12,304 research outputs found
On Whittaker modules over a class of algebras similar to
Motivated by the study of invariant rings of finite groups on the first Weyl
algebras (\cite{AHV}) and finding interesting families of new
noetherian rings, a class of algebras similar to 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 of finite dimensional semisimple
Lie algebras 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 and . We show that each algebra endomorphism of \A
is an algebra automorphism if implies . When
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 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 of
Starting from a Hecke matrix, Jing and Zhang constructed a new
deformation of , 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
.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
In this paper, we construct families of irreducible representations for a
class of quantum groups . First, we give a natural
construction of irreducible weight representations for using
methods in spectral theory developed by Rosenberg. Second, we study the
Whittaker model for the center of . 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
We construct families of irreducible representations for a class of quantum
groups . First, we realize these quantum groups as Hyperbolic
algebras. Such a realization yields natural families of irreducible weight
representations for . Second, we study the relationship
between and . As a result, any finite
dimensional weight representation of is proved to be
completely reducible. Finally, we study the Whittaker model for the center of
, and a classification of all irreducible Whittaker
representations of is obtained.Comment: Some minor modifications to the first versio
Je\'{s}manowicz' conjecture and Fermat numbers
Let be relatively prime positive integers such that
In 1956, Je\'{s}manowicz conjectured that for any positive
integer , the only solution of in positive
integers is . Let be an integer and
be a Fermat number. In this paper, we show that Je\'{s}manowicz' conjecture is
true for Pythagorean triples .Comment: we correct some mistakes in the first version and revised the title
of pape
Strain-engineered A-type antiferromagnetic order in YTiO: a first-principles calculation
The epitaxial strain effects on the magnetic ground state of YTiO films
grown on LaAlO substrates have been studied using the first-principles
density-functional theory. With the in-plane compressive strain induced by
LaAlO (001) substrate, A-type antiferromagnetic order emerges against the
original ferromagnetic order. This phase transition from ferromagnet to A-type
antiferromagnet in YTiO 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
TiO 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
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