  We review the polyhedral realizations of crystal bases in the former half and
in the latter half, we introduce braid-type isomorphisms for some rank 2 finite
type crystals. Using this isomorphisms, for semi-simple Lie algebra we can show
that polyhedral realizations can be obtained in the lattice of finite rank
which coincides with the length of the longest element in the associated Weyl
group.

Quantum Algebra;Combinatorics;Representation Theory