Class Of Algebras Research Articles

On non-counital Frobenius algebras

A Frobenius algebra is a finite-dimensional algebra [Formula: see text] which comes equipped with a coassociative, counital comultiplication map [Formula: see text] that is an [Formula: see text]-bimodule map. Here, we examine comultiplication maps for generalizations of Frobenius algebras: finite-dimensional self-injective (quasi-Frobenius) algebras. We show that large classes of such algebras, including finite-dimensional weak Hopf algebras, come equipped with a nonzero map [Formula: see text] as above that is not necessarily counital. We also conjecture that this comultiplicative structure holds for self-injective algebras in general.

A classification of endo-commutative curled algebras of dimension 2 over a nontrivial field

An endo-commutative algebra is a nonassociative algebra in which the square mapping preserves multiplication. In this paper, we give a complete classification of endo-commutative curled algebras of dimension 2 over an arbitrary nontrivial field, where a curled algebra satisfies the condition that the square of any element is a scalar multiple of that element. We list all multiplication tables of the algebras up to isomorphism.

The conceptual difficulties Saudi female ELL students encountered in learning algebra: A case of a college preparatory program

This research explores the type of conceptual difficulties that female Saudi students encounter in Introductory Algebra courses at college level and identifies pedagogical practices that might impact students’ learning of algebra. Instructional tasks were designed and implemented for 12 weeks to collect and analyze data. The research sample consisted of a study group of 28 students from an Introductory Algebra class. The findings identified and classified the types of difficulties that the students encountered and suggested instructional models to overcome or minimize them and a theme to teach and learn algebra.

Explicit algebraic classification of vacuum, shearfree and non-twisting spacetimes at Large D

Explicit algebraic classification of vacuum, shearfree and non-twisting spacetimes at Large D

2-generated axial algebras of Monster type (2β,β)

Axial algebras of Monster type (α,β) are a class of non-associative algebras that includes, besides associative algebras, other important examples such as the Jordan algebras and the Griess algebra. 2-generated primitive axial algebras of Monster type (α,β) naturally split into three cases: the case when α∉{2β,4β}, the case α=4β and α=2β. In this paper we give a complete classification all 2-generated primitive axial algebras of Monster type (2β,β).

Evolution algebras satisfying train identity of degree 2 and exponent 3

In this paper, we give the necessary and sufficient conditions for an evolution algebra to satisfy a train identity of degree 2 and exponent 3. We show that this class of algebras is a subclass of the Bernstein algebras of order 2. Then, we study the relations existing between these algebras and the Bernstein algebras as well as the power associative algebras. Moreover we give a classification in dimension ≤ 4 of evolution algebras satisfying strictly a train identity of degree 2 and exponent 3. Finally, we describe the derivations and automorphisms of these algebras.

Left-symmetric superalgebras on special linear Lie superalgebras

In this paper, we study the existence and classification problems of left-symmetric superalgebras on special linear Lie superalgebras sl(m|n) with m≠n. The main three results of this paper are: (i) a complete classification of the left-symmetric superalgebras on sl(2|1), (ii) sl(m|1) does not admit left-symmetric superalgebras for m≥3, and (iii) sl(m+1|m) admits a left-symmetric superalgebra for every m≥1. To prove these results we combine previous results on the existence and classification of left-symmetric algebras on the Lie algebras glm with a detailed analysis of small representations of the Lie superalgebras sl(m|1). We also conjecture that sl(m|n) admits left-symmetric superalgebras if and only if m=n+1.

Countably Generated Algebras of Analytic Functions on Banach Spaces

In the paper, we study various countably generated algebras of entire analytic functions on complex Banach spaces and their homomorphisms. Countably generated algebras often appear as algebras of symmetric analytic functions on Banach spaces with respect to a group of symmetries, and are interesting for their possible applications. Some conditions of the existence of topological isomorphisms between such algebras are obtained. We construct a class of countably generated algebras, where all normalized algebraic bases are equivalent. On the other hand, we find non-isomorphic classes of such algebras. In addition, we establish the conditions of the hypercyclicity of derivations in countably generated algebras of entire analytic functions of the bounded type. We use methods from the theory of analytic functions of several variables, the theory of commutative Fréchet algebras, and the theory of linear dynamical systems.

