General Algebra Research Articles

The hom-associative Weyl algebras in prime characteristic

We introduce the first hom-associative Weyl algebras over a field of prime characteristic as a generalization of the first associative Weyl algebra in prime characteristic. First, we study properties of hom-associative algebras constructed from associative algebras by a general ``twisting'' procedure. Then, with the help of these results, we determine the commuter, center, nuclei, and set of derivations of the first hom-associative Weyl algebras. We also classify them up to isomorphism, and show, among other things, that all nonzero endomorphisms on them are injective, but not surjective. Last, we show that they can be described as a multi-parameter formal hom-associative deformation of the first associative Weyl algebra, and that this deformation induces a multi-parameter formal hom-Lie deformation of the corresponding Lie algebra, when using the commutator as bracket.

Level -1/2 Realization of Quantum N-Toroidal Algebra in Type Cn

We construct a level -1/2 vertex representation of the quantum [Formula: see text]-toroidal algebra of type [Formula: see text], which is a natural generalization of the usual quantum toroidal algebra. The construction also provides a vertex representation of the quantum toroidal algebra for type [Formula: see text] as a by-product.

Bumping operators and insertion algorithms for queer supercrystals

Results of Morse and Schilling show that the set of increasing factorizations of reduced words for a permutation is naturally a crystal for the general linear Lie algebra. Hiroshima has recently constructed two superalgebra analogues of such crystals. Specifically, Hiroshima has shown that the sets of increasing factorizations of involution words and fpf-involution words for a self-inverse permutation are each crystals for the queer Lie superalgebra. In this paper, we prove that these crystals are normal and identify their connected components. To accomplish this, we study two insertion algorithms that may be viewed as shifted analogues of the Edelman-Greene correspondence. We prove that the connected components of Hiroshima's crystals are the subsets of factorizations with the same insertion tableau for these algorithms, and that passing to the recording tableau defines a crystal morphism. This confirms a conjecture of Hiroshima. Our methods involve a detailed investigation of certain analogues of the Little map, through which we extend several results of Hamaker and Young.

Hopf-Galois algebras and their Poisson structures

Hopf-Galois objects are natural generalizations of Hopf algebras with a Galois-theoretic flavor. In this paper, we prove a criterion for an Ore extension of a Hopf-Galois algebra to be a Hopf-Galois algebra, introduce a concept of Poisson Hopf-Galois algebra and establish a relationship between Poisson Hopf-Galois algebras and Poisson Hopf algebras. More importantly, we study Poisson Hopf-Galois structures on Poisson polynomial algebras, and give a necessary and sufficient condition for the Poisson enveloping algebra of a Poisson Hopf-Galois algebra to be a Hopf-Galois algebra.

Minimal Mahler measures for generators of some fields

We prove that for each odd integer d \geq 3 there are infinitely many number fields K of degree d such that each generator \alpha of K has Mahler measure greater than or equal to d^{-d}|\Delta_K|^{\frac{d+1}{d(2d-2)}} , where \Delta_K is the discriminant of the field K . This, combined with an earlier result of Vaaler and Widmer for composite d , answers negatively a question of Ruppert raised in 1998 about ‘small’ algebraic generators for every d \geq 3 . We also show that for each d \geq 2 and any \varepsilon>0 , there exist infinitely many number fields K of degree d such that every algebraic integer generator \alpha of K has Mahler measure greater than (1-\varepsilon)|\Delta_K|^{{1}/{d}} . On the other hand, every such field K contains an algebraic integer generator \alpha with Mahler measure smaller that |\Delta_K|^{{1}/{d}} . This generalizes the corresponding bounds recently established by Eldredge and Petersen for d=3 .

A note on the $\mathcal{A}$-generators of the polynomial algebra of six variables and applications

