Class Of Algebras Research Articles

Graded homotopy classification of Leavitt path algebras over finite primitive graphs

We show that the graded Grothendieck group classifies unital Leavitt path algebras of primitive graphs up to graded homotopy equivalence. To this end, we further develop classification techniques for Leavitt path algebras by means of (graded, bivariant) algebraic K -theory.

A REMARKABLE CONTRIBUTION TO SOFT INT-GROUP THEORY VIA A COMPREHENSIVE VIEW OF SOFT COSETS

This paper aims to expand soft int-group theory by analyzing its many aspects and structural properties regarding soft cosets and soft quotient groups, which are crucial concepts of the theory. All the characteristics of soft cosets are given in accordance with the properties of classical cosets in abstract algebra, and many interesting analogous results are obtained. It is proved that if an element is in the e-set, then its soft left and right cosets are the same and equal to the soft set itself. The main and remarkable contribution of this paper to the theory is that the relation between the e-set and the normality of the soft int-group is obtained, and it is proved that if the e-set has an element other than the identity of the group, then the soft int-group is normal. Based on this significant fact, it is revealed that if the soft set is not normal, then there do not exist any equal soft left (right) cosets. These relations are quite striking for the theory, since based on these facts, we show that the normality condition on the soft int-group is unnecessary in many definitions, propositions, and theorems given before. Furthermore, we come up with a fascinating result, unlike classical algebra that to construct a soft quotient group and to hold the fundamental homomorphism theorem, the soft int-group needs not to be normal. It is also demonstrated that the soft int-group is an abelian (normal) int-group if and only if the soft quotient group of G relative to the soft group is abelian. Finally, the torsion soft-int group and 𝑝-soft int-group are introduced, and we show that soft int-group f_G is a torsion soft-int group (𝑝-soft int-group) if and only if the soft quotient group G⁄f_G is a torsion (𝑝-group), respectively.

Deformation Quantization of Nonassociative Algebras

We investigate formal deformations of certain classes of nonassociative algebras including classes of K[Σ3]-associative algebras, Lie-admissible algebras and anti-associative algebras. In a process which is similar to Poisson algebra for the associative case, we identify for each type of algebras (A,μ) a type of algebras (A,μ,ψ) such that formal deformations of (A,μ) appear as quantizations of (A,μ,ψ). The process of polarization/depolarization associates to each nonassociative algebra a couple of algebras which products are respectively commutative and skew-symmetric and it is linked with the algebra obtained from the formal deformation. The anti-associative case is developed with a link with the Jacobi–Jordan algebras.

Root of unity quantum cluster algebras and discriminants

AbstractWe describe a connection between the subjects of cluster algebras, polynomial identity algebras, and discriminants. For this, we define the notion of root of unity quantum cluster algebras and prove that they are polynomial identity algebras. Inside each such algebra we construct a (large) canonical central subalgebra, which can be viewed as a far reaching generalization of the central subalgebras of big quantum groups constructed by De Concini, Kac, and Procesi and used in representation theory. Each such central subalgebra is proved to be isomorphic to the underlying classical cluster algebra of geometric type. When the root of unity quantum cluster algebra is free over its central subalgebra, we prove that the discriminant of the pair is a product of powers of the frozen variables times an integer. An extension of this result is also proved for the discriminants of all subalgebras generated by the cluster variables of nerves in the exchange graph. These results can be used for the effective computation of discriminants. As an application we obtain an explicit formula for the discriminant of the integral form over of each quantum unipotent cell of De Concini, Kac, and Procesi for arbitrary symmetrizable Kac–Moody algebras, where is a root of unity.

On a variety of Lie-admissible algebras

The aim of this paper is to propose the study of a class of Lie-admissible algebras. It is the class (variety) of all the (not-necessarily associative) algebras $M$ over a commutative ring $k$ with identity $1_k$ for which $(x,y,z)=(y,x,z)+(z,y,x)$ for every $x,y,z\in M$. Here $(x,y,z)$ denotes the associator of $M$. We call such algebras {\em algebras of type} $\Cal{V}_2$. Very little is known about these algebras.

