Associative Research Articles

On the Gleason-Kahane-Żelazko Theorem for Associative Algebras

The classical Gleason-Kahane-Żelazko Theorem states that a linear functional on a complex Banach algebra not vanishing on units, and such that Lambda (textbf{1})=1, is multiplicative, that is, Lambda (ab)=Lambda (a)Lambda (b) for all a,bin A. We study the GKŻ property for associative unital algebras, especially for function algebras. In a GKŻ algebra A over a field of at least 3 elements, and having an ideal of codimension 1, every element is a finite sum of units. A real or complex algebra with just countably many maximal left (right) ideals, is a GKŻ algebra. If A is a commutative algebra, then the localization A_{P} is a GKŻ-algebra for every prime ideal P of A. Hence the GKŻ property is not a local-global property. The class of GKŻ algebras is closed under homomorphic images. If a function algebra Asubseteq {mathbb {F}}^{X} over a subfield {mathbb {F}} of {mathbb {C}}, contains all the bounded functions in {mathbb {F}}^{X}, then each element of A is a sum of two units. If A contains also a discrete function, then A is a GKŻ algebra. We prove that the algebra of periodic distributions, and the unitisation of the algebra of distributions with support in (0,infty ) satisfy the GKŻ property, while the algebra of compactly supported distributions does not.

Classification of naturally graded nilpotent associative algebras

In the paper, we consider dimensional naturally graded nilpotent associative algebras with the characteristic sequence over an algebraically closed field of characteristic zero. There are two types of such algebras. We give the algebraic classification up to isomorphism of both types of algebras. Moreover, we prove that there are no such algebras of the second type for .

Bernstein algebras that are algebraic and the Kurosh problem

We study the class of Bernstein algebras that are algebraic, in the sense that each element generates a finite-dimensional subalgebra. Every Bernstein algebra has a maximal algebraic ideal, and the quotient algebra is a zero-multiplication algebra. Several equivalent conditions for a Bernstein algebra to be algebraic are given. In particular, known characterizations of Bernstein train algebras in terms of nilpotency are generalized to the case of locally train algebras. Along the way, we show that if a Banach Bernstein algebra is algebraic (respectively, locally train), then it is of bounded degree (respectively, train).Then we investigate the Kurosh problem for Bernstein algebras: whether a finitely generated Bernstein algebra which is algebraic of bounded degree is finite-dimensional. This problem turns out to have a closed link with a question about associative algebras. In particular, when the bar-ideal is nil, the Kurosh problem asks whether a finitely generated Bernstein train algebra is finite-dimensional. We prove that the answer is positive for some specific cases and for low degrees, and construct counter-examples in the general case.On the other hand, by results of Yagzhev the Jacobian conjecture is equivalent to a certain statement about Engel and nilpotence identities of multioperator algebras. We show that the generalized Jacobian conjecture for quadratic mappings holds for Bernstein algebras.

The structure of symmetric tensor powers of composition algebras

Let [Formula: see text] be a composition algebra which is either the Hamilton quaternion algebra [Formula: see text] or the Cayley octonion algebra [Formula: see text] over [Formula: see text]. In a previous work, the nth symmetric power [Formula: see text] of [Formula: see text] is shown to be a direct sum of central simple algebras, corresponding to the partitions of [Formula: see text] of length [Formula: see text], such that the component corresponding to the partition [Formula: see text] is isomorphic to the component [Formula: see text] of [Formula: see text] corresponding to the partition [Formula: see text] of [Formula: see text]. In this work, we study the building blocks [Formula: see text] of these decompositions. We show that the “local” structure of [Formula: see text], i.e. the complex-like subfields of [Formula: see text], determine both the complement of [Formula: see text] in [Formula: see text] and the trace map of [Formula: see text], induced from the trace map of [Formula: see text]. We also derive a recursive trace formula on the [Formula: see text]’s. We use the “local-global” results to define positive definite symmetric bilinear forms on the vector space [Formula: see text], which has a natural structure of a commutative and associative algebra. Finally, the structure of the central simple algebra [Formula: see text] is described.

