Class Of Algebras Research Articles

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.

On Four Dimensional Absolute Valued Algebras With nonzero Omnipresent Idempotent

n this paper, we studies the absolute valued algebras of dimension four, containing nonzero omnipresent idempotent. And we construct algebraically some news classes of algebras.

Metrics related to oscillator algebras

A Lie algebra is said to be metric if it admits a symmetric, invariant, and non-degenerate bilinear form. The harmonic oscillator algebra, which arises in the quantum mechanical description of a harmonic oscillator, is the smallest solvable non-abelian metric example. This algebra is the first step of a countable series of solvable Lie algebras that support invariant Lorentzian forms. Generalizing this situation, in this paper, we arrive at the oscillator Lie -algebras as double extensions of metric spaces. The aim of this paper is to present some structural features, invariant metrics, and derivations of this class of algebras and to explore their possibilities of being extended to mixed metric Lie algebras.

Quasi-free states on a class of algebras of multicomponent commutation relations

Multicomponent commutations relations (MCRs) describe plektons, i.e. multicomponent quantum systems with a generalized statistics. In such systems, exchange of quasiparticles is governed by a unitary matrix [Formula: see text] that depends on the position of quasiparticles. For such an exchange to be possible, the matrix must satisfy several conditions, including the functional Yang–Baxter equation. The aim of the paper is to give an appropriate definition of a quasi-free state on an MCR algebra, and construct such states on a class of MCR algebras. We observe a significant difference between the classical setting for bosons and fermions and the setting of MCR algebras. We show that the developed theory is applicable to systems that contain quasiparticles of opposite type. An example of such a system is a two-component system in which two quasiparticles, under exchange, change their respective types to the opposite ones ([Formula: see text], [Formula: see text]). Fusion of quasiparticles means intuitively putting several quasiparticles in an infinitely small box and identifying the statistical behavior of the box. By carrying out fusion of an odd number of particles from the two-component system as described above, we obtain further examples of quantum systems to which the developed theory is applicable.

A Study on the Centroid of a Class of Solvable Lie Algebras

A Study on the Centroid of a Class of Solvable Lie Algebras

Counterfactuals as modal conditionals, and their probability

In this paper we propose a semantic analysis of Lewis' counterfactuals. By exploiting the structural properties of the recently introduced boolean algebras of conditionals, we show that counterfactuals can be expressed as formal combinations of a conditional object and a normal necessity modal operator. Specifically, we introduce a class of algebras that serve as modal expansions of boolean algebras of conditionals, together with their dual relational structures. Moreover, we show that Lewis' semantics based on sphere models can be reconstructed in this framework. As a consequence, we establish the soundness and completeness of a slightly stronger variant of Lewis' logic for counterfactuals with respect to our algebraic models. In the second part of the paper, we present a novel approach to the probability of counterfactuals showing that it aligns with the uncertainty degree assigned by a belief function, as per Dempster-Shafer theory, to its associated conditional formula. Furthermore, we characterize the probability of a counterfactual in terms of Gärdenfors' imaging rule for the probabilistic update.

Essential Fierz identities for a fermionic field

For a single fermionic field, an interpretation of the Fierz identities (which establish relations between the bilinear field observables) is given. They appear closely related to the algebraic class (regular or singular) of the spin 2-form S associated to the spinor field. If S ≠ 0, the Fierz identities follow from the 3 + 1 decomposition of the eigenvector equations for S with respect to an inertial laboratory, which makes this interpretation suitable for fermionic particle physics models. When S = 0, the Fierz identities reduce to three constraints on the current densities associated with the spinor field, saying that they are orthogonal, equimodular, the vector current being time-like and the axial one being space-like.

Encoding de Finetti's coherence within Łukasiewicz logic and MV-algebras

The present paper investigates proof-theoretical and algebraic properties for the probability logic FP(Ł,Ł), meant for reasoning on the uncertainty of Łukasiewicz events. Methodologically speaking, we will consider a translation function between formulas of FP(Ł,Ł) to the propositional language of Łukasiewicz logic that allows us to apply the latter and the well-developed theory of MV-algebras directly to probabilistic reasoning. More precisely, leveraging on such translation map, we will show proof-theoretical properties for FP(Ł,Ł) and introduce a class of algebras with respect to which FP(Ł,Ł) will be proved to be locally sound and complete. Finally, we will apply these previous results to investigate what we called “probabilistic unification problem”. In this respect, we will prove that Ghilardi's algebraic view on unification can be extended to our case and, on par with the Łukasiewicz propositional case, we show that probabilistic unification is of nullary type.

Weight modules over Bell–Rogalski algebras

We study a class of Z-graded algebras introduced by Bell and Rogalski. Their construction generalizes in large part that of rank one generalized Weyl algebras (GWAs). We establish certain ring-theoretic properties of these algebras and study their connection to GWAs. We classify the simple weight modules in the infinite orbit case and provide a partial classification in the case of orbits of finite order.

Kac-Moody symmetry in the light front of gauge theories

We discuss the emergence of a new symmetry generator in a Hamiltonian realisation of four-dimensional gauge theories in the flat space foliated by retarded (advanced) time. It generates an asymptotic symmetry that acts on the asymptotic fields in a way different from the usual large gauge transformations. The improved canonical generators, corresponding to gauge and asymptotic symmetries, form a classical Kac-Moody charge algebra with a non-trivial central extension. In particular, we describe the case of electromagnetism, where the charge algebra is the U(1) current algebra with a level proportional to the coupling constant of the theory, κ = 4π2/e2. We construct bilinear generators yielding Virasoro algebras on the null boundary. We also provide a non-Abelian generalization of the previous symmetries by analysing the evolution of Yang-Mills theory in Bondi coordinates.

Hypercyclic Generalized Shift Operators

In this paper, we study the linear dynamical properties of shift operators on some classes of Segal algebras. Moreover, we characterize hypercyclic generalized bilateral shift operators on the standard Hilbert module.

Levi-flat CR structures on compact Lie groups

Pittie (Proc Indian Acad Sci Math Sci 98:117-152, 1988) proved that the Dolbeault cohomology of all left-invariant complex structures on compact Lie groups can be computed by looking at the Dolbeault cohomology induced on a conveniently chosen maximal torus. We generalized Pittie’s result to left-invariant Levi-flat CR structures of maximal rank on compact Lie groups. The main tools we used was a version of the Leray–Hirsch theorem for CR principal bundles and the algebraic classification of left-invariant CR structures of maximal rank on compact Lie groups (Charbonnel and Khalgui in J Lie Theory 14:165-198, 2004) .