On Whittaker modules over a class of algebras similar to U(sl2)U(sl_{2})

Motivated by the study of invariant rings of finite groups on the first Weyl algebras $A_{1}$ (\cite{AHV}) and finding interesting families of new noetherian rings, a class of algebras similar to $U(sl_{2})$ were introduced and studied by Smith in \cite{S}. Since the introduction of these algebras, research efforts have been focused on understanding their weight modules, and many important results were already obtained in \cite{S} and \cite{Ku}. But it seems that not much has been done on the part of nonweight modules. In this note, we generalize Kostant's results in \cite{K} on the Whittaker model for the universal enveloping algebras $U(\frak g)$ of finite dimensional semisimple Lie algebras $\frak g$ to Smith's algebras. As a result, a complete classification of irreducible Whittaker modules (which are definitely infinite dimensional) for Smith's algebras is obtained, and the submodule structure of any Whittaker module is also explicitly described.Comment: 11 page

Algebra endomorphisms and Derivations of Some Localized Down-Up Algebras

We study algebra endomorphisms and derivations of some localized down-up algebras \A. First, we determine all the algebra endomorphisms of \A under some conditions on $r$ and $s$. We show that each algebra endomorphism of \A is an algebra automorphism if $r^{m}s^{n}=1$ implies $m=n=0$. When $r=s^{-1}=q$ is not a root of unity, we give a criterion for an algebra endomorphism of \A to be an algebra automorphism. In either case, we are able to determine the algebra automorphism group for \A. We also show that each surjective algebra endomorphism of the down-up algebra $A(r+s, -rs)$ is an algebra automorphism in either case. Second, we determine all the derivations of \A and calculate its first degree Hochschild cohomology group

On Irreducible weight representations of a new deformation Uq(sl2)U_{q}(sl_{2}) of U(sl2)U(sl_{2})

Starting from a Hecke $R-$matrix, Jing and Zhang constructed a new deformation $U_{q}(sl_{2})$ of $U(sl_{2})$, and studied its finite dimensional representations in \cite{JZ}. Especically, this algebra is proved to be just a bialgebra, and all finite dimensional irreducible representations are constructed in \cite{JZ}. In addition, an example is given to show that not every finite dimensional representation of this algebra is completely reducible. In this note, we take a step further by constructing more irreducible representations for this algebra. We first construct points of the spectrum of the category of representations over this new deformation by using methods in noncommutative algebraic geometry. Then applied to the study of representations, our construction recovers all finite dimensional irreducible representations as constructed in \cite{JZ}, and yields new families of infinite dimensional irreducible weight representations of this new deformation $U_{q}(sl_{2})$.Comment: 8 page

Automorphisms of the two-parameter Hopf algebra \V

We determine the group of algebra automorphisms for the two-parameter quantized enveloping algebra \V. As an application, we prove that the group of Hopf algebra automorphisms for \V is isomorphic to a torus of rank two

Constructing irreducible representations of quantum groups Uq(fm(K))U_{q}(f_{m}(K))

In this paper, we construct families of irreducible representations for a class of quantum groups $U_{q}(f_{m}(K))$. First, we give a natural construction of irreducible weight representations for $U_{q}(f_{m}(K))$ using methods in spectral theory developed by Rosenberg. Second, we study the Whittaker model for the center of $U_{q}(f_{m}(K))$. As a result, the structure of Whittaker representations is determined, and all irreducible Whittaker representations are explicitly constructed. Finally, we prove that the annihilator of a Whittaker representation is centrally generated.Comment: 19 pages, some modifications made to the first versio

On representations of quantum groups Uq(fm(K,H))U_{q}(f_{m}(K,H))

We construct families of irreducible representations for a class of quantum groups $U_{q}(f_{m}(K,H)$. First, we realize these quantum groups as Hyperbolic algebras. Such a realization yields natural families of irreducible weight representations for $U_{q}(f_{m}(K,H))$. Second, we study the relationship between $U_{q}(f_{m}(K,H))$ and $U_{q}(f_{m}(K))$. As a result, any finite dimensional weight representation of $U_{q}(f_{m}(K,H))$ is proved to be completely reducible. Finally, we study the Whittaker model for the center of $U_{q}(f_{m}(K,H))$, and a classification of all irreducible Whittaker representations of $U_{q}(f_{m}(K,H))$ is obtained.Comment: Some minor modifications to the first versio

Je\'{s}manowicz' conjecture and Fermat numbers

Let $a,b,c$ be relatively prime positive integers such that $a^{2}+b^{2}=c^{2}.$ In 1956, Je\'{s}manowicz conjectured that for any positive integer $n$, the only solution of $(an)^{x}+(bn)^{y}=(cn)^{z}$ in positive integers is $(x,y,z)=(2,2,2)$. Let $k\geq 1$ be an integer and $F_k=2^{2^k}+1$ be a Fermat number. In this paper, we show that Je\'{s}manowicz' conjecture is true for Pythagorean triples $(a,b,c)=(F_k-2,2^{2^{k-1}+1},F_k)$.Comment: we correct some mistakes in the first version and revised the title of pape

Strain-engineered A-type antiferromagnetic order in YTiO3_3: a first-principles calculation

The epitaxial strain effects on the magnetic ground state of YTiO$_3$ films grown on LaAlO$_3$ substrates have been studied using the first-principles density-functional theory. With the in-plane compressive strain induced by LaAlO$_3$ (001) substrate, A-type antiferromagnetic order emerges against the original ferromagnetic order. This phase transition from ferromagnet to A-type antiferromagnet in YTiO$_3$ film is robust since the energy gain is about 7.64 meV per formula unit despite the Hubbard interaction and modest lattice changes, even though the A-type antiferromagnetic order does not exist in any $R$TiO$_3$ bulks.Comment: 3 pages, 2 figures. Proceeding of the 12th Joint MMM/Intermag Conference. Accepted by JA

Laser and Microwave Excitations of Rabi Oscillations of a Single Nitrogen-Vacancy Electron Spin in Diamond

A collapse and revival shape of Rabi oscillations of a single Nitrogen-Vacancy (NV) center electron spin has been observed in diamond at room temperature. Because of hyperfine interaction between the host 14N nuclear spin and NV center electron spin, different orientation of the 14N nuclear spin leads to a triplet splitting of the transition between the ground ms=0 and excited states ms=1. Microwave can excite the three transitions equally to induce three independent nutations and the shape of Rabi oscillations is a combination of the three nutations. This result provides an innovative view of electron spin oscillations in diamond.Comment: This manuscript was submitted to Physical Review Letters on June 08, 201

Experience-based Optimization: A Coevolutionary Approach

This paper studies improving solvers based on their past solving experiences, and focuses on improving solvers by offline training. Specifically, the key issues of offline training methods are discussed, and research belonging to this category but from different areas are reviewed in a unified framework. Existing training methods generally adopt a two-stage strategy in which selecting the training instances and training instances are treated in two independent phases. This paper proposes a new training method, dubbed LiangYi, which addresses these two issues simultaneously. LiangYi includes a training module for a population-based solver and an instance sampling module for updating the training instances. The idea behind LiangYi is to promote the population-based solver by training it (with the training module) to improve its performance on those instances (discovered by the sampling module) on which it performs badly, while keeping the good performances obtained by it on previous instances. An instantiation of LiangYi on the Travelling Salesman Problem is also proposed. Empirical results on a huge testing set containing 10000 instances showed LiangYi could train solvers that perform significantly better than the solvers trained by other state-of-the-art training method. Moreover, empirical investigation of the behaviours of LiangYi confirmed it was able to continuously improve the solver through training