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Wensheng Cao

Publications and source records attributed to Wensheng Cao.

16 recordsLinked to original sources
arXiv2026

Conformal Barycenters in Quaternionic Hyperbolic Balls

We extend the notion of conformal barycenter, recently introduced by Jačimović and Kalaj for the complex hyperbolic ball, to the quaternionic unit ball $\BH$. The quaternionic conformal barycenter of a measurable set $D$ with finite hyperbolic measure and finite first moment is defined as the unique point $c$ such that $\int_D Φ_c(q)\, \dLam(q) = \mathbf{0}$, where $Φ_c$ is the quaternionic Hua involution exchanging $0$ and $c$. Equivalently, it is the unique minimum of the energy functional $G(x) = \int_D \log\cosh^2\!\big(\frac12 d_H(x,y)\big)\, \dLam(y)$. We prove existence and uniqueness using the strict geodesic convexity of $G$, which is established by a direct computation along geodesics. The barycenter is invariant under the full isometry group $\mathrm{Sp}(n,1)$. We also treat finite point sets and provide explicit examples.

math-ph↗
arXiv2024

Quadratic unilateral polynomials over split quaternions

In this paper, we derive explicit formulas for computing the roots of $ax^{2}+bx+c=0$ with $a$ being not invertible in split quaternion algebra. We also imitate the approach developed by Opfer, Janovska and Falcao etc. to verify our results when the corresponding companion polynomials are not identically vanishing.

math.AG↗
arXiv2022

On left spectrum of a split quaternionic matrix

In noncommutative and nondivision algebra, left spectrum of matrices are less known and is not easy to handle. Split quaternion algebra is a noncommutative and nondivision algebra. In this paper, by the formulas of solving the equations $ax=b$ and $ax^2+bx+c=0$ over split quaternions, we make an attempt to understand the left spectrum of a split quaternion matrix of order 2.

math.RA↗
arXiv2022

The Moore-Penrose Inverses of Clifford Algebra $C\ell_{1,2}$

In this paper, we introduce a ring isomorphism between the Clifford algebra $C\ell_{1,2}$ and a ring of matrices. By such a ring isomorphism, we introduce the concept of the Moore-Penrose inverse in Clifford algebra $C\ell_{1,2}$. Using the Moore-Penrose inverse, we solve the linear equation $axb=d$ in $C\ell_{1,2}$. We also obtain necessary and sufficient conditions for two numbers in $C\ell_{1,2}$ to be similar.

math.RA↗
arXiv2020

Quadratic formulas for split quaternions

Unlike the Hamilton quaternion algebra, the split-quaternions contain nontrivial zero divisors. In general speaking, it is hard to find the solutions of equations in algebras containing zero divisor. In this paper, we manage to derive explicit formulas for computing the roots of $x^{2}+bx+c=0$ in split quaternion algebra.

math.RA↗
arXiv2019

The Moore-Penrose inverses of split quaternions

In this paper, we find the roots of lightlike quaternions. By introducing the concept of the Moore-Penrose inverse in split quaternions, we solve the linear equations $axb=d$, $xa=bx$ and $xa=b\bar{x}$. Also we obtain necessary and sufficient conditions for two split quaternions to be similar or consimilar.

math.RA↗
arXiv2017

The moduli space of points in quaternionic projective space

Let $\mathcal{M}(n,m;\F \bp^n)$ be the configuration space of $m$-tuples of pairwise distinct points in $\F \bp^n$, that is, the quotient of the set of $m$-tuples of pairwise distinct points in $\F \bp^n$ with respect to the diagonal action of ${\rm PU}(1,n;\F)$ equipped with the quotient topology. It is an important problem in hyperbolic geometry to parameterize $\mathcal{M}(n,m;\F \bp^n)$ and study the geometric and topological structures on the associated parameter space. In this paper, by mainly using the rotation-normalized and block-normalized algorithms, we construct the parameter spaces of both $\mathcal{M}(n,m; \bhq)$ and $\mathcal{M}(n,m;\bp(V_+))$, respectively.

