Tame quivers and affine bases II: nonsimply-laced cases
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866929235101220864 |
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| author | Xiao, Jie Xu, Han |
| author_facet | Xiao, Jie Xu, Han |
| contents | 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 $\mathbf{B}'=\{C(\mathbf{c},t_λ)|(\mathbf{c},t_λ)\in\mathcal{G}^a\}$ 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 $\mathbf{B}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_03739 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Tame quivers and affine bases II: nonsimply-laced cases Xiao, Jie Xu, Han Representation Theory Quantum Algebra Rings and Algebras 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 $\mathbf{B}'=\{C(\mathbf{c},t_λ)|(\mathbf{c},t_λ)\in\mathcal{G}^a\}$ 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 $\mathbf{B}$. |
| title | Tame quivers and affine bases II: nonsimply-laced cases |
| topic | Representation Theory Quantum Algebra Rings and Algebras |
| url | https://arxiv.org/abs/2402.03739 |