We introduce non-associative skew Laurent polynomial rings over unital, non-associative rings and prove simplicity results for these. We also generalize an already existing Hilbert’s basis theorem for non-associative Ore extensions and show that it implies a Hilbert’s basis theorem for non-associative skew Laurent polynomial rings. Moreover, we show that the above theorem can be extended to non-associative generalizations of both skew power series rings and skew Laurent series rings. We provide several examples to illustrate the applications of our results.
We prove a hom-associative version of Hilbert’s basis theorem, which includes as special cases both a non-associative version and the classical Hilbert’s basis theorem for associative Ore extensions. Along the way, we develop hom-module theory. We conclude with some examples of both non-associative and hom-associative Ore extensions which are all noetherian by our theorem.
We introduce hom-associative versions of the higher order Weylalgebras, generalizing the construction of the first hom-associative Weyl alge-bras. We then show that the higher order hom-associative Weyl algebras aresimple, and that all their one-sided ideals are principal.
The first (associative) Weyl algebra is formally rigid in the classical sense. In this paper, we show that it can however be formally deformed in a nontrivial way when considered as a so-called hom-associative algebra, and that this deformation preserves properties such as the commuter, while deforming others, such as the center, power associativity, the set of derivations, and some commutation relations. We then show that this deformation induces a formal deformation of the corresponding Lie algebra into what is known as a hom-Lie algebra, when using the commutator as bracket. We also prove that all homomorphisms between any two purely hom-associative Weyl algebras are in fact isomorphisms. In particular, all endomorphisms are automorphisms in this case, hence proving a hom-associative analogue of the Dixmier conjecture to hold true.
We introduce hom-associative Ore extensions as non-associative, non-unital Ore extensions with a hom-associative multiplication, as well as give some necessary and sufficient conditions when such exist. Within this framework, we also construct a family of hom-associative Weyl algebras as generalizations of the classical analogue, and prove that they are simple.
We introduce hom-associative Ore extensions as non-unital, nonassociative Ore extensions with a hom-associative multiplication, and give some necessary and sucient conditions when such exist. Within this framework, we construct families of hom-associative quantum planes, universal enveloping algebras of a Lie algebra, and Weyl algebras, all being hom-associative generalizations of their classical counterparts, as well as prove that the latter are simple. We also provide a way of embedding any multiplicative hom-associative algebra into a multiplicative, weakly unital hom-associative algebra, which we call a weak unitalization.
Background: Managers and professionals in health and social care are required to implement evidence-based methods. Despite this, they generally lack training in implementation. In clinical settings, implementation is often a team effort, so it calls for team training. The aim of this study was to evaluate the effects of the Building Implementation Capacity (BIC) intervention that targets teams of professionals, including their managers. Methods: A non-randomized design was used, with two intervention cases (each consisting of two groups). The longitudinal, mixed-methods evaluation included pre-post and workshop-evaluation questionnaires, and interviews following Kirkpatrick's four-level evaluation framework. The intervention was delivered in five workshops, using a systematic implementation method with exercises and practical working materials. To improve transfer of training, the teams' managers were included. Practical experiences were combined with theoretical knowledge, social interactions, reflections, and peer support. Results: Overall, the participants were satisfied with the intervention (first level), and all groups increased their self-rated implementation knowledge (second level). The qualitative results indicated that most participants applied what they had learned by enacting new implementation behaviors (third level). However, they only partially applied the implementation method, as they did not use the planned systematic approach. A few changes in organizational results occurred (fourth level). Conclusions: The intervention had positive effects with regard to the first two levels of the evaluation model; that is, the participants were satisfied with the intervention and improved their knowledge and skills. Some positive changes also occurred on the third level (behaviors) and fourth level (organizational results), but these were not as clear as the results for the first two levels. This highlights the fact that further optimization is needed to improve transfer of training when building teams' implementation capacity. In addition to considering the design of such interventions, the organizational context and the participants' characteristics may also need to be considered to maximize the chances that the learned skills will be successfully transferred to behaviors.
