Geometry of Riccati equations over normed division algebras

Abstract

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.