Algebraic Objects Research Articles

Book review: “Tertions: Strange Algebraic Objects”

Book review: “Tertions: Strange Algebraic Objects”

On generalized Sidon spaces

Sidon spaces have been introduced by Bachoc, Serra and Zémor as the q-analogue of Sidon sets, classical combinatorial objects introduced by Simon Szidon. In 2018 Roth, Raviv and Tamo introduced the notion of r-Sidon spaces, as an extension of Sidon spaces, which may be seen as the q-analogue of Br-sets, a generalization of classical Sidon sets. Thanks to their work, the interest on Sidon spaces has increased quickly because of their connection with cyclic subspace codes they pointed out. This class of codes turned out to be of interest since they can be used in random linear network coding. In this work we focus on a particular class of them, the one-orbit cyclic subspace codes, through the investigation of some properties of Sidon spaces and r-Sidon spaces, providing some upper and lower bounds on the possible dimension of their r-span and showing explicit constructions in the case in which the upper bound is achieved. Moreover, we provide further constructions of r-Sidon spaces, arising from algebraic and combinatorial objects, and we show examples of Br-sets constructed by means of them.

Mini-Workshop: Bridging Number Theory and Nichols Algebras via Deformations

Nichols algebras are graded Hopf algebra objects in braided tensor categories. They appeared first in a paper by Nichols in 1978 in the search for new examples of Hopf algebras. Rediscovered later several times, they also provide a conceptual explanation of the construction of quantum groups. The aim of the workshop is to review recent developments in the field, initiate collaborations, and discuss new approaches to open problems.

On fusing matrices associated with conformal boundary conditions

In the context of rational conformal field theories (RCFT) we look at the fusing matrices that arise when a topological defect is attached to a conformal boundary condition. We call such junctions open topological defects. One type of fusing matrices arises when two open defects fuse while another arises when an open defect passes through a boundary operator. We use the topological field theory approach to RCFTs based on Frobenius algebra objects in modular tensor categories to describe the general structure associated with such matrices and how to compute them from a given Frobenius algebra object and its representation theory. We illustrate the computational process on the rational free boson theories. Applications to boundary renormalisation group flows are briefly discussed.

Model i implementacja dwurdzeniowego sterownika programowalnego opartego na maszynie wirtualnej

The concept, semantic model, and prototype implementation of a dual-core programmable controller have been presented. The controller’s design concept involves parallel processing of two execution programs through a virtual machine, utilizing a shared memory area for global variables. The presented model provides a formal description of the execution of portable binary programs created based on the languages of the IEC 61131-3 standard in the CPDev programming environment. The architecture described outlines the operation of the virtual machine using abstract algebraic objects. The solution has been implemented in C/C++ on a dual-core microcontroller platform.

Local Non Abelian Class Field Theory

The “local class field theory”, which can be defined as the description of the extensions of a given local field K in terms of the algebraic and analytic objects depending ony on the base K is one of the central problems of modern number theory. The theory developed for the abelian extensions, around the fundamental works of Artin and Hasse in the first quarter of the 20th century. It is natural to ask if one could construct this theory including the non-abelian extensions of the base field. There are two approches to this problem. One approach is based on the ideas of Langlands, and the other on Koch. Koch’s method was later generalized by Fesenko and Koch-de Shalit for some non-abelian extensions of the base field. On the other hand, Ikeda and Serbest extended Fesenko’s works to construct a non-abelian local class field theory. But there is a restriction for the base field K. In this study, we extended Ikeda-Serbest’s construction of the local reciprocity map for a certain local field to any local field. Also we have shown that the extended map satisfies the certain functoriality and ramification theoretic properties.