Algebraic classification of the gravitational field in Weyl-Cartan spacetimes

Algebraic classification of the gravitational field in Weyl-Cartan spacetimes

Pedagogical discourse markers in online algebra learning: Unraveling instructor's communication using natural language processing

Despite the proliferation of video-based instruction and its benefits—such as promoting student autonomy and self-paced learning—the complexities of online teaching remain a challenge. To be effective, educators require extensive training in digital teaching methodologies. As such, there's a pressing need to examine and comprehend the intricacies of instructors' communication patterns within this context. This research addresses the pressing need to understand pedagogical discourse in online video lectures in Algebra classes by employing computational linguistic tools and natural language processing (NLP). Using transcripts from 125 Algebra 1 video lectures—comprising 4962 instances of pedagogical discourse—from five instructors at Math Nation, a virtual math learning environment, we analyzed the conveyance of linguistic, attitudinal, and emotional nuances. With the aid of 26 Coh-Metrix and SÉANCE features, we classified educators' language choices, achieving an accuracy of 86.7%. Furthermore, variations in language choices, as signified by discourse markers, were examined through a K-means clustering approach. The resulting 17 clusters were grouped into interpersonal, structural, and cognitive pedagogic functions. Through this exploration, we demonstrate the promising potential of NLP in efficiently deciphering pedagogical communication patterns in video lectures. These insights open a new avenue for research, aimed at assessing the efficacy of digital instruction by scrutinizing pedagogical discourse characteristics in computer-based learning environments.

Alkaline: A Simplified Post-Quantum Encryption Algorithm for Classroom Use

This paper describes Alkaline, a size-reduced version of Kyber, which has recently been announced as a prototype NIST standard for post-quantum public-key cryptography. While not as simple as RSA, I believe that Alkaline can be used in an undergraduate classroom to effectively teach the techniques and principles behind Kyber and post-quantum cryptography in general. Classroom experiences with individual concepts used in Alkaline support this belief. In addition to cryptography, linear algebra and abstract algebra classes would be good candidates for the use of Alkaline. A few exercises suitable for use in these classes are included.

Quadratic Lie algebras with 2-plectic structures

We study the existence of 2-plectic structures on Lie algebras which admit an ad-invariant non-degenerate symmetric bilinear form, frequently called quadratic Lie algebras. It is well-known that every centerless quadratic Lie algebra admits a 2-plectic form but not many quadratic examples with nontrivial center are known. In this paper we give several constructions to obtain large families of 2-plectic quadratic Lie algebras with nontrivial center, many of them among the class of nilpotent Lie algebras. We give some sufficient conditions to assure that certain extensions of 2-plectic quadratic Lie algebras result to be 2-plectic as well. We prove that every quadratic and symplectic Lie algebra with dimension greater than 4 also admits a 2-plectic form. Further, conditions to assure that one may find a 2-plectic which is exact on certain quadratic Lie algebras are also obtained.

AN ASSESSMENT-DIRECTED, STUDENT-CENTERED, AND MASTERY-BASED MODEL OF TEACHING AND LEARNING IN MATHEMATICS

This is a descriptive article illustrating a multi-year research study in mathematics education of college students in the U.S. It focuses on the first two years in the undergraduate mathematics curriculum. The study presents the theory, practice, and results of an innovative instructional program (Keystone model) applied to algebra classes at the college. The research investigated two areas of student learning outcomes: performance in mathematics and improvements in students’ concentration skills. Under the application of this program, students not only achieved significant outcomes in mathematics, as evidenced by formative and summative assessments but also improved their concentration skills, as demonstrated in the gains in national reading comprehension tests. In the following, we present, in extensive detail, the elements of the Keystone approach and its implementation in a learning environment, such as a college mathematics classroom.

