Commutativity and ideals in algebraic crossed products

We investigate properties of commutative subrings and ideals in non-commutative algebraic crossed products for actions by arbitrary groups. A description of the commutant of the base coefficient subring in the crossed product ring is given. Conditions for commutativity and maximal commutativity of the commutant of the base subring are provided in terms of the action as well as in terms of the intersection of ideals in the crossed product ring with the base subring, specially taking into account both the case of base rings without non-trivial zero-divisors and the case of base rings with non-trivial zero-divisors

Interaction of the Laws of Electrodynamics in the Huber Effect

A complex physical phenomenon, first discovered by engineer J. Huber in 1951, is investigated. From the perspective of an external observer, the phenomenon is as follows: an electric current is passed through the wheel pairs of the car moving from the rail to the rail. The current, passing through the movable contacts of the wheels and rails, creates an additional (up to the moment of inertia) torque. The research task is to explain the reason for the occurrence of torque. Based on the analysis of individual components of the electrodynamic phenomenon discovered by Huber, an algorithm for the successive interaction of the individual components of the effect is found on the basis of the laws of classical electrodynamics: electric, ferromagnetic, and mechanical.The identity of the effect is explained, both for the wheel pair and for the bearing (Kosyrev-Milroy engine). For the first time, the cause of the appearance of the torque is revealed: relative movement of surface charges in the region of the movable electrical contact to the wheel body and the rails (or balls and guides). Moving charges unevenly magnetized ferromagnetic bodies according to the Biot-Savart-Laplace law. Due to the reduction in the clearance of the oncoming side of the wheel (or balls) and the increase on the trailing side, the pulling force from the oncoming side and, accordingly, the moment are more than on trailing side. The presented theoretical explanations completely correspond to the experimental investigation of the effect carried out by different scientists at different times

Maximal commutative subrings and simplicity of Ore extensions

The aim of this article is to describe necessary and sufficient conditions for simplicity of Ore extension rings, with an emphasis on differential polynomial rings. We show that a differential polynomial ring, R[x;id,\delta], is simple if and only if its center is a field and R is \delta-simple. When R is commutative we note that the centralizer of R in R[x;\sigma,\delta] is a maximal commutative subring containing R and, in the case when \sigma=id, we show that it intersects every non-zero ideal of R[x;id,\delta] non-trivially. Using this we show that if R is \delta-simple and maximal commutative in R[x;id,\delta], then R[x;id,\delta] is simple. We also show that under some conditions on R the converse holds.Comment: 16 page

Charge accumulation at the boundaries of a graphene strip induced by a gate voltage: Electrostatic approach

Distribution of charge induced by a gate voltage in a graphene strip is investigated. We calculate analytically the charge profile and demonstrate a strong(macroscopic) charge accumulation along the boundaries of a micrometers-wide strip. This charge inhomogeneity is especially important in the quantum Hall regime where we predict the doubling of the number of edge states and coexistence of two different types of such states. Applications to graphene-based nanoelectronics are discussed.Comment: 5 pages, 6 figures, Title changed due to Edito

Brackets with (τ,σ)(\tau,\sigma)-derivations and (p,q)(p,q)-deformations of Witt and Virasoro algebras

The aim of this paper is to study some brackets defined on $(\tau,\sigma)$-derivations satisfying quasi-Lie identities. Moreover, we provide examples of $(p,q)$-deformations of Witt and Virasoro algebras as well as $\mathfrak{sl}(2)$ algebra. These constructions generalize the results obtained by Hartwig, Larsson and Silvestrov on $\sigma$-derivations, arising in connection with discretizations and deformations of algebras of vector fields.Comment: 30 page

Representations and Cocycle Twists of Color Lie Algebras

We study relations between finite-dimensional representations of color Lie algebras and their cocycle twists. Main tools are the universal enveloping algebras and their FCR-properties (finite-dimensional representations are completely reducible.) Cocycle twist preserves the FCR-property. As an application, we compute all finite dimensional representations (up to isomorphism) of the color Lie algebra $sl_2^c$.Comment: 18 pages, with an concrete exampl

Ergodipotent maps and commutativity of elements in noncommutative rings and algebras with twisted intertwining

AbstractA property of algebraic dependence between two commuting elements is shown to hold in a more general setting than that in which it has previously been established. Key conditions are identified and some methods for establishing them are given.Moreover the class of algebras with a generalised Weyl structure, generalising the so-called Generalised Weyl Algebras (GWAs) or hyperbolic rings, is introduced and studied. We also present an interesting class of algebras which are not GWAs but share many of their properties by virtue of their generalised Weyl structure. For these classes of algebras, centralisers and algebraic dependence are investigated