This article is devoted to a multi-parametric family of associative complex algebras defined by commutation relations associated with group or semigroup actions of dynamical systems and iterated function systems. General reordering and nested commutator formulas for arbitrary elements in these families are presented, generalizing some well-known results in mathematics and physics. A generalization of this family in three generators is also considered.
A general class of multi-parametric families of unital associative complex algebras, defined by commutation relations associated with group or semigroup actions of dynamical systems and iterated function systems, is considered. A generalization of these commutation relations in three generators is also considered, modifying Lie algebra type commutation relations, typical for usual differential or difference operators, to relations satisfied by more general twisted difference operators associated with general twisting maps. General reordering and nested commutator formulas for arbitrary elements in these algebras are presented, and some special cases are considered, generalizing some well-known results in mathematics and physics.
Operator representations of deformed Lie type commutation relations, associated with group or semigroup actions of dynamical systems and iterated function systems are considered. In particular, it is shown that some multi-parameter deformed symmetric difference and multiplication operators satisfy these commutation relations. The operator representations are considered also in the context of twisted derivations.
We introduce non-associative Ore extensions, S = R[X; σ, δ], for any nonassociative unital ring R and any additive maps σ, δ: R → R satisfying σ(1) = 1 and δ(1) = 0. In the special case when δ is either left or right Rδ-linear, where Rδ = ker(δ), and R is δ-simple, i.e. {0} and R are the only δ-invariant ideals of R, we determine the ideal structure of the nonassociative differential polynomial ring D = R[X; idR, δ]. Namely, in that case, we show that all non-zero ideals of D are generated by monic polynomials in the center Z(D) of D. We also show that Z(D) = Rδ[p] for a monic p ∈ Rδ[X], unique up to addition of elements from Z(R)δ. Thereby, we generalize classical results by Amitsur on differential polynomial rings defined by derivations on associative and simple rings. Furthermore, we use the ideal structure of D to show that D is simple if and only if R is δ-simple and Z(D) equals the field Rδ ∩ Z(R). This provides us with a non-associative generalization of a result by Öinert, Richter and Silvestrov. This result is in turn used to show a non-associative version of a classical result by Jordan concerning simplicity of D in the cases when the characteristic of the field Rδ ∩ Z(R) is either zero or a prime. We use our findings to show simplicity results for both non-associative versions of Weyl algebras and non-associative differential polynomial rings defined by monoid/group actions on compact Hausdorff spaces.
Given a non-associative unital ring R, a monoid G and a set π of additive maps R→R, we introduce the Ore monoid ring R[π;G], and, in a special case, the differential monoid ring. We show that these structures generalize, in a natural way, not only the classical Ore extensions and differential polynomial rings, but also the constructions, introduced by Cojuhari, defined by so-called D-structures π. Moreover, for commutative monoids, we give necessary and sufficient conditions for differential monoid rings to be simple. We use this in a special case to obtain new and shorter proofs of classical simplicity results for differential polynomial rings in several variables previously obtained by Voskoglou and Malm by other means. We also give examples of new Ore-like structures defined by finite commutative monoids.
For any n-dimensional Hom-Lie algebra, a system of polynomial equations is obtained from the Hom-Jacobi identity, containing both structure constants of the skew-symmetric bilinear map and constants describing the twisting linear endomorphism. The equations are linear in the constants representing the endomorphism and non-linear in the structure constants. The space of possible endomorphisms has minimum dimension 6, and we describe the possible endomorphisms in that case. We further give families of 3-dimensional Hom-Lie algebras arising from a general nilpotent linear endomorphism constructed upto isomorphism.
We derive conditions for an arbitrary n-dimensional algebra to be a Hom-Lie algebra, in the form of a system of polynomial equations, containing both structure constants of the skew-symmetric bilinear map and constants describing the twisting linear endomorphism. The equations are linear in the constants representing the endomorphism and non-linear in the structure constants. When the algebra is 3 or 4-dimensional we describe the space of possible endomorphisms with minimum dimension. For the 3-dimensional case we give families of 3-dimensional Hom-Lie algebras arising from a general nilpotent linear endomorphism constructed upto isomorphism together with non-isomorphic canonical representatives for all the families in that case. We further give a list of 4-dimensional Hom-Lie algebras arising from a general nilpotent linear endomorphisms.
