Polyhedral Realizations of Crystal Bases for Modified Quantum Algebras   of Type A
  We describe the crystal bases of modified quantum algebras and its connected component containing ``zero vector''by the polyhedral realization method for the types A_n and A^(1)_1. We also present the explicit form of the unique highest weight vector in the connected component. 
Ayumu Hoshino and Toshiki Nakashima