Vortex Filaments in Hermitian Reductive Lie Algebras
It is well-known that the investigation of vortex filaments (i.e., moving curves) in the Euclidean 3-space $\mathbb R^3$ is an attractive topic both in physics and mathematics. The theory consists mainly of the three vortex models, up to the third-order approximation. Such a theory has been successfully extended to Hermitian symmetric Lie algebras in mathematics with physical and geometrical backgrounds. This article is devoted to developing it to Hermitian reductive Lie algebras in a purely geometric way. The three vortex models obtained in this article fulfill that when the Hermitian reductive Lie algebra ${\mathfrak g}$ equi-collapses to a Hermitian symmetric Lie algebra ${\mathfrak h}$, they revert respectively to those in ${\mathfrak h}$.