of K-flat complexes in terms of the pure derived category of R $R$ .

Negative cluster categories from simple minded collection quadruples

Fomin and Zelevinsky’s definition of cluster algebras laid the foundation for cluster theory. The various categorifications and generalizations of the original definition led to Iyama and Yoshino’s generalized cluster categories coming from positive-Calabi-Yau triples Jin later defined simple minded collection quadruples where the special case is the analogue of Iyama and Yang’s triples: negative-Calabi-Yau triples. In this paper, we further study the quotient categories coming from simple minded collection quadruples. Our main result uses limits and colimits to describe Hom-spaces over in relation to the easier to understand Hom-spaces over Moreover, we apply our theorem to give a different proof of a result by Jin: if we have a negative-Calabi-Yau triple, then is a negative cluster category.

Cherednik algebra for the normalizer

Ginzburg, Guay, Opdam and Rouquier established an equivalence of categories between a quotient category of the category 𝒪 for the rational Cherednik algebra and the category of finite dimension modules of the Hecke algebra of a complex reflection group W. We announce a generalization of this result to the extension of the Hecke algebra associated to the normalizer of a reflection subgroup.

Abelian subcategories of triangulated categories induced by simple minded systems

If k is a field, A a finite dimensional k-algebra, then the simple A-modules form a simple minded collection in the derived category \(\mathscr {D}^{ {\text {b}}}( {\text {mod}}\,A )\). Their extension closure is \({\text {mod}}\,A\); in particular, it is abelian. This situation is emulated by a general simple minded collection \(\mathscr {S}\) in a suitable triangulated category \(\mathscr {C}\). In particular, the extension closure \(\langle \mathscr {S}\rangle \) is abelian, and there is a tilting theory for such abelian subcategories of \(\mathscr {C}\). These statements follow from \(\langle \mathscr {S}\rangle \) being the heart of a bounded t-structure. It is a defining characteristic of simple minded collections that their negative self extensions vanish in every degree. Relaxing this to vanishing in degrees \(\{ -w+1, \ldots , -1 \}\) where w is a positive integer leads to the rich, parallel notion of w-simple minded systems, which have recently been the subject of vigorous interest. If \(\mathscr {S}\) is a w-simple minded system for some \(w \geqslant 2\), then \(\langle \mathscr {S}\rangle \) is typically not the heart of a t-structure. Nevertheless, using different methods, we will prove that \(\langle \mathscr {S}\rangle \) is abelian and that there is a tilting theory for such abelian subcategories. Our theory is based on Quillen’s notion of exact categories, in particular a theorem by Dyer which provides exact subcategories of triangulated categories. The theory of simple minded systems can be viewed as “negative cluster tilting theory”. In particular, the result that \(\langle \mathscr {S}\rangle \) is an abelian subcategory is a negative counterpart to the result from (higher) positive cluster tilting theory that if \(\mathscr {T}\) is a cluster tilting subcategory, then \(( \mathscr {T}* \Sigma \mathscr {T})/[ \mathscr {T}]\) is an abelian quotient category.

Categorical properties of some algorithms of differentiation for equipped posets

In this paper it is proved that the algorithms of differentiation VIII-X (introduced by A.G. Zavadskij to classify equipped posets of tame representation type) induce categorical equivalences between some quotient categories, in particular, analgorithm Az is introduced to build equipped posets with a pair ofpoints (a, b) suitable for differentiation VII such that the subset of strong points related with the weak pointais not empty.

Penalaran Visuospasial Siswa Kategori Intelligence Quotient (IQ) Superior

The purpose of this study was to describe the superior intelligence quotient (IQ) category students' visuospatial reasoning. This research was conducted at MAN 1 Trenggalek because the school had already conducted an IQ test. This research uses a qualitative approach with a descriptive exploratory type. The subjects in this study were students who had already done an IQ test with a superior IQ. This study will describe 2 subjects based on relatively the same tendency. Data collection techniques used documentary studies on IQ, test questions, think aloud, and semi-structured interviews. The data analysis in this study refers to Tversky B's research on visuospatial reasoning. The validity of the data in this study used technical triangulation. The results of this study indicate that students with superior IQ are able to meet the indicators of visuospatial reasoning, namely external representation, analysis, synthesis and conclusions as a whole. This is indicated by students with superior IQ who are able to complete the questions according to the procedure and completely.

Resolving by a free action linear category and applications to Hochschild-Mitchell (co)homology

Let G be a group acting on a small category C over a field k, that is C is a G-k-category. We first obtain an unexpected result: C is resolvable by a category which is G-k-equivalent to it, on which G acts freely on objects.This resolving category enables to show that if the coinvariants and the invariants functors are exact, then the coinvariants and invariants of the Hochschild-Mitchell (co)homology of C are isomorphic to the trivial component of the Hochschild-Mitchell (co)homology of the skew category C[G].If the action of G is free on objects, then there is a canonical decomposition of the Hochschild-Mitchell (co)homology of the quotient category C/G along the conjugacy classes of G. This way we provide a general frame for monomorphisms which have been described previously in low degrees.

Higher-dimensional Auslander–Reiten theory on (d+2)-angulated categories

AbstractLet $\mathscr{C}$ be a $(d+2)$ -angulated category with d-suspension functor $\Sigma^d$ . Our main results show that every Serre functor on $\mathscr{C}$ is a $(d+2)$ -angulated functor. We also show that $\mathscr{C}$ has a Serre functor $\mathbb{S}$ if and only if $\mathscr{C}$ has Auslander–Reiten $(d+2)$ -angles. Moreover, $\tau_d=\mathbb{S}\Sigma^{-d}$ where $\tau_d$ is d-Auslander–Reiten translation. These results generalize work by Bondal–Kapranov and Reiten–Van den Bergh. As an application, we prove that for a strongly functorially finite subcategory $\mathscr{X}$ of $\mathscr{C}$ , the quotient category $\mathscr{C}/\mathscr{X}$ is a $(d+2)$ -angulated category if and only if $(\mathscr{C},\mathscr{C})$ is an $\mathscr{X}$ -mutation pair, and if and only if $\tau_d\mathscr{X} =\mathscr{X}$ .