Ore Extensions Research Articles

Structure and arithmetic of multivariate Ore extensions

We give the basic structure of the multivariable Ore extensions S = A [ t ¯ ; σ , δ ¯ ] introduced in the work of Martínez-Peñas and Kschischang. The Pseudo multilinear transformations (PMT’s) are introduced and correspond to modules over S. These maps are strongly connected to the evaluation of polynomials in S. A general product formula is obtained. PMT’s help to put some structure on the set of roots of a polynomial f ( t ) ∈ S .

A note on the Jacobson radical of Ore extensions

Let R be a ring with a monomorphism α and an α-derivation δ. In this article, we give a simple and different proof about the semiprimitivity of Ore extensions which states that the skew polynomial ring R [ x ; α , δ ] is semiprimitive reduced if and only if R is α-rigid. This unifies and extends a number of known results on the Jacobson radical in the special cases. Also, as an application of our results, by imposing constraints on α and δ, we completely identify the Jacobson radical of rings whose the set of all nilpotent elements has special conditions. Important examples are provided to illustrate the applications of the results.

Differential Calculi on Quantum Principal Bundles Over Projective Bases

We propose a sheaf-theoretic approach to the theory of differential calculi on quantum principal bundles over non-affine bases. After recalling the affine case we define differential calculi on sheaves of comodule algebras as sheaves of covariant bimodules together with a morphism of sheaves -the differential- such that the Leibniz rule and surjectivity hold locally. The main class of examples is given by covariant calculi over quantum flag manifolds, which we provide via an explicit Ore extension construction. In a second step we introduce principal covariant calculi by requiring a local compatibility of the calculi on the total sheaf, base sheaf and the structure Hopf algebra in terms of exact sequences. In this case Hopf–Galois extensions of algebras lift to Hopf–Galois extensions of exterior algebras with compatible differentials. In particular, the examples of principal (covariant) calculi on the quantum principal bundles Oq(SL2(C))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O}_q(\ extrm{SL}_2(\\mathbb {C}))$$\\end{document} and Oq(GL2(C))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O}_q(\ extrm{GL}_2(\\mathbb {C}))$$\\end{document} over the projective space P1(C)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{P}^1(\\mathbb {C})$$\\end{document} are discussed in detail.

Radicals in Ore extensions

Let [Formula: see text] be a ring, [Formula: see text] be an automorphism of [Formula: see text] and [Formula: see text] be a [Formula: see text]-derivation of [Formula: see text]. We use [Formula: see text] to denote the set of all words composed of [Formula: see text], [Formula: see text] and [Formula: see text]. A [Formula: see text]-ideal [Formula: see text] of [Formula: see text] is [Formula: see text]-prime if whenever [Formula: see text] are such that [Formula: see text] for any [Formula: see text], we have [Formula: see text] or [Formula: see text]. In this paper, we first introduce the [Formula: see text]-prime ideal and the [Formula: see text]-prime radical of a ring [Formula: see text], to obtain connections between the prime radical of the Ore extension [Formula: see text] and the [Formula: see text]-prime radical of the base ring [Formula: see text]. Based on these results, we next give definitions of the [Formula: see text]-LS-prime ideal, the [Formula: see text]-strongly prime ideal and the [Formula: see text]-uniformly strongly prime ideal of a ring [Formula: see text] to provide formulas for the LS-prime radical, the strongly prime radical and the uniformly strongly prime radical of the Ore extension.

Noncommutative Coding Theory and Algebraic Sets for Skew PBW Extensions

The classical commutative coding theory has been recently extended to noncommutative rings of polynomial type. There are many interesting works in coding theory over single Ore extensions. In this review article we present the most relevant algebraic tools and properties of single Ore extensions used in noncommutative coding theory. The last section represents the novelty of the paper. We will discuss the algebraic sets arising in noncommutative coding theory but for skew PBW extensions. These extensions conform a general class of noncommutative rings of polynomial type and cover several algebras arising in physics and noncommutative algebraic geometry, in particular, they cover the Ore extensions of endomorphism injective type and the polynomials rings over fields.

