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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2412.00841 |
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| _version_ | 1866912139404378112 |
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| author | Li, Yiyu Peng, Liangang |
| author_facet | Li, Yiyu Peng, Liangang |
| contents | Let $\mathcal{A}$ be an arbitrary hereditary abelian category. Lu and Peng defined the semi-derived Ringel-Hall algebra $SH(\mathcal{A})$ of $\mathcal{A}$ and proved that $SH(\mathcal{A})$ has a natural basis and is isomorphic to the Drinfeld double Ringel-Hall algebra of $\mathcal{A}$. In this paper, we introduce a coproduct formula on $SH(\mathcal{A})$ with respect to the basis of $SH(\mathcal{A})$ and prove that this coproduct is compatible with the product of $SH(\mathcal{A})$, thereby the semi-derived Ringel-Hall algebra of $\mathcal{A}$ is endowed with a bialgebra structure which is identified with the bialgebra structure of the Drinfeld double Ringel-Hall algebra of $\mathcal{A}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_00841 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Semi-derived Ringel-Hall bialgebras Li, Yiyu Peng, Liangang Representation Theory Quantum Algebra 18E30, 16E60, 17B37 Let $\mathcal{A}$ be an arbitrary hereditary abelian category. Lu and Peng defined the semi-derived Ringel-Hall algebra $SH(\mathcal{A})$ of $\mathcal{A}$ and proved that $SH(\mathcal{A})$ has a natural basis and is isomorphic to the Drinfeld double Ringel-Hall algebra of $\mathcal{A}$. In this paper, we introduce a coproduct formula on $SH(\mathcal{A})$ with respect to the basis of $SH(\mathcal{A})$ and prove that this coproduct is compatible with the product of $SH(\mathcal{A})$, thereby the semi-derived Ringel-Hall algebra of $\mathcal{A}$ is endowed with a bialgebra structure which is identified with the bialgebra structure of the Drinfeld double Ringel-Hall algebra of $\mathcal{A}$. |
| title | Semi-derived Ringel-Hall bialgebras |
| topic | Representation Theory Quantum Algebra 18E30, 16E60, 17B37 |
| url | https://arxiv.org/abs/2412.00841 |