Cluster-tilted algebras

We introduce a new class of algebras, which we call cluster-tilted. They are by definition the endomorphism algebras of tilting objects in a cluster category. We show that their representation theory is very close to the representation theory of hereditary algebras. As an application of this, we prove a generalised version of so-called APR-tilting.

     

Clusters and seeds in acyclic cluster algebras

Cluster algebras are commutative algebras that were introduced by Fomin and Zelevinsky in order to model the dual canonical basis of a quantum group and total positivity in algebraic groups. Cluster categories were introduced as a representation-theoretic model for cluster algebras. In this article we use this representation-theoretic approach to prove a conjecture of Fomin and Zelevinsky, that for cluster algebras with no coefficients associated to quivers with no oriented cycles, a seed is determined by its cluster. We also obtain an interpretation of the monomial in the denominator of a non-polynomial cluster variable in terms of the composition factors of an indecomposable exceptional module over an associated hereditary algebra.

     

Relativistic Landau‐He‐McKellar‐Wilkens quantization induced by noninertial effects

We show that the relativistic analogue of the Landau‐He‐McKellar‐Wilkens quantization can be achieved through the noninertial effects of the Fermi‐Walker reference frame without assuming the existence of a magnetic charge density and discuss the nonrelativistic limit of the energy levels. We also obtain the Dirac spinors for positive‐energy values parallel and antiparallel to the z axis of the spacetime and obtain the Gordon decomposition of the Dirac probability current J^μ.     

Rashba coupling induced by Lorentz symmetry breaking effects

The arising of a Rashba‐like coupling, a Zeeman‐like term and a Darwin‐like term induced by Lorentz symmetry breaking effects in the non‐relativistic quantum dynamics of a spin‐1/2 neutral particle interacting with external fields is studied. Moreover, the influence of a Rashba‐like coupling induced by a Lorentz symmetry violation background on the Tan–Inkson potential [W.‐C. Tan and J. C. Inkson, Semicond. Sci. Technol. 11 , 1635 (1996)] is discussed.     

A geometric approach to confining a Dirac neutral particle in analogous way to a quantum dot

Based on the transfer-matrix method, the spin transport properties through a graphene-based multi-barrier nanostructure with an exchange field and Rashba spin orbit coupling (SOC), have been investigated. It is found that if Rashba SOC equals to the exchange field, the multi-barrier nanostructure is an efficient way to achieve spin rotators and spin filters. In addition, it is also found that the shot noise of a spin state can be enhanced by electrostatic potential, and plateaus of the Fano factor is formed.