  We have found an exact analytical solution of the Bogoliubov-de Gennes
equations for the Tkachenko modes of the vortex lattice in the lowest Landau
level (LLL) in the thermodynamic limit at any momenta and calculated their
damping rates. At finite temperatures both Beliaev and Landau damping leads to
momentum independent damping rates in the low-energy limit, which shows that at
sufficiently low energies Tkachenko modes become strongly damped. We then found
that the mean square fluctuations of the density grow logarithmically at large
distances, which indicates that the state is ordered in the vortex lattice only
on a finite (although exponentially large) distance scale and introduces a
low-momentum cut-off. Using this circumstance we showed that at finite
temperatures the one-body density matrix undergoes an exponential decay at
large distances.
