Tame quivers and affine bases II: nonsimply-laced cases

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 ${\mathbf{B}}^{\prime }=\{C(\mathbf{c},t_{\lambda })|(\mathbf{c},t_{\lambda })\in {\mathcal{G}}^a\}$ in the generic composition algebra ${\mathcal{C}}^{\ast }$ and prove that ${\mathcal{B}}^{\prime }={\mathbf{B}}^{\prime }\bigsqcup (-{\mathbf{B}}^{\prime })$ coincides with Lusztig's signed canonical basis $\mathcal{B}$. Moreover, in type ${\tilde{B}}_n,{\tilde{C}}_n$, ${\mathbf{B}}^{\prime }$ is the canonical basis $\mathbf{B}$.

33 pages