Semisimple Flat F-Manifolds in Higher Genus

In this paper, we generalize the Givental theory for Frobenius manifolds and cohomological field theories to flat F-manifolds and F-cohomological field theories. In particular, we define a notion of Givental cone for flat F-manifolds, and we provide a generalization of the Givental group as a matrix loop group acting on them. We show that this action is transitive on semisimple flat F-manifolds. We then extend this action to F-cohomological field theories in all genera. We show that, given a semisimple flat F-manifold and a Givental group element connecting it to the constant flat F-manifold at its origin, one can construct a family of F-CohFTs in all genera, parameterized by a vector in the associative algebra at the origin, whose genus 0 part is the given flat F-manifold. If the flat F-manifold is homogeneous, then the associated family of F-CohFTs contains a subfamily of homogeneous F-CohFTs. However, unlike in the case of Frobenius manifolds and CohFTs, these homogeneous F-CohFTs can have different conformal dimensions, which are determined by the properties of a certain metric associated to the flat F-manifold.

Around Van den Bergh’s double brackets for different bimodule structures

A double Poisson bracket, in the sense of M. Van den Bergh, is an operation on an associative algebra A which induces a Poisson bracket on each representation space in an explicit way. In this note, we study the impact of changing the Leibniz rules underlying a double bracket. This change amounts to make a suitable choice of A-bimodule structure on . In the most important cases, we describe how the choice of A-bimodule structure fixes an analogue to Jacobi identity, and we obtain induced Poisson brackets on representation spaces. The present theory also encodes a formalization of the widespread tensor notation used to write Poisson brackets of matrices in mathematical physics. Communicated by P. Kolesnikov

On the identical relations of associative and Lie algebras equipped with an action

Let R be a unitary associative algebra over a field. We call an algebra A a generalized R-algebra when A is endowed with an R-module action with the property that, for each r∈R, there exists finitely many elements r+=(r1+,r2+)∈R2 and r−=(r1−,r2−)∈R2 such that, for all a1,a2∈A,r⋅(a1a2)=∑r+(r1+⋅a1)(r2+⋅a2)+∑r−(r2−⋅a2)(r1−⋅a1). Suppose an associative generalized R-algebra A satisfies an identical relation of the formx1⋯xd−∑1≠σ∈Sd∑r(r1⋅xσ(1))⋯(rd⋅xσ(d))≡0, where Sd denotes the symmetric group of degree d and the inner sum runs over finitely many r=(r1,…,rd)∈Rd. We prove: if the algebra of endomorphisms on A defined by the action of R is m-dimensional, then A satisfies a classical polynomial identity of degree bounded by an explicit function of d and m only. We also prove the analogous result holds when A is a Lie algebra, thus extending a collection of results in associative and Lie PI-theory.

Generalization of Jordan-Lie inner ideals of finite dimensional associative algebras

We generalize Baranov and Shlaka’s results about bar-minimal Jordan-Lie and regular inner ideals of finite dimensional associative algebras. Let [Formula: see text] be a finite dimensional [Formula: see text]-perfect associative algebras [Formula: see text] over an algebraically closed field [Formula: see text] of arbitrary characteristic [Formula: see text] and let [Formula: see text] be a subalgebra of [Formula: see text]. We prove that for any bar-minimal Jordan-Lie inner ideal [Formula: see text] a [Formula: see text], there is a bar-minimal Jordan-Lie inner ideal of [Formula: see text] that contains [Formula: see text] and if [Formula: see text] is regular, then is a regular inner ideal of [Formula: see text] that contains [Formula: see text]. We also prove that for any strict orthogonal pair [Formula: see text] in [Formula: see text], there is a strict orthogonal idempotent pair [Formula: see text] in [Formula: see text] such that [Formula: see text].

Reverse ∗-Jordan type maps on Jordan ∗-algebras

Let [Formula: see text] and [Formula: see text] be two ∗-Jordan algebras with identities [Formula: see text] and [Formula: see text], respectively, and [Formula: see text] a nontrivial ∗-idempotent in [Formula: see text]. In this paper, we study the characterization of multiplicative ∗-Jordan-type maps. In particular, we provide a characterization in the case of unital prime associative algebra endowed with an involution.

