  Quantum-gravity (QG) effects might generate Lorentz invariance violation by
the interaction of energetic particles with the foamy structure of the
space-time. As a consequence, particles may not travel at the universal speed
of light. We propose to constrain Lorentz invariance violation for energetic
neutrinos exploiting the $\nu_e$ neutronization burst from the next galactic
supernova (SN). This prompt signal is expected to produce a sharp peak in the
SN $\nu_e$ light curve with a duration of $\sim 25$ ms. However presence of
energy-dependent Lorentz invariance violation would significantly spread out
the time structure of this signal. We find that the detection the SN $\nu_e$
burst from a typical galactic explosion at $d=10$ kpc in a Mton-class water
Cerenkov detector, would be sensitive to a quantum-gravity mass scale $M_{\rm
QG} \sim 10^{12}$ GeV ($2 \times10^{5}$ GeV) for the linear (quadratic) energy
dependence of Lorentz invariance violation. These limits are rather independent
of the neutrino mass hierarchy and whether the neutrino velocity is super or
subluminal.