math.AG↗
arXiv2017

On volumes of quaternionic hyperbolic n-orbifolds

By use of H. C. Wang's bound on the radius of a ball embedded in the fundamental domain of a lattice of a semisimple Lie group, we construct an explicit lower bound for the volume of a quaternionic hyperbolic orbifold that depends only on dimension.

math.GT↗
arXiv2015

Congruence classes of points in quaternionic hyperbolic spaces

An important problem in quaternionic hyperbolic geometry is to classify ordered $m$-tuples of pairwise distinct points in the closure of quaternionic hyperbolic n-space, $\overline{{\bf H}_\bh^n}$, up to congruence in the holomorphic isometry group ${\rm PSp}(n,1)$ of ${\bf H}_\bh^n$. In this paper we concentrate on two cases: $m=3$ in $\overline{{\bf H}_\bh^n}$ and $m=4$ on $\partial{\bf H}_\bh^n$ for $n\geq 2$. New geometric invariants and several distance formulas in quaternionic hyperbolic geometry are introduced and studied for this problem. The congruence classes are completely described by quaternionic Cartan's angular invariants and the distances between some geometric objects for the first case. The moduli space is constructed for the second case.

math.AG↗
arXiv2011

Algebraic Characterization of Isometries of the Complex and Quaternionic Hyperbolic Planes

It is of interest to characterize algebraically the dynamical types of isometries of the complex and quaternionic hyperbolic planes. In the complex case, such a characterization is known from the work of Giraud-Goldman. In this paper, we offer an algebraic characterization of the isometries of the two-dimensional quaternionic hyperbolic space. Our result restricts to the complex case and provides another characterization of the isometries of the complex hyperbolic plane which is different from the characterization due to Giraud-Goldman. Two elements in a group G are said to be in the same z-class if their centralizers are conjugate in G. The $z$-classes provide a finite partition of the isometry group. In this paper we describe the centralizers of the isometries, and determine the z-classes.

math.GT↗
arXiv2010

Commuting Isometries of the Complex Hyperbolic Space

Let $H^n$ denote the complex hyperbolic space of dimension $n$. The group $U(n,1)$ acts as the group of isometries of $H^n$. In this paper we investigate when two isometries of the complex hyperbolic space commute. Along the way we determine the centralizers.

math.DG↗
arXiv2010

Discreteness criterion in SL(2,$\bc$) by a test map

In the paper (Osaka J. Math. {\bf 46}: 403-409, 2009), Yang conjectured that a non-elementary subgroup $G$ of $SL(2, \bc)$ containing elliptic elements is discrete if for each elliptic element $g\in G$ the group $< f, g >$ is discrete, where $f\in SL(2,\bc)$ is a test map which is loxodromic or elliptic. The purpose of this paper is to give an affirmative answer to this question.

math.AG↗
arXiv2010

Jorgensen's inequality for quaternionic hyperbolic n-space

Jorgensen's inequality gives a necessary condition for a non-elementary two generator group of isometries of real hyperbolic 2-space to be discrete. We give analogues of Jorgensen's inequality for non-elementary groups of isometries of quaternionic hyperbolic n-space generated by two elements, one of which is loxodromic.

math.AT↗
arXiv2010

Jorgensen's Inequalities and Collars in n-dimensional Quaternionic Hyperbolic Space

In this paper, we obtain analogues of Jorgensen's inequality for non-elementary groups of isometries of quaternionic hyperbolic $n$-space generated by two elements, one of which is loxodromic. Our result gives some improvement over earlier results of Kim [10] and Markham [15]}. These results also apply to complex hyperbolic space and give improvements on results of Jiang, Kamiya and Parker [7] As applications, we use the quaternionic version of Jørgensen's inequalities to construct embedded collars about short, simple, closed geodesics in quaternionic hyperbolic manifolds. We show that these canonical collars are disjoint from each other. Our results give some improvement over earlier results of Markham and Parker and answer an open question posed in [16].

math.GT↗