标题： 赋范可除代数上的Riccati方程的几何学 显示英文标题

标题： Geometry of Riccati equations over normed division algebras

摘要： 本文提出并研究了有限维赋范可除代数上的Riccati方程。 我们证明，有限维赋范可除代数上的Riccati方程$A$是欧几里得空间上共形Riccati方程的一个特例，可以被视为Lie代数中向量场的曲线$V\simeq\mathfrak{so}(\dim A+1,1)$。从新的视角恢复了已知类型Riccati方程的先前结果。 一种新的Riccati方程类型——八元数Riccati方程被推广到八元数射影直线$\mathbb{O}{\rm P}^1$上。作为一个新的物理应用，四元数Riccati方程被用于研究1+1维上的四元数Schrödinger方程。 显示英文摘要

摘要： This work presents and studies Riccati equations over finite-dimensional normed division algebras. We prove that a Riccati equation over a finite-dimensional normed division algebra $A$ is a particular case of conformal Riccati equation on a Euclidean space and it can be considered as a curve in a Lie algebra of vector fields $V\simeq\mathfrak{so}(\dim A+1,1)$. Previous results on known types of Riccati equations are recovered from a new viewpoint. A new type of Riccati equations, the octonionic Riccati equations, are extended to the octonionic projective line $\mathbb{O}{\rm P}^1$. As a new physical application, quaternionic Riccati equations are applied to study quaternionic Schr\"odinger equations on 1+1 dimensions.