Associative Research Articles

Local and 2-local derivations on filiform associative algebras

This paper is devoted to the study of local and 2-local derivations of null-filiform, filiform and naturally graded quasi-filiform associative algebras. We prove that these algebras as a rule admit local derivations which are not derivations. We show that filiform and naturally graded quasi-filiform associative algebras admit 2-local derivations which are not derivations and any 2-local derivation of null-filiform associative algebras is a derivation.

Semiassociative algebras over a field

An associative central simple algebra is a form of a matrix algebra, because a maximal étale subalgebra acts on the algebra faithfully by left and right multiplication. In an attempt to extract and isolate the full potential of this point of view, we study nonassociative algebras whose nucleus contains an étale subalgebra bi-acting faithfully on the algebra. These algebras, termed semiassociative, are shown to be the forms of skew matrix algebras, which we are led to define and investigate. Semiassociative algebras modulo skew matrix algebras compose a Brauer monoid, which contains the Brauer group of the field as a unique maximal subgroup.

Automorphisms of categories of free algebras with unit for classical varieties of non-associative algebras

The goal of this paper is to describe the group of strongly stable automorphisms for categories of free finitely generated non-associative algebras with unit for classical varieties of non-associative algebras, in particular for the varieties of commutative, Jordan, alternative and power associative algebras. The motivation of this research is tightly connected with some problems in Universal Algebraic Geometry.

L ∞ -structures and cohomology theory of compatible O-operators and compatible dendriform algebras

The notion of O-operator is a generalization of the Rota–Baxter operator in the presence of a bimodule over an associative algebra. A compatible O-operator is a pair consisting of two O-operators satisfying a compatibility relation. A compatible O-operator algebra is an algebra together with a bimodule and a compatible O-operator. In this paper, we construct a graded Lie algebra and an L∞-algebra that respectively characterize compatible O-operators and compatible O-operator algebras as Maurer–Cartan elements. Using these characterizations, we define cohomology of these structures and as applications, we study formal deformations of compatible O-operators and compatible O-operator algebras. Finally, we consider a brief cohomological study of compatible dendriform algebras and find their relationship with the cohomology of compatible associative algebras and compatible O-operators.

Associations between allelic differences of the fifth group chromosome loci and a complex of agronomically valuable traits in common wheat (Triticum aestivum L.)

Aim. Characterization of recombinant inbred wheat lines by agronomically relevant traits and detection of asso- ciations between their expression level and allelic differences of microsatellite loci of the fifth group chromosomes. Methods. Phenological observations, morphometric indicators, and elements of crop structure. Statistical analysis of the obtained data was carried out in Microsoft Excel. The significance of the difference between samples was assessed by Fisher’s F test. A difference of p < 0.05 was considered statistically significant for all indicators. Results. The poly- morphism of the parental genotypes and the population of 47 recombinant-inbred lines (RILs) F7 from the crossing of the winter wheat varieties Luzanivka odeska and Odeska chervonokolosa over three years (2016–2018) in conditions of the Southern Steppe (Odesa) was evaluated according to eight characteristics: duration of a period prior to heading, plant height, productive tillering, grain number per spike, grain weight per spike, thousand-grain weight, productive tiller number per unit area and grain yield. The comparison of the data of line evaluation by agronomic traits against the data of microsatellite analysis of lines by 14 microsatellite loci of the fifth group chromosomes demonstrated the associations between several loci and grain yield and between two loci and plant height. Conclusions. The changes in grain yield in the most unfavorable year, 2018, were associated with allelic differences of lines at locus Xbarc319- 5A, and in the most favorable year, 2017, – with another locus of chromosome 5A, Xbarc330-5A. In both cases, the lines that had an allele from Odeska chervonokolosa were characterized by a significantly higher level of grain yield compared to the lines that had an allele from the Luzanivka odeska variety. There are at least two QTLs on 5B chromo- some, associated with winter wheat grain yield in favorable growing years. The first of them is located in the region of locus Xbarc88-5B, and the second – in the region of loci Xwmc415-5B, Xgpw3191-5B, and Xcfd7-5B. Based on the alleles of the specified microsatellite loci, which are linked QTLs of grain yield, it is possible to select genotypes at the early stages of selection that will give a higher yield in years with favorable conditions for growing wheat.

