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(∞) of a quantized enveloping algebra explicitly. This method can be applied to symmetrizable Kac-Moody types. We can also apply this method to the crystal bases B(λ) of integrable highest weight modules and of modified quantum algebras. But, the explicit forms of the polyhedral realizations of crystal bases B(∞) and B(λ) are only given in the case of arbitrary rank 2, of An and of An(1). So, we will give the polyhedral realizations of crystal bases B(∞) and B(λ) for all simple Lie algebras in this paper.