GeoGebra—A great platform for experiential learning, explorations and creativity in mathematics

In this paper we are presenting some examples of how GeoGebra is used in: a) explaining concepts of the first derivative, monotony, extremums; b) studying the properties of the function (strictly increasing/decreasing); c) demonstrating the mean value theorem. The results and the conclusions are based on the experiment carried out in the teaching and learning process in the chapter of derivatives in a third-year class of a secondary school in Albania. Also, there are some encouraging facts got by the use of GeoGebra: the double representation and the dynamic features of GeoGebra allow the students to quickly grasp the mathematical concepts and properties and be actively involved in further explorations. The use of GeoGebra tools is similar to the use of the tools of virtual games, and this is a great advantage to stimulate the students to learn mathematics and master their mathematical performance the same way they play games. On the other side, using GeoGebra, it is easier for the teachers to explain mathematical concepts, the properties of algebraic objects, to discuss about different situations and aspects of the subject under study and to methodically reason the results got. GeoGebra provides a very commodious environment for the students to effectively interoperate with each other. Our results showed that GeoGebra is effective in teaching and learning mathematics. GeoGebra software contributed in enhancing students’ understanding of mathematical concepts and improved students’ interest to learn mathematics. Also, we admit that not all the stages of implementing GeoGebra software in the classroom are flowing smoothly. Based on our experience and the other researchers, it is observed that the effectiveness is dependent on the way GeoGebra is integrated in the teaching and learning process, implying that the research must continue with the commitment of many researchers of mathematics, physics and other sciences.

On homology groups for pairwise comparisons method

In this study, we introduce pairwise comparisons matrix classification based on homology groups of graphs with unique vertices. Algebraic topology transforms a sequence of topological objects (such as graphs associated with pairwise comparison matrices) into algebraic objects such as homology groups. It is the first attempt to use this tool to classify matrices of pairwise comparisons based on the triads in which the inconsistency occurs. The Koczkodaj inconsistency indicator was used in this study.

Knotted 4-regular graphs. II. Consistent application of the Pachner moves

A common choice for the evolution of the knotted graphs in loop quantum gravity is to use the Pachner moves, adapted to graphs from their dual triangulations. Here, we show that the natural way to consistently use these moves is on framed graphs with edge twists, where the Pachner moves can only be performed when the twists, and the vertices the edges are incident on, meet certain criteria. For other twists, one can introduce an algebraic object, which allow any knotted graph with framed edges to be written in terms of a generalized braid group.

Brauer Analysis of Some Cayley and Nilpotent Graphs and Its Application in Quantum Entanglement Theory

Cayley and nilpotent graphs arise from the interaction between graph theory and algebra and are used to visualize the structures of some algebraic objects as groups and commutative rings. On the other hand, Green and Schroll introduced Brauer graph algebras and Brauer configuration algebras to investigate the algebras of tame and wild representation types. An appropriated system of multisets (called a Brauer configuration) induces these algebras via a suitable bounded quiver (or bounded directed graph), and the combinatorial properties of such multisets describe corresponding indecomposable projective modules, the dimensions of the algebras and their centers. Undirected graphs are examples of Brauer configuration messages, and the description of the related data for their induced Brauer configuration algebras is said to be the Brauer analysis of the graph. This paper gives closed formulas for the dimensions of Brauer configuration algebras (and their centers) induced by Cayley and nilpotent graphs defined by some finite groups and finite commutative rings. These procedures allow us to give examples of Hamiltonian digraph constructions based on Cayley graphs. As an application, some quantum entangled states (e.g., Greenberger–Horne–Zeilinger and Dicke states) are described and analyzed as suitable Brauer messages.

Module categories, internal bimodules, and Tambara modules

AbstractWe use the theory of Tambara modules to extend and generalize the reconstruction theorem for module categories over a rigid monoidal category to the nonrigid case. We show a biequivalence between the 2‐category of cyclic module categories over a monoidal category and the bicategory of algebra and bimodule objects in the category of Tambara modules on . Using it, we prove that a cyclic module category can be reconstructed as the category of certain free module objects in the category of Tambara modules on , and give a sufficient condition for its reconstructability as module objects in . To that end, we extend the definition of the Cayley functor to the nonclosed case, and show that Tambara modules give a proarrow equipment for ‐module categories, in which ‐module functors are characterized as 1‐morphisms admitting a right adjoint. Finally, we show that the 2‐category of all ‐module categories embeds into the 2‐category of categories enriched in Tambara modules on , giving an “action via enrichment” result.

