29 research outputs found
A characterization of the unitary highest weight modules by Euclidean Jordan algebras
Let be the conformal algebra of a simple Euclidean Jordan
algebra . We show that a (non-trivial) unitary highest weight
-module has the smallest positive Gelfand-Kirillov dimension
if and only if a certain quadratic relation is satisfied in the universal
enveloping algebra . In particular, we find
an quadratic element in . A prime ideal in
equals the Joseph ideal if and only if it
contains this quadratic element.Comment: 34pages, accepted by Journal of Lie Theor
Kazhdan-Lusztig right cells and associated varieties of highest weight modules
Let be a simple Lie algebra with a Weyl group . Let
be a simple module with highest weight . By using a conjecture of
Tanisaki, we show that there is a bijection between the right cells and
associated varieties of highest weight modules with infinitesimal character
. When is a simple integral highest weight module of
with the minimal Gelfand-Kirillov dimension
, we will show that its associated variety is irreducible. In particular,
its associated variety will be given in the information of . When
is a simple highest weight module of
in a given parabolic category with maximal
Gelfand-Kirillov dimension, we will show that its associated variety is also
irreducible.Comment: 9page
A combinatorial characterization of the annihilator varieties of highest weight modules for classical Lie algebras
Let be a classical Lie algebra. Let be a highest
weight module of with highest weight , where
is half the sum of positive roots. In 1985, Joseph proved that the
associated variety of a primitive ideal is the Zariski closure of a nilpotent
orbit in . In this paper, we will give some combinatorial
characterizations of the annihilator varieties of highest weight modules for
classical Lie algebras. In fact, we will give two algorithms, i.e., bipartition
algorithm and partition algorithm.Comment: 40page
Irreducible representations of of minimal Gelfand-Kirillov dimension
In this article, by studying the Bernstein degrees and Goldie rank
polynomials, we establish a comparison between the irreducible representations
of possessing the minimal Gelfand-Kirillov
dimension and those induced from finite-dimensional representations of the
maximal parabolic subgroup of of type . We give the transition
matrix between the two bases for the corresponding coherent families.Comment: To appear in Acta Mathematica Sinica, English Serie
Quantum PT-Phase Diagram in a Non-Hermitian Photonic Structure
Photonic structures have an inherent advantage to realize PT-phase transition
through modulating the refractive index or gain-loss. However, quantum PT
properties of these photonic systems have not been comprehensively studied yet.
Here, in a bi-photonic structure with loss and gain simultaneously existing, we
analytically obtained the quantum PT-phase diagram under the steady state
condition. To characterize the PT-symmetry or -broken phase, we define an
Hermitian exchange operator expressing the exchange between quadrature
variables of two modes. If inputting several-photon Fock states into a
PT-broken bi-waveguide splitting system, most photons will concentrate in the
dominant waveguide with some state distributions. Quantum PT-phase diagram
paves the way to the quantum state engineering, quantum interferences, and
logic operations in non-Hermitian photonic systems.Comment: 6 pages, 3 figure