Equivariant cohomology and deformation for algebras - an operadic approach

Let P be an operad with a multiplication, and later with a (higher) derivation. This paper introduces and develops a notion of (partial) group action and corresponding equivariant cohomology and deformation theory on P. It generalizes existing results for equivariant cohomology and deformation for associative algebras, dialgebras, and dendriform algebras, among others. At the end, we discuss compatibilities of group action and some operadic constructions.

Associative 2-algebras and nonabelian extensions of associative algebras

Associative 2-algebras and nonabelian extensions of associative algebras

Twisted representations of ℕ-graded vertex algebras over a good field

Let [Formula: see text] be an [Formula: see text]-graded vertex algebra over a field [Formula: see text], and let [Formula: see text] be an automorphism of [Formula: see text] with a finite order [Formula: see text]. Assume that the field [Formula: see text] contains a [Formula: see text]th primitive root of unity. We construct two associative algebras, [Formula: see text] and [Formula: see text], and investigate their properties and interconnections. As applications, we establish the following results: (1) If [Formula: see text] is finite-dimensional, then [Formula: see text] is also finite-dimensional; (2) If [Formula: see text] is strongly finitely generated, then [Formula: see text] is stably noetherian; (3) If [Formula: see text] is strongly finitely generated and [Formula: see text] is an irreducible admissible [Formula: see text]-twisted [Formula: see text]-module, then the endomorphism algebra [Formula: see text] is finite-dimensional.

Octonions as Clifford-like algebras

The associative Cayley-Dickson algebras over the field of real numbers are also Clifford algebras. The alternative but nonassociative real Cayley-Dickson algebras, notably the octonions and split octonions, share with Clifford algebras an involutary anti-automorphism and a set of mutually anticommutative generators. On the basis of these similarities, we introduce Kingdon algebras: alternative Clifford-like algebras over vector spaces equipped with a symmetric bilinear form. Over three-dimensional vector spaces, our construction quantizes an alternative non-associative analogue of the exterior algebra. The octonions and split octonions, along with other real generalized Cayley-Dickson algebras in Albert's sense, arise as Kingdon algebras. Our construction gives natural characterizations of the octonion and split octonion algebras by a universality property endowing them with a selected superalgebra structure.

A look at FPF rings

An error in Corollary 9.32 of [C. Faith, Rings and Things and a Fine Array of Twentieth Century Associative Algebra, Mathematical Surveys and Monographs, Vol. 65 (American Mathematical Society, Providence, RI, 2004)], motivated us to consider again FPF rings which were initiated by Faith in the 1970s. In this paper, it is shown that a commutative ring [Formula: see text] is reduced FPF if and only if it is [Formula: see text]-semihereditary. We show that when a semiperfect ring with a strongly right bounded basic ring with right and left Ore conditions, is an FPF ring. After some general results, the article focuses on rings of continuous functions. We give some algebraic characterizations for a [Formula: see text] to be FPF and retrieve a result of Jorge Martinez. Also, we show that a space [Formula: see text] is fraction-dense if and only if [Formula: see text] is a continuous ring.

Lie structure of the Heisenberg-Weyl algebra

As an associative algebra, the Heisenberg--Weyl algebra $\HWeyl$ is generated by two elements $A$, $B$ subject to the relation $AB-BA=1$. As a Lie algebra, however, where the usual commutator serves as Lie bracket, the elements $A$ and $B$ are not able to generate the whole space $\HWeyl$. We identify a non-nilpotent but solvable Lie subalgebra $\coreLie$ of $\HWeyl$, for which, using some facts from the theory of bases for free Lie algebras, we give a presentation by generators and relations. Under this presentation, we show that, for some algebra isomorphism $\isoH:\HWeyl\into\HWeyl$, the Lie algebra $\HWeyl$ is generated by the generators of $\coreLie$, together with their images under $\isoH$, and that $\HWeyl$ is the sum of $\coreLie$, $\isoH(\coreLie)$ and $\lbrak \coreLie,\isoH(\coreLie)\rbrak$.

