Related papers: Noncommutative coding theory and algebraic sets fo…
This dissertation considers new constructions and decoding approaches for error-correcting codes based on non-conventional polynomials, with the objective of providing new coding solutions to the applications mentioned above. With skew…
We observe \cite[Proposition 4.1]{LaLe} that Poisson polynomial extensions appear as semiclassical limits of a class of Ore extensions. As an application, a Poisson generalized Weyl algebra $A_1$ considered as a Poisson version of the…
We investigate deformations of a skew group algebra that arise from a finite group acting on a polynomial ring. When the characteristic of the underlying field divides the order of the group, a new type of deformation emerges that does not…
Using the language of operated algebras, we construct and investigate a class of operator rings and enriched modules induced by a derivation or Rota-Baxter operator. In applying the general framework to univariate polynomials, one is led to…
In this paper we investigate extended modules for a special class of Ore extensions. We will assume that $R$ is a ring and $A$ will denote the Ore extension $A:=R[x_1,\dots,x_n;\sigma]$ for which $\sigma$ is an automorphism of $R$,…
We investigate the diameter and girth of the nilpotent graph for skew PBW extensions over $2$-primal rings, generalizing similar results on skew polynomial rings. Under certain compatibility conditions, we establish bounds for the diameter…
We give the basic structure of the multivariable Ore extensions $S=A[\underline{t} ; \sigma, \underline{\delta}]$ introduced in the work of Mart\'inez-Pe\~nas and Kschischang. The Pseudo multilinear transformations (PMT's) are introduced…
The aim of the paper is twofold. First, we introduce analogs of (partial) derivatives on certain Noncommutative algebras, including some enveloping algebras and their "braided counterparts", namely, the so-called modified Reflection…
A one parameter set of noncommutative complex algebras is given. These may be considered deformation quantisation algebras. The commutative limit of these algebras correspond to the algebra of polynomial functions over a manifold or…
We generalize the notion of cyclic codes by using generator polynomials in (non commutative) skew polynomial rings. Since skew polynomial rings are left and right euclidean, the obtained codes share most properties of cyclic codes. Since…
We begin by reviewing a classical result on the algebraic dependence of commuting elements in Weyl algebras. We proceed by describing generalizations of this result to various classes of Ore extensions, both results that have already been…
We investigate algebraic properties of weakly commutative triples, appearing in the theory of integrable nonlinear partial differential equations. Algebraic technique of skew fields of formal pseudodifferential operators as well as skew Ore…
With a small suitable modification, dropping the projectivity condition, we extend the notion of a Frobenius algebra to grant that a Frobenius algebra over a Frobenius commutative ring is itself a Frobenius ring. The modification introduced…
This article provides a synthesis of recent advances in the study of the PI property in various classes of noncommutative algebras of polynomial type.
In the previous three papers in this series, [WKO1]-[WKO3] (arXiv:1405.1956, arXiv:1405.1955, and to appear), Z. Dancso and I studied a certain theory of "homomorphic expansions" of "w-knotted objects", a certain class of knotted objects in…
We study a noncommutative deformation of the commutative Hopf algebra of rooted trees which was shown by Connes and Kreimer to be related to the mathematical structure of renormalization in quantum field theories. The requirement of the…
This is a survey on the theory of skew-cyclic codes based on skew-polynomial rings of automorphism type. Skew-polynomial rings have been introduced and discussed by Ore (1933). Evaluation of skew polynomials and sets of (right) roots were…
The aim of this paper is to establish necessary and sufficient algorithmic conditions to guarantee that an algebra is actually a 3-dimensional skew polynomial algebra in the sense of Bell and Smith.
Poincare-Birkhoff-Witt (PBW) Theorems have attracted significant attention since the work of Drinfeld (1986), Lusztig (1989), and Etingof-Ginzburg (2002) on deformations of skew group algebras $H \ltimes {\rm Sym}(V)$, as well as for other…
Structural properties of unitary groups over local, not necessarily commutative, rings are developed, with applications to the computation of the orders of these groups (when finite) and to the degrees of the irreducible constituents of the…