Some results and questions on r-amenability

The notion of an r-amenable Banach lattice algebra was introduced by M. Roelands as an order counterpart to the well-known notion of amenability for Banach algebras. In this note, we consider the implications of r-amenability for various classes of Banach lattice algebras, and in particular, the question of whether r-amenability implies amenability. We also pose some questions arising from this research.

Polynomial Algorithms for Primality Testing in Algebraic Number Fieldswith Class Number 1

Polynomial Algorithms for Primality Testing in Algebraic Number Fieldswith Class Number 1

Anti-Leibniz algebras: A non-commutative version of mock-Lie algebras

Leibniz algebras are non skew-symmetric generalization of Lie algebras. In this paper we introduce the notion of anti-Leibniz algebras as a “non commutative version” of mock-Lie algebras. Low dimensional classification of such algebras is given. Then we investigate the notion of averaging operators and more general embedding tensors to build some new algebraic structures, namely anti-associative dialgebras, anti-associative trialgebras and anti-Leibniz trialgebras.

A note on injectivity of monomial algebras

We show that a monomial algebra Λ over an algebraically closed field K is self-injective if and only if it is socle-injective, i.e., each map soc ( Λ Λ ) → Λ Λ can be extended to an endomorphism of Λ Λ , and provide a complete classification of such algebras. As a consequence, we show that the class of socle-injective monomial algebras is a subclass of Nakayama algebras.

Non-associative Frobenius algebras of type [formula omitted] with trivial Tits algebras

Very recently, Maurice Chayet and Skip Garibaldi have introduced a class of commutative non-associative algebras, for each simple linear algebraic group over an arbitrary field (with some minor restriction on the characteristic). In a previous paper, we gave an explicit description of these algebras for groups of type G2 and F4 in terms of the octonion algebras and the Albert algebras, respectively. In this paper, we attempt a similar approach for type E6.

Classification of three dimensional anti-dendriform algebras

This article is devoted to the classification of anti-dendriform algebras that are associated with associativity. In particular, the paper is devoted to classifying anti-dendriform algebras associated with null-filiform associative algebras and three-dimensional algebras. It is known that any finite-dimensional associative algebra without a nonzero idempotent element is nilpotent. If an associative algebra has a nonzero idempotent element, then there does not exist a compatible anti-dendriform algebra structure associated with associative algebras.

A Complete Finite Axiomatisation of the Equational Theory of Common Meadows

We analyse abstract data types that model numerical structures with a concept of error. Specifically, we focus on arithmetic data types that contain an error value \(\bot\) whose main purpose is to always return a value for division. To rings and fields, we add a division operator \(x/y\) and study a class of algebras called common meadows wherein \(x/0=\bot\) . The set of equations true in all common meadows is named the equational theory of common meadows . We give a finite equational axiomatisation of the equational theory of common meadows and prove that it is complete and that the equational theory is decidable.

Classification of dynamical Lie algebras of 2-local spin systems on linear, circular and fully connected topologies

Much is understood about 1-dimensional spin chains in terms of entanglement properties, physical phases, and integrability. However, the Lie algebraic properties of the Hamiltonians describing these systems remain largely unexplored. In this work, we provide a classification of all Lie algebras generated by the terms of 2-local spin chain Hamiltonians, or so-called dynamical Lie algebras, on 1-dimensional linear and circular lattice structures. We find 17 unique dynamical Lie algebras. Our classification includes some well-known models such as the transverse-field Ising model and the Heisenberg chain, and we also find more exotic classes of Hamiltonians that appear new. In addition to the closed and open spin chains, we consider systems with a fully connected topology, which may be relevant for quantum machine learning approaches. We discuss the practical implications of our work in the context of variational quantum computing, quantum control and the spin chain literature.

Yang-Baxter Equations and Relative Rota-Baxter Operators for Left-Alia Algebras Associated to Invariant Theory

Left-Alia algebras are a class of algebras with symmetric Jacobi identities. They contain several typical types of algebras as subclasses, and are closely related to the invariant theory. In this paper, we study the construction theory of left-Alia bialgebras. We introduce the notion of the left-Alia Yang-Baxter equation. We show that an antisymmetric solution of the left-Alia Yang-Baxter equation gives rise to a left-Alia bialgebra that we call triangular. The notions of relative Rota-Baxter operators of left-Alia algebras and pre-left-Alia algebras are introduced to provide antisymmetric solutions of the left-Alia Yang-Baxter equation.