The Relation between constants in generic and degenerate subspaces of free unital associative complex algebra

From the study of the constants in the generic and the degenerate weight subspaces of the free unitary associative complex algebra B, it follows that the constants in the degenerate weight subspaces of the algebra B can be constructed from the corresponding constants in the generic case by a certain specialization procedure. Here we consider that each constant in each generic weight subspace of the algebra B can be expressed by certain iterated q-commutators.

What conjugate phase retrieval complex vectors can do in quaternion Euclidean spaces

Abstract Quaternion algebra ℍ {\mathbb{H}} is a noncommutative associative algebra. In recent years, quaternionic Fourier analysis has received increasing attention due to its applications in signal analysis and image processing. This paper addresses conjugate phase retrieval problem in the quaternion Euclidean space ℍ M {\mathbb{H}^{M}} with M ≥ 2 {M\geq 2} . Write ℂ η = { ξ : ξ = ξ 0 + β ⁢ η , ξ 0 , β ∈ ℝ } {\mathbb{C}_{\eta}=\{\xi:\xi=\xi_{0}+\beta\eta,\,\xi_{0},\,\beta\in\mathbb{R}\}} for η ∈ { i , j , k } {\eta\in\{i,\,j,\,k\}} . We remark that not only ℂ η M {\mathbb{C}_{\eta}^{M}} -vectors cannot allow traditional conjugate phase retrieval in ℍ M {\mathbb{H}^{M}} , but also ℂ i M ∪ ℂ j M {\mathbb{C}_{i}^{M}\cup\mathbb{C}_{j}^{M}} -complex vectors cannot allow phase retrieval in ℍ M {\mathbb{H}^{M}} . We are devoted to conjugate phase retrieval of ℂ i M ∪ ℂ j M {\mathbb{C}_{i}^{M}\cup\mathbb{C}_{j}^{M}} -complex vectors in ℍ M {\mathbb{H}^{M}} , where “conjugate” is not the traditional conjugate. We introduce the notions of conjugation, maximal commutative subset and conjugate phase retrieval. Using the phase lifting techniques, we present some sufficient conditions on complex vectors allowing conjugate phase retrieval. And some examples are also provided to illustrate the generality of our theory.

Representation rings of extensions of Hopf algebra of Kac-Paljutkin type

<p>In this paper, we focus on studying two classes of finite dimensional $ \Delta $-associative algebras, which are extensions of a family of $ 2n^2 $-dimensional Kac-Paljutkin type semi-simple Hopf algebras $ H_{2n^2} $. All their indecomposable modules are classified. Furthermore, their representation rings are described by generators with suitable relations.</p>

On Schur-type theorem for Leibniz 3-algebras

One of the classic results of group theory is the so-called Schur theorem. It states that if the central factor-group G/ζ(G) of a group G is finite, then its derived subgroup [G,G] is also finite. This theorem was proved by B. Neumann in 1951. This result has numerous generalizations and modifications in group theory. At the same time, similar investigations were conducted in other algebraic structures, namely in modules, linear groups, topological groups, n-groups, associative algebras, Lie algebras, Lie n-algebras. In 2016, L.A. Kurdachenko, J. Otal and O.O. Pypka proved an analogue of Schur theorem for Leibniz algebras: if central factor-algebra L/ζ(L) of Leibniz algebra L has finite dimension, then its derived ideal [L,L] is also finite-dimensional. Moreover, they also proved a slightly modified analogue of Schur theorem: if the codimensions of the left ζ^l (L) and right ζ^r (L) centers of Leibniz algebra L are finite, then its derived ideal [L,L] is also finite-dimensional. One of the generalizations of Leibniz algebras is the so-called Leibniz n-algebras. It is worth noting that Leibniz n-algebra theory is currently much less developed than Leibniz algebra theory. One of the directions of development of the general theory of Leibniz n-algebras is the search for analogies with other types of algebras. Therefore, the question of proving analogs of the above results for this type of algebras naturally arises. In this article, we prove the analogues of the two mentioned theorems for Leibniz n-algebras for the case n=3. The obtained results indicate the prospects of further research in this direction.

