Crystallized Peter-Weyl type decomposition for level 0 part of modified quantum algebra widetilde U_q(widehat{sl₂})₀

We shall give some criteria for the existence of the Peter-Weyl type decomposition for modified quantum algebra, which is motivated by the fact that modified quantum algebra has commutative two crystal structures. We shall apply the criteria to the crystal base of the level 0 part of the modified quantum affine algebra $\widetilde U_q(\widehat{sl_2})_0$ and give more precise description of Peter-Weyl type decomposition.