Polyhedral Realizations of Crystal Bases for Quantum Algebras of Finite Types

Polyhedral realization of crystal bases is one of the methods for describing the crystal base $B(\mathrm{\infty })$ explicitly. This method can be applied to symmetrizable Kac-Moody types. We can also apply this method to the crystal bases $B(\lambda )$ of integrable highest weight modules and of modified quantum algebras. But, the explicit forms of the polyhedral realizations of crystal bases $B(\mathrm{\infty })$ and $B(\lambda )$ are only given in the case of arbitrary rank 2, of $A_n$ and of $A_n^{(1)}$. So, we will give the polyhedral realizations of crystal bases $B(\mathrm{\infty })$ and $B(\lambda )$ for all simple Lie algebras in this paper.