  Recent experiments with ultracold Rydberg-excited atoms have shown that
long-range interactions can give rise to spatially ordered structures.
Observation of crystalline phases in a system with Rydberg atoms loaded into an
optical lattice seems also within reach. Here we investigate a bosonic model on
a triangular lattice suitable for description of such experiments. Numerical
simulations based on bosonic dynamical mean-field theory reveal a rich phase
diagram with different supersolid phases. Comparison with the results obtained
for a square lattice geometry shows qualitatively similar results in a wide
range of parameters, however, on a triangular lattice we do not observe the
checkerboard supersolid. Moreover, unlike on a square lattice we did not find a
phase transition from uniform superfluid to supersolid induced by increase of
the hopping amplitude on a triangular lattice. Based on our results we propose
an intuitive interpretation of the nature of different supersolid phases. We
also propose parameters for the experimental realization.
