Extensions of Modules over Hopf Algebras Arising from Lie Algebras of Cartan Type

In this paper we prove that there are no self-extensions of simple modules over restricted Lie algebras of Cartan type. The proof given by Andersen for classical Lie algebras not only uses the representation theory of the Lie algebra, but also representations of the corresponding reductive algebraic group. The proof presented in the paper follows in the same spirit by using the construction of a infinite-dimensional Hopf algebra D(G) u( ) containing u( ) as a normal Hopf subalgebra, and the representation theory of this algebra developed in our previous work. Finite-dimensional hyperalgebra analogs D(G _r ) u( ) have also been constructed, and the results are stated in this setting.     

Decomposition Numbers and Canonical Bases

We obtain some simple relations between decomposition numbers of quantized Schur algebras at an nth root of unity (over a field of characteristic 0). These relations imply that every decomposition number for such an algebra occurs as a decomposition number for some Hecke algebra of type A. We prove similar relations between coefficients of the canonical basis of the q-deformed Fock space representation of . It follows that these coefficients can all be expressed in terms of those of the global crystal basis of the irreducible subrepresentation generated by the vacuum vector. As a consequence, using the works of Ariki and Varagnolo and Vasserot, it is possible to give a new proof of Lusztig"s character formula for the simple U _v (sl _r )-modules at roots of unity, which does not involve representations of of negative level.     

On the Classification of q-Algebras

The problem is the classification of the ideals of free differential algebras, or the associated quotient algebras, the q-algebras; being finitely generated, unital C-algebras with homogeneous relations and a q-differential structure. This family of algebras includes the quantum groups, or at least those that are based on simple (super) Lie or Kac–Moody algebras. Their classification would encompass the so far incompleted classification of quantized (super) Kac–Moody algebras and of the (super) Kac–Moody algebras themselves. These can be defined as singular limits of q-algebras, and it is evident that to deal with the q-algebras in their full generality is more rational than the examination of each singular limit separately. This is not just because quantization unifies algebras and superalgebras, but also because the points q=1 and q=–1 are the most singular points in parameter space. In this Letter, one of two major hurdles in this classification program has been overcome. Fix a set of integers n ₁,...,n _k, and consider the space of homogeneous polynomials of degree n ₁ in the generator e ₁, and so on. Assume that there are no constants among the polynomials of lower degree, in any one of the generators; in this case all constants in the space have been classified. The task that remains, the more formidable one, is to remove the stipulation that there are no constants of lower degree.     

Plasmonic enhancing nanoantennas for photodetection

This paper discusses the use of plasmonic nanostructured systems as nanoantennas for photodetection. Even though semiconductors and their heterostructures have many useful properties and widely used in photodetection, their electron density is very small compared to that of metals and, therefore, they have low absorption cross sections. The idea of using metal nanostructured antennas is to combine the high optical responses of metals with the functional electric properties of semiconductors.     

AFM nanooxidation process - Technology perspective for mesoscopic structures

We study the technology of local anodic oxidation (LAO) by the AFM tip applied to semiconductor heterostructures with two-dimensional electron gas. The aim is to design mesoscopic rings with persistent current and one subband occupied. For this purpose the need is to oxidize narrow lines that represent energy barriers high enough. Using the electrostatic model, we explain the electric field distribution in the system tip-sample just before LAO starts. We study the influence of the conductivity of the cap layer on LAO and explain the origin of the saddle-like profile lines, observed in the experiment. Using Monte Carlo simulation we show that the carrier redistribution in the system with LAO energy barriers effectively lowers the barrier height. In the experimental part we have grown InGaP/AlGaAs/GaAs heterostructures by organometalic vapor phase epitaxy with an active layer only 31 nm below the surface. We have prepared oxide lines on the heterostructures by LAO and characterized them by the temperature-dependent transport measurement.     

Generalized Digroups

The concept of digroup has been proposed as a generalization of continuous groups whose tangent space is a Leibniz algebra. We study a further generalization of the digroup structure in which we do not demand that the inverses are necessarily bilateral. We characterize a generalized digroup as a set and as a product. We show an analogous to the first isomorphism theorem. Additionally, we explore the concept of dialgebra digroup.     

Local time-stepping procedures for the space-time conservation element and solution element method

A local time-stepping procedure for the space-time conservation element and solution element (CESE) method has been developed. This new procedure allows for variation of time-step size in both space and time, and can also be extended to become multi-dimensional solvers with structured/unstructured spatial grids. Moreover, it differs substantially in concept and methodology from the existing approaches. By taking full advantage of key concepts of the CESE method, in a simple and efficient manner it can enforce flux conservation across an interface separating grid zones of different time-step sizes. In particular, no correction pass is needed. Numerical experiments show that, for a variety of flow problems involving moving shock and flame discontinuities, accurate and robust numerical simulations can be achieved even with a reduction in time-step size on the order of 10 or higher for grids across a single interface.     

On the local fractional derivative

We present the necessary conditions for the existence of the Kolwankar-Gangal local fractional derivatives (KG-LFD) and introduce more general but weaker notions of LFDs by using limits of certain integral averages of the difference-quotient. By applying classical results due to Stein and Zygmund (1965) [16] we show that the KG-LFD is almost everywhere zero in any given intervals. We generalize some of our results to higher dimensional cases and use integral approximation formulas obtained to design numerical schemes for detecting fractional dimensional edges in signal processing.