Polyhedral Realizations of Crystal Bases for Quantum Algebras of Finite Types

Abstract

Polyhedral realization of crystal bases is one of the methods for describing the crystal base $B(\infty)$ 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 $B(\infty)$ and $B(\lambda)$ are only given in the case of arbitrary rank 2, of $A_n$ and of $A^{(1)}_n$. So, we will give the polyhedral realizations of crystal bases $B(\infty)$ and $B(\lambda)$ for all simple Lie algebras in this paper.