Polyhedral Realizations of Crystal Bases for Modified Quantum Algebras   of Arbitrary Rank 2 Cases
  We describe the crystal bases of the modified quantum algebras and give the explicit form of the highest (or lowest) weight vector of its connected component $B_0(\lambda)$ containing the unit element for arbitrary rank 2 cases. We also present the explicit form of $B_0(\lambda)$ containing the highest (or lowest) weight vector by the polyhedral realization method. 
Ayumu Hoshino