Let $ \mathcal P_{n}:=H^{*}((\mathbb{R}P^{\infty})^{n}) \cong \mathbb Z_2[x_{1},x_{2},\ldots,x_{n}]$ be the polynomial algebra of $n$ generators $x_1, x_2, \ldots, x_n$ with the degree of each $x_i$ being 1. We investigate the Peterson hit problem for the polynomial algebra $ \mathcal P_{n},$ regarded as a module over the mod-$2$ Steenrod algebra, $ \mathcal{A}.$ For $n>4,$ this problem remains unsolvable, even with the aid of computers in the case of $n=5.$ In this article, we study the hit problem for the case $n=6$ in degree $d_s=6(2^s -1)+3.2^s,$ with $s$ an arbitrary nonnegative integer. By considering $ \mathbb Z_2$ as a trivial $ \mathcal A$-module, then the hit problem is equivalent to the problem of finding a basis of $ \mathbb Z_2$-vector space $ \mathbb Z_2 {\otimes}_{\mathcal{A}}\mathcal P_{n}.$ The main goal of the current article is to explicitly determine an admissible monomial basis of the $ \mathbb Z_2$vector space $ \mathbb Z_2{\otimes}_{\mathcal{A}}\mathcal P_n$ for $n=6$ in some degrees. One of the most important applications of the hit problem is to investigate homomorphism introduced by Singer, which is a homomorphism $$ \varphi_n :\text{Tor}^{\mathcal A}_{n, n+d} (\mathbb Z_2,\mathbb Z_2) \longrightarrow (\mathbb{Z}_{2}{\otimes}_{\mathcal{A}}\mathcal P_{n})_d^{GL(n; \mathbb Z_2)}$$ from the homology of the Steenrod algebra to the subspace of $(\mathbb{Z}_{2}{\otimes}_{\mathcal{A}}\mathcal P_{n})_d$ consisting of all the $GL(n; \mathbb Z_2)$-invariant classes. It is a useful tool in describing the homology groups of the Steenrod algebra, $\text{Tor}^{\mathcal A}_{n, n+d}(\mathbb Z_2,\mathbb Z_2).$ The behavior of the sixth Singer algebraic transfer in degree $d_s=6(2^s -1)+3.2^s$ was also discussed at the end of this paper.

Козображення матричних алгебр Манна

Let A be an algebra over a field K, m and n natural numbers and P = (pji) a fixed n x m matrix over A. The K-vector space of all m x n matrices over the algebra A can be made into an algebra with respect to the following operation (o): B o C = BPC. This algebra is called the Munn matrix algebra over A with sandwich matrix P. The algebras of such type arose as generalizations of semigroup algebras of Rees matrix semigroups which in turn are closely related to simple semigroups. This article describes the generators and defining relations of Mann matrix algebras with a regular sandwich matrix.

Clifford Algebra and Hypercomplex Number as well as Their Applications in Physics

The Clifford algebra is a unification and generalization of real number, complex number, quaternion, and vector algebra, which accurately and faithfully characterizes the intrinsic properties of space-time, providing a unified, standard, elegant, and open language and tools for numerous complicated mathematical and physical theories. So it is worth popularizing in the teaching of undergraduate physics and mathematics. Clifford algebras can be directly generalized to 2n-ary associative algebras. In this generalization, the matrix representation of the orthonormal basis of space-time plays an important role. The matrix representation carries more information than the abstract definition, such as determinant and the definition of inverse elements. Without this matrix representation, the discussion of hypercomplex numbers will be difficult. The zero norm set of hypercomplex numbers is a closed set of special geometric meanings, like the light-cone in the realistic space-time, which has no substantial effect on the algebraic calculus. The physical equations expressed in Clifford algebra have a simple formalism, symmetrical structure, standard derivation, complete content. Therefore, we can hope that this magical algebra can complete a new large synthesis of modern science.

Statistical modelling for a new family of generalized distributions with real data applications.

The modern trend in distribution theory is to propose hybrid generators and generalized families using existing algebraic generators along with some trigonometric functions to offer unique, more flexible, more efficient, and highly productive G-distributions to deal with new data sets emerging in different fields of applied research. This article aims to originate an odd sine generator of distributions and construct a new G-family called "The Odd Lomax Trigonometric Generalized Family of Distributions". The new densities, useful functions, and significant characteristics are thoroughly determined. Several specific models are also presented, along with graphical analysis and detailed description. A new distribution, "The Lomax cosecant Weibull" (LocscW), is studied in detail. The versatility, robustness, and competency of the LocscW model are confirmed by applications on hydrological and survival data sets. The skewness and kurtosis present in this model are explained using modern graphical methods, while the estimation and statistical inference are explored using many estimation approaches.