On properties of some nilpotent evolution algebras and their enveloping algebras

In this paper, we consider [Formula: see text]-dimensional nilpotent finite-dimensional evolution algebras with non-maximal index of nilpotency. We give complete invariants of these algebras in order to classify them. Then, we describe their associative enveloping algebras. We also give a positive answer to the question whether a derivation of this nilpotent evolution algebra is possible to be extended to its enveloping algebra.

The Association of Gender with the Patterns of Impactions and Associated Radiolucency’s in Mandible and Maxilla

OBJECTIVES To find the association between gender and the patterns of impactions and associated radiolucency in Mandibleand Maxilla. METHODOLOGY A cross-sectional descriptive study was conducted on 385 Orthopantomograms (OPG) of patients visiting SharifMedical and Dental College (SMDC), Services Institute of Medical Sciences (SIMS) and Institute of Dentistry,CMH, Lahore Medical College, Lahore over a period of three months from December 2020 to February 2021.The angle of the impacted third molar was classified using Winters classification. The Pell and Gregoryclassification was used for depth of third molar impaction and ramus relationship of mandibular third molars.Statistical analysis was done using SPSS 23 and a P value ≤ 0.05 was considered significant. RESULTS The angle of impaction and gender in impacted mandibular third molars were significantly associated with eachother (p=0.011) while the depth of impaction (p=0.729), and ramus relationship(p=0.318) were not significantlyassociated with gender. The associa tion between gender and periapical radiolucency (p=0.605), peri coronalradiolucency (p=0.591), caries (p=0.499) and external resorption (p=0.499) were non-significant. In maxillaryimpacted third molars the association between gender and depth of impaction (p=0.121), angle of impaction(p=0.284), periapical radiolucency (p=0.111), peri coronal radiolucency (p=0.805) and caries (p=1.000) werenon-significant. CONCLUSION In impacted mandibular third molars the periapical, peri coronal radiolucency,caries and external resorption weremore in the males as compared to the females. In maxillary third molar impactions, the periapical radiolucencyradiolucency was greater in males while peri coronal radiolucency was greater in females.

Gradings induced by nilpotent elements

An element a is nilpotent last-regular if it is nilpotent and its last nonzero power is von Neumann regular. In this paper we show that any nilpotent last-regular element a in an associative algebra R over a ring of scalars Φ gives rise to a complete system of orthogonal idempotents that induces a finite Z-grading on R; we also show that such element gives rise to an sl2-triple in R with semisimple adjoint map adh, and that the grading of R with respect to the complete system of orthogonal idempotents is a refinement of the Φ-grading induced by the eigenspaces of adh. These results can be adapted to nilpotent elements a with all their powers von Neumann regular, in which case the element a can be completed to an sl2-triple and a is homogeneous of degree 2 both in the Z-grading of R and in the Φ-grading given by the eigenspaces of adh.Conversely, when there are enough invertible elements in Φ, we will show that if a nilpotent element a can be completed to an sl2-triple, then all powers ak are von Neumann regular.

Double Multiplicative Poisson Vertex Algebras

Abstract We develop the theory of double multiplicative Poisson vertex algebras. These structures, defined at the level of associative algebras, are shown to be such that they induce a classical structure of multiplicative Poisson vertex algebra on the corresponding representation spaces. Moreover, we prove that they are in one-to-one correspondence with local lattice double Poisson algebras, a new important class among Van den Bergh’s double Poisson algebras. We derive several classification results, and we exhibit their relation to non-abelian integrable differential-difference equations. A rigorous definition of double multiplicative Poisson vertex algebras in the non-local and rational cases is also provided.

Double Lie algebras of a nonzero weight

We introduce the notion of λ-double Lie algebra, which coincides with usual double Lie algebra when λ=0. We show that every λ-double Lie algebra for λ≠0 provides the structure of modified double Poisson algebra on the free associative algebra. In particular, it confirms the conjecture of S. Arthamonov (2017). We prove that there are no simple finite-dimensional λ-double Lie algebras.

