Affine cellularity of quantum affine algebras

Cui (arXiv:1405.6441) has shown that the modified quantum affine algebra $\tilde{\mathbf{U}}$ (more precisely its quotients, BLN algebras) is affine cellular in the sense of Koenig and Xi. The proof is based on the structure of cells of $\tilde{\mathbf{U}}$, studied previously in [http://arxiv.org/abs/math/0212253], the author’s joint work with Beck. We here give more direct proof based on Lemma 6.17 in [http://arxiv.org/abs/math/0212253], together with a property of the bilinear form introduced in [http://arxiv.org/abs/math/0204183]. We also prove that cell ideals are idempotent, and hence Theorem 4.4 in [Koenig-Xi] is applicable.