Level 1 Perfect Crystals and Path Realizations of Basic Representations at q=0

Abstract

We present a uniform construction of level 1 perfect crystals $\mathcal{B}$ for all affine Lie algebras. We also introduce the notion of a crystal algebra and give an explicit description of its multiplication. This allows us to determine the energy function on $\mathcal{B}\otimes \mathcal{B}$ completely and thereby give a path realization of the basic representations at $q=0$ in the homogeneous picture.