We describe the dimension of the space of possible linear endomorphisms that turn skew-symmetric three-dimensional algebras into Hom-Lie algebras. We find a correspondence between the rank of a matrix containing the structure constants of the bilinear product and the dimension of the space of Hom-Lie structures. Examples from classical complex Lie algebras are given to demonstrate this correspondence.
We begin by reviewing a classical result on the algebraic dependence of commuting elements in the Weyl algebra. We proceed by describing generalizations of this result to various classes of Ore extensions, including both results that are already known and one new result.
This paper generalizes a proof of certain results by Hellström and Silvestrov on centralizers in graded algebras. We study centralizers in certain algebras with valuations. We prove that the centralizer of an element in these algebras is a free module over a certain ring. Under further assumptions we obtain that the centralizer is also commutative.
In this paper we consider centralizers of single elements in Ore extensions of the ring of polynomials in one variable over a field. We show that they are commutative and finitely generated as an algebra. We also show that for certain classes of elements their centralizer is singly generated as an algebra.
In this expository paper we discuss a way of computing the Burchnall-Chaundy polynomial of two commuting differential operators using a determinant.We describe how the algorithm can be generalized to general Ore extensions, andwhich properties of the algorithm that are preserved.
In the present article we continue investigating the algebraic dependence of commutingelements in q-deformed Heisenberg algebras. We provide a simple proof that the0-chain subalgebra is a maximal commutative subalgebra when q is of free type and thatit coincides with the centralizer (commutant) of any one of its elements dierent fromthe scalar multiples of the unity. We review the Burchnall-Chaundy-type construction forproving algebraic dependence and obtaining corresponding algebraic curves for commutingelements in the q-deformed Heisenberg algebra by computing a certain determinantwith entries depending on two commuting variables and one of the generators. The coecients in front of the powers of the generator in the expansion of the determinant arepolynomials in the two variables dening some algebraic curves and annihilating the twocommuting elements. We show that for the elements from the 0-chain subalgebra exactlyone algebraic curve arises in the expansion of the determinant. Finally, we present severalexamples of computation of such algebraic curves and also make some observations onthe properties of these curves.
In this paper we consider algebras of functions that are constant on the sets of a partition. We describe the crossed product algebras of the mentioned algebras with Z. We show that the function algebra is isomorphic to the algebra of all functions on some set. We also describe the commutant of the function algebra and finish by giving an example of piece-wise constant functions on a real line.
In this paper we consider crossed product algebras of algebras of piecewiseconstant functions on the real line with Z. For an increasing sequence of algebras (in which case the commutants form a decreasing sequence), we describe the set difference between the corresponding commutants.
In this paper we consider commutants in crossed product algebras, for algebras of piece-wise constant functions on the real line acted on by the group of integers Z. The algebra of piece-wise constant functions does not separate points of the real line, and interplay of the action with separation properties of the points or subsets of the real line by the function algebra become essential for many properties of the crossed product algebras and their subalgebras. In this article, we deepen investigation of properties of this class of crossed product algebras and interplay with dynamics of the actions. We describe the commutants and changes in the commutants in the crossed products for the canonical generating commutative function subalgebras of the algebra of piece-wise constant functions with common jump points when arbitrary number of jump points are added or removed in general positions, that is when corresponding constant value set partitions of the real line change, and we give complete characterization of the set difference between commutants for the increasing sequence of subalgebras in crossed product algebras for algebras of functions that are constant on sets of a partition when partition is refined.
In this article we consider the Ore extension algebra for the algebra A of functions with finite support on a countable set. We derive explicit formulas for twisted derivations on A, give a description for the centralizer of A, and the center of the Ore extension algebra under specific conditions.