A noncommutative spacetime realization of quantum black holes, Regge trajectories and holography

It is shown that the radial spectrum associated with a fuzzy sphere in a noncommutative phase space characterized by the Yang algebra, leads exactly to a Regge-like spectrum GMl2=l=1,2,3,…, for all positive values of l, and which is consistent with the extremal quantum Kerr black hole solution that occurs when the outer and inner horizon radius coincide r+=r−=GM. The condition GMl2=l is tantamount to the mass-angular momentum relation Ml2=lMp2 implying a mass-squared quantization in multiples of the Planck-mass-squared. Another important feature (also pointed out by Tanaka) is the holographic nature of these results that are based in recasting the Yang algebra associated with an 8D noncommuting phase space, involving xμ,pν,μ,ν=0,1,2,3, in terms of the undeformed realizations of the Lorentz algebra generators JAB corresponding to a 6D-spacetime, and associated to a 12D-phase-space with coordinates XA,PA;A=0,1,2,…,5. We finalize with a discussion of the noncommuting 3D isotropic and Born oscillators. Finding solutions to these oscillators merit investigation because they introduce explicit dynamics to the quantum black holes. We hope that the findings in this work, relating the Regge-like spectrum l=GM2 and the quantized area of black hole horizons in Planck bits, via the Yang algebra in Noncommutative phase spaces, will help us elucidate some of the impending issues pertaining the black hole information paradox and the role that string theory and quantum information will play in its resolution.

On Quantum-MV Algebras - Part I: The Orthomodular Algebras

We prove that almost all the properties of quantum-MV algebras are verified by orthomodular algebras, the new algebras introduced in a previous paper. We put a special insight on transitive antisymmetric orthomodular (taOM) algebras, generalizations of MV algebras. We make the connection with IMTL and NM algebras. In memoriam Dragos. Vaida (1933 – 2020)

Menger systems of idempotent cyclic and weak near-unanimity multiplace functions

Multiplace functions, which are also called functions of many variables, and their algebras called Menger algebras have been studied in various fields of mathematics. Based on the theory of many-sorted algebras, the primary aim of this paper is to present the ideas of Menger systems and Menger systems of full multiplace functions which are natural generalizations of Menger algebras and Menger algebras of [Formula: see text]-ary operations, respectively. Two specific types of [Formula: see text]-ary operations, which are called idempotent cyclic and weak near-unanimity generated by cyclic and weak near-unanimity terms, are provided. The Menger algebras under consideration have a two-element universe, the elements of which are two specific [Formula: see text]-ary operations. Additionally, we provide necessary and sufficient conditions in which the abstract Menger algebra and the Menger algebras of these two [Formula: see text]-ary operations are isomorphic. An abstract characterization of unitary Menger systems via systems of idempotent cyclic and weak near-unanimity multiplace functions is generally investigated. A strong connection between clone of terms and Menger systems of full multiplace functions is also investigated.

The Basis Invariant of Generalized n-Cube Symmetries Group with Odd Degrees

The basis polynomial invariants with even degrees relatively to the symmetries group were described in cited literature. Here, the polynomial invariants with odd degrees are constructed. We give an explicit construction of all the basic polynomial invariants as algebra generators of odd degrees relatively to the symmetries group. All calculations are presented in detail.

Lie Algebra Representation, Conservation Laws and Some Invariant Solutions for a Generalized Emden-Fowler Equation

All generators of the optimal algebra associated with a generalization of the Endem-Fowler equation are showed; some of them allow to give invariant solutions. Variational symmetries and the respective conservation laws are also showed. Finally, a representation of Lie symmetry algebra is showed by groups of matrices.

Economic-environmental analysis of renewable-based microgrid under a CVaR-based two-stage stochastic model with efficient integration of plug-in electric vehicle and demand response

• Developing the electric vehicles and demand response in the MG’ operation. • Analyzing the economic-environmental performance of flexible MG. • Developing a risk-aversion strategy in the MG performance with multiple technologies. • Applying the CVaR technique to manage the risk of the proposed probabilistic model. The optimal scheduling of microgrid systems as a promising milieu to improve energy efficiency and environmental benefits is faced with several challenges related to strong uncertainties. Hence, control the risk in the daily operation of the microgrid to overcome the effects of random variables such as renewable power, load, and energy prices in the proposed scheduling are urgently required. Motivated by mentioned challenges, this paper focuses on the optimal risk assessment of the microgrid incorporated with an electric vehicle parking lot, demand response program (as two emerging flexible resources), and high penetration of renewable energy (wind and solar energies) to enhance financial and environmental goals. To indicate the taken strategies (risk-taker or risk-aversion strategies) by the system operator in the daily scheduling in facing PV and wind power variation, real-time power market, load variation, and behavior of the electric vehicle drivers, the conditional value-at-risk as a risk measure criterion is employed. The proposed model is formulated as a mixed-integer linear two-stage stochastic and modeled in General Algebra Modeling System. A 33-bus smart microgrid test system is considered to demonstrate the model's effectiveness, and simulation results are presented for several cases. Results indicate that the simultaneous integration of electric vehicles and demand response in the microgrid reduces the total operation cost by 9.97%. Besides, the emission pollution and renewable power curtailment are respectively reduced up to 12.34% and 8.49% under the proposed approach. Also, under the risk-aversion strategy, the low expected operation cost is obtained in the higher risk level.