Coideal subalgebras of pointed and connected Hopf algebras

Let H H be a pointed Hopf algebra with abelian coradical. Let A ⊇ B A\supseteq B be left (or right) coideal subalgebras of H H that contain the coradical of H H . We show that A A has a PBW basis over B B , provided that H H satisfies certain mild conditions. In the case that H H is a connected graded Hopf algebra of characteristic zero and A A and B B are both homogeneous of finite Gelfand-Kirillov dimension, we show that A A is a graded iterated Ore extension of B B . These results turn out to be conceptual consequences of a structure theorem for each pair S ⊇ T S\supseteq T of homogeneous coideal subalgebras of a connected graded braided bialgebra R R with braiding satisfying certain mild conditions. The structure theorem claims the existence of a well-behaved PBW basis of S S over T T . The approach to the structure theorem is constructive by means of a combinatorial method based on Lyndon words and braided commutators, which is originally developed by V. K. Kharchenko [Algebra Log. 38 (1999), pp. 476–507, 509] for primitively generated braided Hopf algebras of diagonal type. Since in our context we don’t priorilly assume R R to be primitively generated, new methods and ideas are introduced to handle the corresponding difficulties, among others.

Iterated Ore polynomial maps

We study a natural notion of polynomial maps attached to elements of an iterated Ore extension [Formula: see text]. We develop some tools to analyze these maps such as good points, multilinear transformations and P-independence. We also present polynomial maps arising in the Multivariate extensions [Formula: see text], introduced by Martínez-Peñas and Kschischang [Evaluation and interpolation over multivariate skew polynomial rings, J. Algebra 525 (2019) 111–139], through the use of Multilinear polynomial maps.

The Ore Extension of Group-Cograded Hopf Coquasigroups

The aim of this paper is the Ore extension of group-cograded Hopf coquasigroups. This paper first shows a categorical interpretation and some examples of group-cograded Hopf coquasigroups, and then provides a necessary and sufficient condition for the Ore extension of group-cograded Hopf coquasigroups to be group-cograded Hopf coquasigroups. Finally, a certain isomorphism between Ore extensions are considered.

Homological dimensions of analytic Ore extensions

If [Formula: see text] is an algebra with finite right global dimension, then for any automorphism [Formula: see text] and [Formula: see text] the right global dimension of [Formula: see text] satisfies [Formula: see text] We extend this result to the case of holomorphic Ore extensions and smooth crossed products by [Formula: see text] of [Formula: see text]-algebras.

Nonassociative Algebras, Rings and Modules over Them

The review is devoted to nonassociative algebras, rings and modules over them. The main actual and recent trends in this area are described. Works are reviewed on radicals in nonassociative rings, nonassociative algebras related with skew polynomials, commutative nonassociative algebras and their modules, nonassociative cyclic algebras, rings obtained as nonassociative cyclic extensions, nonassociative Ore extensions of hom-associative algebras and modules over them, and von Neumann finiteness for nonassociative algebras. Furthermore, there are outlined nonassociative algebras and rings and modules over them related to harmonic analysis on nonlocally compact groups, nonassociative algebras with conjugation, representations and closures of nonassociative algebras, and nonassociative algebras and modules over them with metagroup relations. Moreover, classes of Akivis, Sabinin, Malcev, Bol, generalized Cayley–Dickson, and Zinbiel-type algebras are provided. Sources also are reviewed on near to associative nonassociative algebras and modules over them. Then, there are the considered applications of nonassociative algebras and modules over them in cryptography and coding, and applications of modules over nonassociative algebras in geometry and physics. Their interactions are discussed with more classical nonassociative algebras, such as of the Lie, Jordan, Hurwitz and alternative types.

