arXiv subjects

Zhanqiang Bai

Publications and source records attributed to Zhanqiang Bai.

25 records · Page 2Linked to original sources

arXiv2021

Gelfand-Kirillov dimensions and associated varieties of highest weight modules

In this paper, we present a uniform formula of Lusztig's $ \mathbf{a}$-functions on classical Weyl groups. Then we obtain an efficient algorithm for the Gelfand-Kirillov dimensions of simple highest weight modules of classical Lie algebras, whose highest weight is not necessarily regular or integral. To deal with type $ D $, we prove an interesting property about domino tableaux by introducing an invariant, called the hollow tableau. As an application, the associated varieties of all the simple highest weight Harish-Chandra modules are explicitly determined, including the exceptional cases.

math.RT↗

arXiv2018

Gelfand-Kirillov Dimensions of Highest Weight Harish-Chandra Modules for $SU(p,q)$

Let $ (G,K) $ be an irreducible Hermitian symmetric pair of non-compact type with $G=SU(p,q)$, and let $ λ$ be an integral weight such that the simple highest weight module $ L(λ) $ is a Harish-Chandra $ (\mathfrak{g},K) $-module. We give a combinatoric algorithm for the Gelfand-Kirillov dimension of $ L(λ) $. This enables us to prove that the Gelfand-Kirillov dimension of $ L(λ) $ decreases as the integer $ \langleλ+ρ,β^\vee\rangle $ increases, where $ρ$ is the half sum of positive roots and $β$ is the maximal noncompact root. As a byproduct, we obtain a description on the associated variety of $ L(λ) $.

math.RT↗

arXiv2017

Gelfand-Kirillov Dimensions of the Z-graded Oscillator Representations of $\mathfrak{o}(n,\mathbb{C})$ and $\mathfrak{sp}(2n,\mathbb{C})$

In this paper, we compute the Gelfand-Kirillov dimensions for the infinite-dimensional explicit irreducible $\mathfrak{o}(n, \mathbb{C})$-modules and $\mathfrak{sp}(2n, \mathbb{C})$-modules that appeared in the $\mathbb{Z}$-graded oscillator generalizations of the classical theorem on harmonic polynomials established by Luo and Xu. We also found that some of the modules have the secondly minimal GK-dimension, and some of them have the larger GK-dimension than the maximal GK-dimension of unitary highest-weight modules.

math.RT↗

arXiv2015

Gelfand-Kirillov dimensions of the $Z^2$-graded oscillator representations of sl(n)

We find an exact formula of Gelfand-Kirillov dimensions for the infinite-dimensional explicit irreducible sl(n, F)-modules that appeared in the Z^2-graded oscillator generalizations of the classical theorem on harmonic polynomials established by Luo and Xu. Three infinite subfamilies of these modules have the minimal Gelfand-Kirillov dimension. They contain weight modules with unbounded weight multiplicities and completely pointed modules.

math.RT↗

arXiv2015

Particle Motion in Generalized Dirac's Monopoles of dimension 2k+1

By using Meng's idea in his generalization of the classical MICZ-Kepler problem, we obtained the equations of motion of a charged particle in the field of generalized Dirac monopole in odd dimensional Euclidean spaces. The main result is that for every particle trajectory $\mathbf{r}: I\to\mathbb{R}^{2k+1} \setminus \{0\}$, there is a 2-dimensional cone with vertex at the origin on which $\mathbf{r}$ is a~geodesic.

math-ph↗

arXiv2013

On the Orbits of the Magnetized Kepler Problems in Dimension 2k+1

It is demonstrated that, for the recently introduced classical magnetized Kepler problems in dimension $2k+1$, the non-colliding orbits in the "external configuration space" $\mathbb R^{2k+1}\setminus\{\mathbf 0\}$ are all conics, moreover, a conic orbit is an ellipse, a parabola, and a branch of a hyperbola according as the total energy is negative, zero, and positive. It is also demonstrated that the Lie group ${\mr {SO}}^+(1,2k+1)\times {\bb R}_+$ acts transitively on both the set of oriented elliptic orbits and the set of oriented parabolic orbits.

math-ph↗

arXiv2012

A characterization of the unitary highest weight modules by Euclidean Jordan algebras

Let $\mathfrak{co}(J)$ be the conformal algebra of a simple Euclidean Jordan algebra $J$. We show that a (non-trivial) unitary highest weight $\mathfrak{co}(J)$-module has the smallest positive Gelfand-Kirillov dimension if and only if a certain quadratic relation is satisfied in the universal enveloping algebra $U(\mathfrak{co}(J)_{\mathbb{C}})$. In particular, we find an quadratic element in $U(\mathfrak{co}(J)_{\mathbb{C}})$. A prime ideal in $U(\mathfrak{co}(J)_{\mathbb{C}})$ equals the Joseph ideal if and only if it contains this quadratic element.

math.RT↗