四元数斜域上的仿射圆几何

A finiteness theorem for special unitary groups of quaternionic skew-hermitian forms with good reduction

Given a field $K$ equipped with a set of discrete valuations $V$, we develop a general theory to relate reduction properties of skew-hermitian forms over a quaternion $K$-algebra $Q$ to quadratic forms over the function field $K(Q)$ obtained via Morita equivalence. Using this we show that if $(K,V)$ satisfies certain conditions, then the number of $K$-isomorphism classes of the universal coverings of the special unitary groups of quaternionic skew-hermitian forms that have good reduction at all valuations in $V$ is finite and bounded by a value that depends on size of a quotient of the Picard …

Surfaces containing two circles through each point and decomposition of quaternionic matrices

We find all analytic surfaces in space R^3 such that through each point of the surface one can draw two circular arcs fully contained in the surface. The proof uses a new decomposition technique for quaternionic matrices.

Cauchy-Riemann equations and Jacobians of quaternion polynomials

A map $f$ from the quaternion skew field $H$ to itself, can also be thought as a transformation $f:R^4 \to R^4$. In this manuscript, the Jacobian $J(f)$ of $f$ is computed, in the case where $f$ is a quaternion polynomial. As a consequence, the Cauchy-Riemman equations for $f$ are derived. It is also shown that the Jacobian determinant of $f$ is non negative over $H$. The above commensurates well with the theory of analytic functions of one complex variable.

类克利福德平行性的自同构

Automorphisms of a Clifford-like parallelism

研究四元数体上三维射影双空间中类克利福德平行性的自同构群，对比其与左平行性自同构群的关系，构造实例说明不同四元数体下两类自同构群的包含或相等情形。

Halving spaces and lower bounds in real enumerative geometry

We develop the theory of halving spaces to obtain lower bounds in real enumerative geometry. Halving spaces are topological spaces with an action of a Lie group $Γ$ with additional cohomological properties. For $Γ=\mathbb{Z}_2$ we recover the conjugation spaces of Hausmann, Holm and Puppe. For $Γ=\mathrm{U}(1)$ we obtain the circle spaces. We show that real even and quaternionic partial flag manifolds are circle spaces leading to non-trivial lower bounds for even real and quaternionic Schubert problems. To prove that a given space is a halving space, we generalize results of Borel and Haeflige…

关于规范原理的旋量群的赫尔曼引理

Hensel's lemma for the norm principle for spinor groups

本文在规范原理成立的条件下，证明了旋量群的赫尔曼引理在四元数代数上的有效性。