Exceptional cycles in triangular matrix algebras

An exceptional cycle in a triangulated category with Serre functor is a generalization of a spherical object. Suppose that A and B are Gorenstein algebras, given a perfect exceptional n-cycle E⁎ in Kb(A-proj) and a perfect exceptional m-cycle F⁎ in Kb(B-proj), we construct an A-B-bimodule N, and prove the product E⁎⊠F⁎ is an exceptional (n+m−1)-cycle in Kb(Λ-proj), where Λ=(AN0B). Using this construction, one gets many new exceptional cycles which is unknown before for certain class of algebras.

On the observability of Galois representations and the Tate conjecture

The Tate conjecture has two parts: i) Tate classes are linear combination of algebraic classes, ii) semisimplicity of Galois representations (for smooth projective varieties). B. Moonen proved that i) implies ii) in characteristic 0, using p-adic Hodge theory.We show that an unconditional result lies behind this implication: the observability of arithmetic monodromy groups of geometric origin (in any characteristic) - which leads to a sharpening of Moonen's result.We also discuss another aspect of the Tate conjecture related to the transcendence of p-adic periods.

ON THE GROUP OF POINTWISE INNER AUTOMORPHISMS

Let $L_{m,c}$ stand for the free metabelian nilpotent Lie algebra of class $c$ of rank $m$ over a field $K$ of characteristic zero.
 Automorphisms of the form $\varphi(x_i)=e^{adu_i}(x_i)$ are called pointwise inner, where $e^{adu_i}$, is the inner automorphism
 induced by the element $u_i\in L_{m,c}$ for each $i=1,\ldots,m$. In the present study, we investigate the group structure of
 the group $\text{\rm PInn}(L_{m,c})$ of pointwise inner automorphisms of $L_{m,c}$ for low nilpotency classes.

Algebras Satisfying Polynomial Identity of Degree Six that are Principal Train

In this paper we study the class of algebras satisfying a polynomial identity of degree six that are principal train algebras of rank $3$ or $4$, for which we give the explicit form of the train equation. If the rank of $A$ is $n\geq 5$ in general, we provide the form of the train equation in some cases.

On 10-dimensional Exceptional Drinfeld algebras

Abstract Based on Mubarakzyanov’s classification of four-dimensional real Lie algebras, we classify ten-dimensional Exceptional Drinfeld algebras (EDAs). The classification is restricted to EDAs whose maximal isotropic (geometric) subalgebras cannot be represented as a product of a 3D Lie algebra and a 1D abelian factor. We collect the obtained algebras into families depending on the dualities found between them. Despite algebras related by a generalized Yang–Baxter deformation we find two algebras related by a different Nambu–Lie U-duality transformation. We show that this duality relates two Type IIA backgrounds.

Biderivations, commuting mappings and 2-local derivations of -graded Lie algebras of maximal class

In Fialowski's classification for algebras of maximal class, there are three Lie algebras of maximal class with one-dimensional homogeneous components: m 0 , L 1 and m 2 . In this paper, we study their biderivations by considering the embedded mapping to derivation algebras. Then we determine commuting mappings on these algebras as an application of biderivations. Finally, local and 2-local derivations for these three algebras are characterized as the given gradings.

The Cluster Multiplication Theorem for Acyclic Quantum Cluster Algebras

Abstract Let $Q$ be a finite acyclic valued quiver. We give the cluster multiplication formulas in the quantum cluster algebra of $Q$ with arbitrary coefficients, by applying certain quotients of derived Hall subalgebras of $Q$. These formulas can be viewed as the quantum version of the cluster multiplication theorem in the classical cluster algebra proved by Caldero–Keller for finite type, Hubery for affine type, and Xiao–Xu for acyclic quivers.