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.