  The Letter considers the ground state and the Tkachenko modes for a rapidly
rotating Bose-Einstein condensate (BEC), when its macroscopic wave function is
a coherent superposition of states analogous to the lowest Landau levels of a
charge in a magnetic field. As well as in type II superconductors close to the
critical magnetic field $H_{c2}$, this corresponds to a periodic vortex
lattice. The exact value of the shear elastic modulus of the vortex lattice,
which was known from the old works on type II superconductors, essentially
exceeds the values calculated recently for BEC. This is important for
comparison with observation of the Tkachenko mode in the rapidly rotating BEC.