Brain structural abnormalities of the associative striatum in adolescents and young adults at genetic high-risk of schizophrenia: Implications for illness endophenotypes

ObjectiveDysfunction in cortico-striatal circuitry represents a core component of the pathophysiology in schizophrenia (SZ) but its potential as a candidate endophenotype of the illness is often confounded by neuroleptic medication. MethodsAccordingly, 26 adolescent and young adult participants at genetic high-risk for schizophrenia, but who were asymptomatic and neuroleptic naïve, and 28 age-matched controls underwent 1.5T structural magnetic resonance imaging of the striatum, manually parcellated into limbic (LST), associative (AST), and sensorimotor (SMST) functional subregions. ResultsIn relation to their age peers, participants at genetic high-risk for schizophrenia showed overall lower striatal gray matter volume with their most pronounced loss, bilaterally in the AST, but not the LST or SMST. Neuropsychological testing revealed reduced executive functioning for genetically at-risk participants, although the groups did not differ significantly in overall intelligence or oral reading. For controls but not for at-genetic high-risk participants, stronger executive functioning correlated with increased bilateral AST volume. ConclusionsReduced bilateral AST volume in genetic high-risk adolescents and young adults, accompanied by heritable loss of higher cognitive brain-behavior relationships, might serve as a useful endophenotype of SZ.

Algebraic Constructions for Novikov–Poisson Algebras

A Novikov–Poisson algebra (A,∘,·) is a vector space with a Novikov algebra structure (A,∘) and a commutative associative algebra structure (A,·) satisfying some compatibility conditions. Give a Novikov–Poisson algebra (A,∘,·) and a vector space V. A natural problem is how to construct and classify all Novikov–Poisson algebra structures on the vector space E=A⊕V such that (A,∘,·) is a subalgebra of E up to isomorphism whose restriction on A is the identity map. This problem is called extending structures problem. In this paper, we introduce the definition of a unified product for Novikov–Poisson algebras, and then construct an object GH2(V,A) to answer the extending structures problem. Note that unified product includes many interesting products such as bicrossed product, crossed product and so on. Moreover, the special case when dim(V)=1 is investigated in detail.

Cohomology and deformations of weighted Rota–Baxter operators

Weighted Rota–Baxter operators on associative algebras are closely related to modified Yang–Baxter equations, splitting of algebras, and weighted infinitesimal bialgebras and play an important role in mathematical physics. For any λ ∈ k, we construct a differential graded Lie algebra whose Maurer–Cartan elements are given by λ-weighted relative Rota–Baxter operators. Using such characterization, we define the cohomology of a λ-weighted relative Rota-Baxter operator T and interpret this as the Hochschild cohomology of a suitable algebra with coefficients in an appropriate bimodule. We study linear, formal, and finite order deformations of T from cohomological points of view. Among others, we introduce Nijenhuis elements that generate trivial linear deformations and define a second cohomology class to any finite order deformation, which is the obstruction to extend the deformation. In the end, we also consider the cohomology of λ-weighted relative Rota–Baxter operators in the Lie case and find a connection with the case of associative algebras.

Corrigendum to “Logarithmic intertwining operators and associative algebras” [J. Pure Appl. Algebra 216 (2012) 1467–1492

Corrigendum to “Logarithmic intertwining operators and associative algebras” [J. Pure Appl. Algebra 216 (2012) 1467–1492

On (t2, t3)-Zakharov–Shabat equations of generalized Kadomtsev–Petviashvili hierarchies

We review the integration of the Kadomtsev–Petviashvili (KP) hierarchy in several non-standard contexts. Specifically, we consider KP in the following associative differential algebras: an algebra equipped with a nilpotent derivation, an algebra of functions equipped with a derivation that generalizes the gradient operator, an algebra of quaternion-valued functions, a differential Lie algebra, an algebra of polynomials equipped with the Pincherle differential, and a Moyal algebra. In all these cases, we can formulate and solve the Cauchy problem of the KP hierarchy. In addition, in each of these cases, we derive different Zakharov–Shabat (t2, t3)-equations—that is, different Kadomtsev–Petviashvili equations—and we prove the existence of solutions arising from solutions to the corresponding KP hierarchy.

Identities for subspaces of a parametric Weyl algebra

In 2013 Benkart, Lopes and Ondrus introduced and studied in a series of papers the infinite-dimensional unital associative algebra Ah generated by elements x,y, which satisfy the relation yx−xy=h for some 0≠h∈F[x]. In this paper we investigate the standard polynomial identities and minimal identities for certain subspaces of Ah over an infinite field of arbitrary characteristic.