Isomorphism problems and groups of automorphisms for Ore extensions 𝐾[𝑥][𝑦;𝛿] (zero characteristic)

Let Λ ( f ) = K [ x ] [ y ; f d d x ] \Lambda (f) = K[x][y; f\frac {d}{dx} ] be an Ore extension of a polynomial algebra K [ x ] K[x] over a field K K of characteristic zero where f ∈ K [ x ] f\in K[x] . For a given polynomial f f , the automorphism group of the algebra Λ ( f ) \Lambda (f) is explicitly described. The polynomial case Λ ( 0 ) = K [ x , y ] \Lambda (0) = K[x,y] and the case of the Weyl algebra A 1 = K [ x ] [ y ; d d x ] A_1= K[x][y; \frac {d}{dx} ] were done by Jung [J. Reine Angew. Math. 184 (1942), pp. 161–174] and van der Kulk [Nieuw Arch. Wisk. (3) 1 (1953), pp. 33–41], and Dixmier [Bul. Soc. Math. France 96 (1968), pp. 209–242], respectively. Alev and Dumas [Comm. Algebra 25 (1997), pp. 1655–1672] proved that the algebras Λ ( f ) \Lambda (f) and Λ ( g ) \Lambda (g) are isomorphic iff g ( x ) = λ f ( α x + β ) g(x) = \lambda f(\alpha x+\beta ) for some λ , α ∈ K ∖ { 0 } \lambda , \alpha \in K\backslash \{ 0\} and β ∈ K \beta \in K . Benkart, Lopes and Ondrus [Trans. Amer. Math. Soc. 367 (2015), pp. 1993–2021] gave a complete description of the set of automorphism groups of algebras Λ ( f ) \Lambda (f) . In this paper we complete the picture, i.e. given the polynomial f f we have the explicit description of the automorphism group of Λ ( f ) \Lambda (f) . The key concepts in finding the automorphism groups are the eigenform, the eigenroot and the eigengroup of a polynomial (introduced in the paper; they are of independent interest).

Unimodular Completions and Orthogonal Complements of Matrices over Univariate Ore Extensions

We generalize an efficient hyper-regularity and unimodularity test from differential Ore polynomial matrices to arbitrary Ore polynomial matrices. The core of the contribution consists of algorithms for unimodular row and column completions of arbitrary univariate Ore polynomial matrices, respectively. After a possible degree reduction using noncommutative companion matrices, this is done by a systematic projection of suitable coordinates with a subsequent elimination. Based on previous work for differential Ore polynomial matrices, we remove rather restrictive conditions, which now allows for the application of this algorithm to a larger class of systems that may contain pure algebraic equations.

Constructing annihilators for parameter estimation in nonlinearly parameterized signals

This paper studies the theoretical challenge of linearly representing a class of nonlinearly parameterized estimation problems. Conventional linear regression estimation methods can then be used. For illustration purposes, in our work, the DREM method is chosen to estimate the frequencies for a multi-sinusoidal uncertain signal in a fixed time. Identifying annihilators for signals generated by exponential polynomials with unknown parameters is the first step to accomplishing the linear representation. The annihilators are obtained as time-delay operators constructed with an Ore extension. The fundamental idea is to use the action of these operators on exponential polynomials to obtain the linear regression in nonlinear unknown quantities. We precisely characterize the annihilator as a principal ideal using appropriate Ore extensions, substantially simplifying further calculations. Some examples illustrate our annihilator's design.

Non-unital Ore extensions

We study Ore extensions of non-unital associative rings. We provide a characterization of simple non-unital differential polynomial rings $R[x;\delta ]$, under the hypothesis that $R$ is $s$-unital and $\ker (\delta )$ contains a non-zero idempotent. This

On derivations of differential Ore extensions

Let [Formula: see text] be a derivation of a ring [Formula: see text], and let [Formula: see text] be the differential Ore extension of [Formula: see text] defined by putting [Formula: see text] for all [Formula: see text]. We describe the derivations of [Formula: see text] extending a derivation of [Formula: see text] and we deduce an explicit description of every derivation of [Formula: see text] in the case where [Formula: see text] is the polynomial ring [Formula: see text] over a field [Formula: see text]. As an application, we compute the first Hochschild cohomology group of [Formula: see text] when [Formula: see text] and [Formula: see text]; for [Formula: see text] a conjectural result is proposed.

