Differential operators preserving the space of harmonic functions (aka higher symmetries of the Laplacian)
The article http://arxiv.org/abs/hep-th/0206233 (published in Ann. of Math. (2) 161 (2005), no. 3) deals with linear differential operators $D$ for which there exists another linear differential ...
The physical meaning of a Lie group symmetry is clear: for example, if you have a quantum system whose states have values in some Hilbert space $H$, then a Lie group symmetry of the system means that ...