C⁎-algebras associated to directed graphs of groups, and models of Kirchberg algebras

We introduce C⁎-algebras associated to directed graphs of groups. In particular, we associate a combinatorial C⁎-algebra to each row-finite directed graph of groups with no sources, and show that this C⁎-algebra is Morita equivalent to the crossed product coming from the corresponding group action on the boundary of a directed tree. Finally, we show that these C⁎-algebras (and their Morita equivalent crossed products) contain the class of stable UCT Kirchberg algebras.

The optimal tennis serve: a mathematical model

Mathematical modelling has several valuable properties that address a number of pedagogical issues. As such, it makes for an excellent classroom exercise. First and foremost, it reinforces the insight that mathematics is nearly ubiquitous. It is all around us, hiding in plain sight. Secondly, in contrast to textbook problems which are often highly artificial, mathematical modelling is decidedly real. Next, it serves to un-silo the curriculum. As mathematics students advance, they populate their toolbox with more and more tools. However, these tools are generally course specific. Algebra tools in algebra class, calculus tools in calculus class, and so on. In attacking a modelling problem, the student needs to decide what tool to pick up and how to apply it. Finally, modelling is empowering in that it allows, indeed it requires, the student to decide for herself what the salient features of a problem are and which features are extraneous and can safely be suppressed.

The algebraic classification of low-dimensional nilpotent n-Lie algebras

In this paper, we study the low-dimensional nilpotent [Formula: see text]-Lie algebras and classify nilpotent [Formula: see text]-Lie algebras of dimension [Formula: see text] and nilpotent [Formula: see text]-Lie algebras of class two with dimension [Formula: see text].

A parametrization of nonassociative cyclic algebras of prime degree

We determine and explicitly parametrize the isomorphism classes of nonassociative quaternion algebras over a field of characteristic different from two, as well as the isomorphism classes of nonassociative cyclic algebras of odd prime degree m when the base field contains a primitive mth root of unity. In the course of doing so, we prove that any two such algebras can be isomorphic only if the cyclic field extension and the chosen generator of the Galois group are the same. As an application, we give a parametrization of nonassociative cyclic algebras of prime degree over a local nonarchimedean field F, which is entirely explicit under mild hypotheses on the residual characteristic. In particular, this gives a rich understanding of the important class of nonassociative quaternion algebras up to isomorphism over nonarchimedean local fields.

On the classification of function algebras on subvarieties of noncommutative operator balls

We study algebras of bounded noncommutative (nc) functions on unit balls of operator spaces (nc operator balls) and on their subvarieties. Considering the example of the nc unit polydisk we show that these algebras, while having a natural operator algebra structure, might not be the multiplier algebra of any reasonable nc reproducing kernel Hilbert space (RKHS). After examining additional subtleties of the nc RKHS approach, we turn to study the structure and representation theory of these algebras using function theoretic and operator algebraic tools. We show that the underlying nc variety is a complete invariant for the algebra of uniformly continuous nc functions on a homogeneous subvariety, in the sense that two such algebras are completely isometrically isomorphic if and only if the subvarieties are nc biholomorphic. We obtain extension and rigidity results for nc maps between subvarieties of nc operator balls corresponding to injective spaces that imply that a biholomorphism between homogeneous varieties extends to a biholomorphism between the ambient balls, which can be modified to a linear isomorphism. Thus, the algebra of uniformly continuous nc functions on nc operator balls, and even its restriction to certain subvarieties, completely determine the operator space up to completely isometric isomorphism.

Skew axial algebras of Monster type II

In our first paper, we looked at 2-generated primitive axial algebras of Monster type with skew axet X′(1+2). We continue our work by focusing on larger skew axets and classifying all such algebras with skew axets. This brings us one step closer to a complete classification of all 2-generated primitive axial algebras of Monster type.