Half Potential on Geometric Crystals and Connectedness of Cellular Crystals

Abstract

For any simple complex algebraic group, we define upper/lower half-decorated geometric crystals and show that their tropicalization will be upper/lower normal Kashiwara's crystals. In particular, we show that the tropicalization of the half-decorated geometric crystal on the big Bruhat cell(=$B_{w_0}^{-}:=B^{-}\cap U{\bar{w}}_{0}U$) is isomorphic to the crystal $B(\infty )$ of the nilpotent subalgebra of quantum group $U_q^{-}(\mathfrak{g})$. As an application, we shall show that any cellular crystal associated with a reduced word is connected in the sense of a crystal graph.

The gap in the proof has been filled. Some theorem was added