Resolutions for truncated Ore extensions

We extend recent results in order to construct projective resolutions for modules over twisted tensor products of truncated polynomial rings. We begin by taking note of the conditions necessary to think of these algebras as a type of Ore extension. We then use this parallel with Ore extensions to develop a method for constructing projective resolutions. Finally, we use the new construction to compute a resolution for a family of examples.

Iterated Hopf Ore extensions in positive characteristic

Iterated Hopf Ore extensions (IHOEs) over an algebraically closed base field \Bbbk of positive characteristic p are studied. We show that every IHOE over \Bbbk satisfies a polynomial identity (PI), with PI-degree a power of p , and that it is a filtered deformation of a commutative polynomial ring. We classify all 2 -step IHOEs over \Bbbk , thus generalising the classification of 2 -dimensional connected unipotent algebraic groups over \Bbbk . Further properties of 2 -step IHOEs are described: for example their simple modules are classified, and every 2 -step IHOE is shown to possess a large Hopf center and hence an analog of the restricted enveloping algebra of a Lie \Bbbk -algebra. As one of a number of questions listed, we propose that such a restricted Hopf algebra may exist for every IHOE over \Bbbk .

The structure of connected (graded) Hopf algebras revisited

Let H be a connected graded Hopf algebra over a field of characteristic zero and K an arbitrary graded Hopf subalgebra of H. We show that there is a family of homogeneous elements of H indexed by a totally order set that satisfy several desirable conditions, which reveal interesting connections between H and K. In particular, the set on non-decreasing products of these elements give a basis for H as a left and right graded K-module. As one of its consequences, we see that H is a graded iterated Hopf Ore extension of K of derivation type provided that H is of finite Gelfand-Kirillov dimension. The main tool of this work is Lyndon words, along the idea developed by Lu, Shen and the second-named author in [24].

Nearring of ore extensions: Zero-divisors and regular elements

Let [Formula: see text] be a symmetric and [Formula: see text]-compatible ring. In this paper, we first specify the zero-divisor elements of the near-ring [Formula: see text]. We prove that if [Formula: see text] is an [Formula: see text]-rigid ring, then the set of all zero-divisor elements of the near-ring [Formula: see text] forms an ideal of [Formula: see text] if and only if [Formula: see text] is an ideal of [Formula: see text] and [Formula: see text] has the right Property [Formula: see text]. Also, we are interested in studying the zero-divisor graph of [Formula: see text] which is denoted by [Formula: see text]. It is shown that [Formula: see text], and if [Formula: see text] is not reduced, then [Formula: see text] for each [Formula: see text] if and only if [Formula: see text]. Moreover, we characterize the units, the clean elements, the regular elements, the nilpotent elements and also the [Formula: see text]-regular elements of the near-ring [Formula: see text], where [Formula: see text] is a reversible and [Formula: see text]-compatible ring and [Formula: see text] is a countable locally nilpotent ideal of [Formula: see text]. Finally, we prove that the set of all [Formula: see text]-regular elements of [Formula: see text] forms a semigroup, where [Formula: see text] is a reversible and [Formula: see text]-compatible ring and [Formula: see text] is a countable locally nilpotent ideal of [Formula: see text].

The Calabi–Yau property of Ore extensions of two-dimensional Artin–Schelter regular algebras and their PBW deformations

Let A be a noncommutative Artin–Schelter regular algebra of dimension 2 with the Nakayama automorphism μ A and U a PBW deformation of A with the Nakayama automorphism μ U . We prove that any graded Ore extension A[z;μ A ,δ] and any filtered Ore extension U[z;μ U ,δ ˜] are 3-Calabi–Yau.