On the structure of graded 3-Lie-Rinehart algebras

We study the structure of a graded 3-Lie-Rinehart algebraLover an associative and commutative graded algebra A. For G an abelian group, we show that if (L,A) is a tight G-graded 3-Lie-Rinehart algebra, then L and A decompose as L = ? i?I Li and A = ? j?J Aj, where any Li is a non-zero graded ideal of L satisfying [Li1 ,Li2 ,Li3] = 0 for any i1, i2, i3 ? I different from each other, and any Aj is a non-zero graded ideal of A satisfying AjAl = 0 for any l, j ? J such that j ?l, and both decompositions satisfy that for any i ? I there exists a unique j ? J such that AjLi ? 0. Furthermore, any (Li,Aj) is a graded 3-Lie-Rinehart algebra. Also, under certain conditions, it is shown that the above decompositions of L and A are by means of the family of their, respectively, graded simple ideals.

Classification of Real Division Algebras

This article aims to offer a unifying approach to the basic theory of division algebras by presenting the research of the German-American mathematician Max August Zorn, who classified alternative division algebras. In section 1 the basic theory of real division algebras is developed. Section 2 presents the Cayley-Dickson Process, which consists of constructing an extension algebra from an algebra provided with a conjugation, similar to the construction of complex numbers from real numbers. In Section 3 presents the classical division algebras R (real), C (complex), H (quaternions) and O (octonions) and mentions some of their applications. In section 4 the main theorem is presented, which establishes that the only (except isomorphism) alternative division algebras are: R, C, H and O (Zorn’s theorem). The classification theorems of associative division algebras (Frobenius) and normed division algebras (Hurwitz) are obtained as corollaries of Zorn’s theorem. Finally in section 5 applications of division algebras to Geometry, Number Theory, Classical Physics, Modern Physics, Quantum Mechanics and Cryptography are mentioned.

Derivations of Mackey algebras

We describe derivations of finitary Mackey algebras over fields of characteristics not equal to $2.$ We prove that an arbitrary derivation of an associative finitary Mackey algebra or one of the Lie algebras $\mathfrak{sl}_{\infty}(V|W)$, $\mathfrak{o}_{\infty}(f)$ is an adjoint operator of an element in the corresponding Mackey algebra. It provides description of derivations of all algebras in the Baranov-Strade classification of finitary simple Lie algebras. The proof is based on N. Jacobson's result on derivations of associative algebras of linear transformations of an infinite-dimensional vector space and the results on Herstein's conjectures.

A categories equivalence of associative bimodules

In this paper we use the classical Wedderburn's Kronecker Factorization Theorem to prove that category of bimodules over $B$ and the category of bimodules over $M_{n}(B)$ are equivalent, where $B$ is some unital associative algebra. In addition to this, we classify the irreducible bimodules over $M_{n}(F).$

Hexa-algebras and related algebraic structures

In this paper, we introduce the notion of hexa-algebra. These are associative algebras where the product can be decomposed as the sum of six operations satisfying 21 axioms. We present the quasi-shuffle algebra and the tensor product of a dendriform and a tridendriform algebra as examples of hexa-algebras. We study the relation with ennea algebras and with other algebraic structures and we construct hexa-algebra structures from Rota–Baxter operators.