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arXiv · 2402.03739

Tame quivers and affine bases II: nonsimply-laced cases

Jie Xiao·Han Xu
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Abstract

In [Tame_quivers_and_affine_bases_I], we give a Ringel-Hall algebra approach to the canonical bases in the symmetric affine cases. In this paper, we extend the results to general symmetrizable affine cases by using Ringel-Hall algebras of representations of a valued quiver. We obtain a bar-invariant basis $\mathbf{B}'=\{C(\mathbf{c},t_\lambda)|(\mathbf{c},t_\lambda)\in\mathcal{G}^a\}$ in the generic composition algebra $\mathcal{C}^*$ and prove that $\mathcal{B}'=\mathbf{B}'\sqcup(-\mathbf{B}')$ coincides with Lusztig's signed canonical basis $\mathcal{B}$. Moreover, in type $\tilde{B}_n,\tilde{C}_n$, $\mathbf{B}'$ is the canonical basis $\mathbf{B}$.

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Jie Xiao, Han Xu. 2024-02-06. Tame quivers and affine bases II: nonsimply-laced cases. https://arxiv.org/abs/2402.03739

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