Spinorial Snyder and Yang models from superalgebras and noncommutative quantum superspaces

The relativistic Lorentz-covariant quantum space-times obtained by Snyder can be described by the coset generators of (anti) de-Sitter algebras. Similarly, the Lorentz-covariant quantum phase spaces introduced by Yang, which contain additionally quantum curved fourmomenta and quantum-deformed relativistic Heisenberg algebra, can be defined by suitably chosen coset generators of conformal algebras. We extend such algebraic construction to the respective superalgebras, which provide quantum Lorentz-covariant superspaces (SUSY Snyder model) and indicate also how to obtain the quantum relativistic phase superspaces (SUSY Yang model). In last Section we recall briefly other ways of deriving quantum phase (super)spaces and we compare the spinorial Snyder type models defining bosonic or fermionic quantum-deformed spinors.

Generalizations of harmonic functions in $${\mathbb R}^m$$

In recent works, arbitrary structural sets in the non-commutative Clifford analysis context have been used to introduce non-trivial generalizations of harmonic Clifford algebra valued functions in $\mathbb{R}^m$. Being defined as the solutions of elliptic (generally non-strongly elliptic) partial differential equations, $(\varphi,\psi)$-inframonogenic and $(\varphi,\psi)$-harmonic functions do not share the good structure and properties of the harmonic ones. The aim of this paper it to show and clarified the relationship between these classes of functions.

Generalized parafermions of orthogonal type

There is an embedding of affine vertex algebras Vk(gln)↪Vk(sln+1), and the coset Ck(n)=Com(Vk(gln),Vk(sln+1)) is a natural generalization of the parafermion algebra of sl2. It was called the algebra of generalized parafermions by the third author and was shown to arise as a one-parameter quotient of the universal two-parameter W∞-algebra of type W(2,3,…). In this paper, we consider an analogous structure of orthogonal type, namely Dk(n)=Com(Vk(so2n),Vk(so2n+1))Z2. We realize this algebra as a one-parameter quotient of the two-parameter even spin W∞-algebra of type W(2,4,…), and we classify all coincidences between its simple quotient Dk(n) and the algebras Wℓ(so2m+1) and Wℓ(so2m)Z2. As a corollary, we show that for the admissible levels k=−(2n−2)+12(2n+2m−1) for soˆ2n the simple affine algebra Lk(so2n) embeds in Lk(so2n+1), and the coset is strongly rational. As a consequence, the category of ordinary modules of Lk(so2n+1) at such a level is a braided fusion category.

Quantum Fokker–Planck Dynamics

The Fokker-Planck equation is a partial differential equation which is a key ingredient in many models in physics. This paper aims to obtain a quantum counterpart of Fokker-Planck dynamics, as a means to describing quantum Fokker-Planck dynamics. Given that relevant models relate to the description of large systems, the quantization of the Fokker-Planck equation should be done in a manner that respects this fact, and is therefore carried out within the setting of non-commutative analysis based on general von Neumann algebras. Within this framework we present a quantization of the generalized Laplace operator, and then go on to incorporate a potential term conditioned to noncommutative analysis. In closing we then construct and examine the asymptotic behaviour of the corresponding Markov semigroups. We also present a noncommutative Csiszar-Kullback inequality formulated in terms of a notion of relative entropy, and show that for more general systems, good behaviour with respect to this notion of entropy ensures similar asymptotic behaviour of the relevant dynamics.

Onsager algebra and algebraic generalization of Jordan-Wigner transformation

Recently, an algebraic generalization of the Jordan-Wigner transformation was introduced and applied to one- and two-dimensional systems. This transformation is composed of the interactions ηi that appear in the Hamiltonian H as H=∑i=1NJiηi, where Ji are coupling constants. In this short note, it is derived that operators that are composed of ηi, or its n-state clock generalizations, satisfy the Dolan-Grady condition and hence obey the Onsager algebra which was introduced in the original solution of the rectangular Ising model and appears in some integrable models.