The formal theory of relative monads

We develop the theory of relative monads and relative adjunctions in a virtual equipment, extending the theory of monads and adjunctions in a 2-category. The theory of relative comonads and relative coadjunctions follows by duality. While some aspects of the theory behave analogously to the non-relative setting, others require new insights. In particular, the universal properties that define the algebra object and the opalgebra object for a monad in a virtual equipment are stronger than the classical notions of algebra object and opalgebra object for a monad in a 2-category. Inter alia, we prove a number of representation theorems for relative monads, establishing the unity of several concepts in the literature, including the devices of Walters, the j-monads of Diers, and the relative monads of Altenkirch, Chapman, and Uustalu. A motivating setting is the virtual equipment ▪ of categories enriched in a monoidal category ▪, though many of our results are new even for ▪.

$\alpha$-induction for bi-unitary connections

The tensor functor called \alpha -induction produces a new unitary fusion category from a Frobenius algebra object, or a Q -system, in a braided unitary fusion category. In the operator algebraic language, it gives extensions of endomorphism of N to M arising from a subfactor N\subset M of finite index and finite depth, which gives a braided fusion category of endomorphisms of N . It is also understood in terms of Ocneanu’s graphical calculus. We study this \alpha -induction for bi-unitary connections, which provides a characterization of finite-dimensional nondegenerate commuting squares, and present certain 4 -tensors appearing in recent studies of 2 -dimensional topological order. We show that the resulting \alpha -induced bi-unitary connections are flat if we start with a commutative Frobenius algebra, or a local Q -system. Examples related to chiral conformal field theory and the Dynkin diagrams are presented.

Detecting algebra objects from NIM-reps in pointed, near-group and quantum group-like fusion categories

In this article we study the possible Morita equivalence classes of algebras in three families of fusion categories (pointed, near-group and (A1,l)12) by studying the Non-negative Integer Matrix representations (NIM-reps) of their underlying fusion ring, and compare these results with existing classification results of algebra objects. Also, in an appendix we include a test of the exponents conjecture for modular tensor categories of rank up to 4.

Gauging non-invertible symmetries: topological interfaces and generalized orbifold groupoid in 2d QFT

Gauging is a powerful operation on symmetries in quantum field theory (QFT), as it connects distinct theories and also reveals hidden structures in a given theory. We initiate a systematic investigation of gauging discrete generalized symmetries in two-dimensional QFT. Such symmetries are described by topological defect lines (TDLs) which obey fusion rules that are non-invertible in general. Despite this seemingly exotic feature, all well-known properties in gauging invertible symmetries carry over to this general setting, which greatly enhances both the scope and the power of gauging. This is established by formulating generalized gauging in terms of topological interfaces between QFTs, which explains the physical picture for the mathematical concept of algebra objects and associated module categories over fusion categories that encapsulate the algebraic properties of generalized symmetries and their gaugings. This perspective also provides simple physical derivations of well-known mathematical theorems in category theory from basic axiomatic properties of QFT in the presence of such interfaces. We discuss a bootstrap-type analysis to classify such topological interfaces and thus the possible generalized gaugings and demonstrate the procedure in concrete examples of fusion categories. Moreover we present a number of examples to illustrate generalized gauging and its properties in concrete conformal field theories (CFTs). In particular, we identify the generalized orbifold groupoid that captures the structure of fusion between topological interfaces (equivalently sequential gaugings) as well as a plethora of new self-dualities in CFTs under generalized gaugings.

