Affine cellularity of quantum affine algebras

TL;DR

This paper provides a more direct proof that certain quantum affine algebras are affine cellular, establishing their structural properties and confirming the idempotency of cell ideals.

Contribution

It offers a new, more straightforward proof of affine cellularity for quantum affine algebras and demonstrates the idempotency of their cell ideals.

Findings

Quantum affine algebras are affine cellular.

Cell ideals in these algebras are idempotent.

The proof utilizes properties of a bilinear form and cell structures.

Abstract

Cui (arXiv:1405.6441) has shown that the modified quantum affine algebra $\widetilde{\mathbf U}$ (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 $\widetilde{\mathbf U}$, 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.