  The hypothesis of Lorentz violation in the neutrino sector has intrigued
scientists for the last two to three decades. A number of theoretical arguments
support the emergence of such violations first and foremost for neutrinos,
which constitute the "most elusive" and "least interacting" particles known to
mankind. It is of obvious interest to place stringent bounds on the
Lorentz-violating parameters in the neutrino sector. In the past, the most
stringent bounds have been placed by calculating the probability of neutrino
decay into a lepton pair, a process made kinematically feasible by Lorentz
violation in the neutrino sector, above a certain threshold. However, even more
stringent bounds can be placed on the Lorentz-violating parameters if one takes
into account, additionally, the possibility of neutrino splitting, i.e., of
neutrino decay into a neutrino of lower energy, accompanied by "neutrino-pair
Cerenkov radiation". This process has negligible threshold and can be used to
improve the bounds on Lorentz-violating parameters in the neutrino sector.
Finally, we take the opportunity to discuss the relation of Lorentz and gauge
symmetry breaking, with a special emphasis on the theoretical models employed
in our calculations.