Tensor Categories for Vertex Operator Superalgebra Extensions

Let V V be a vertex operator algebra with a category C \mathcal {C} of (generalized) modules that has vertex tensor category structure, and thus braided tensor category structure, and let A A be a vertex operator (super)algebra extension of V V . We employ tensor categories to study untwisted (also called local) A A -modules in C \mathcal {C} , using results of Huang-Kirillov-Lepowsky that show that A A is a (super)algebra object in C \mathcal {C} and that generalized A A -modules in C \mathcal {C} correspond exactly to local modules for the corresponding (super)algebra object. Both categories, of local modules for a C \mathcal {C} -algebra and (under suitable conditions) of generalized A A -modules, have natural braided monoidal category structure, given in the first case by Pareigis and Kirillov-Ostrik and in the second case by Huang-Lepowsky-Zhang. Our main result is that the Huang-Kirillov-Lepowsky isomorphism of categories between local (super)algebra modules and extended vertex operator (super)algebra modules is also an isomorphism of braided monoidal (super)categories. Using this result, we show that induction from a suitable subcategory of V V -modules to A A -modules is a vertex tensor functor. Two applications are given: First, we derive Verlinde formulae for regular vertex operator superalgebras and regular 1 2 Z \frac {1}{2}\mathbb {Z} -graded vertex operator algebras by realizing them as (super)algebra objects in the vertex tensor categories of their even and Z \mathbb {Z} -graded components, respectively. Second, we analyze parafermionic cosets C = C o m ( V L , V ) C=\mathrm {Com}(V_L,V) where L L is a positive definite even lattice and V V is regular. If the vertex tensor category of either V V -modules or C C -modules is understood, then our results classify all inequivalent simple modules for the other algebra and determine their fusion rules and modular character transformations. We illustrate both directions with several examples.

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The algorithm for canonical forms of neural ideals

<abstract><p>To elucidate the combinatorial architecture of neural codes, the neural ideal $ J_C $, an algebraic object, was introduced. Represented in its canonical form, $ J_C $ provides a succinct characterization of the inherent receptive field architecture within the code. The polynomials in $ J_C $ are also instrumental in determining the relationships among the neurons' receptive fields. Consequently, the computation of the collection of canonical forms is pivotal. In this paper, based on the study of relations between pseudo-monomials, the authors present a computationally efficient iterative algorithm for the canonical forms of the neural ideal. Additionally, we introduce a new relationship among the neurons' receptive fields, which can be characterized by if-and-only-if statements, relating both to $ J_C $ and to a larger ideal of a code $ I(C) $.</p></abstract>

On tertions and other algebraic objects

The concept of the object called “tertion” is discussed. Some operations over tertions are introduced and their properties are studied. The relationship between tertions, complex numbers are quaternions are discussed.

Creation of Problems by Prospective Teachers to Develop Proportional and Algebraic Reasonings in a Probabilistic Context

To promote optimal learning in their students, mathematics teachers must be proficient in problem posing, making this skill a cornerstone in teacher training programs. This study presents a formative action in which pre-service teachers are required to create and analyze a problem involving proportional reasoning within a probabilistic context. For this problem, they must identify the objects and processes involved in its resolution, recognize the degree of algebraic reasoning implied and identify potential difficulties for students. Subsequently, they need to formulate and analyze a new problem with variation, which mobilizes higher-level algebraic activity. Results indicate that prospective teachers struggle to pose problems that engage proportional reasoning, as well as to identify in their analysis which elements of proportional and algebraic reasoning are present in their solutions. Despite this fact, a significant percentage of participants adequately modify the original problem to address higher levels of algebraic reasoning, identifying in these cases the new algebraic objects and potential difficulties that might arise as the degree of generalization required in the solution increases. The study concludes by underscoring the importance of training in problem posing to enhance the knowledge and competences of prospective teachers concerning proportional and algebraic reasoning.