$\imath$Hall algebras of weighted projective lines and quantum symmetric pairs III: quasi-split type
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| Format: | Preprint |
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2024
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| _version_ | 1866913581803503616 |
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| author | Lu, Ming Ruan, Shiquan |
| author_facet | Lu, Ming Ruan, Shiquan |
| contents | From a category $\mathcal{A}$ with an involution $\varrho$, we introduce $\varrho$-complexes, which are a generalization of (bounded) complexes, periodic complexes and modules of $\imath$quiver algebras. The homological properties of the category $\mathcal{C}_\varrho(\mathcal{A})$ of $\varrho$-complexes are given to make the machinery of semi-derived Ringel-Hall algebras applicable. The $\imath$Hall algebra of the weighted projective line $\mathbb{X}$ is the twisted semi-derived Ringel-Hall algebra of $\mathcal{C}_\varrho({\rm coh}(\mathbb{X}))$, where $\varrho$ is an involution of ${\rm coh}(\mathbb{X})$. This $\imath$Hall algebra is used to realize the quasi-split $\imath$quantum loop algebra, which is a generalization of the $\imath$quantum group arising from the quantum symmetric pair of quasi-split affine type ADE in its Drinfeld type presentation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_13078 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $\imath$Hall algebras of weighted projective lines and quantum symmetric pairs III: quasi-split type Lu, Ming Ruan, Shiquan Quantum Algebra Representation Theory From a category $\mathcal{A}$ with an involution $\varrho$, we introduce $\varrho$-complexes, which are a generalization of (bounded) complexes, periodic complexes and modules of $\imath$quiver algebras. The homological properties of the category $\mathcal{C}_\varrho(\mathcal{A})$ of $\varrho$-complexes are given to make the machinery of semi-derived Ringel-Hall algebras applicable. The $\imath$Hall algebra of the weighted projective line $\mathbb{X}$ is the twisted semi-derived Ringel-Hall algebra of $\mathcal{C}_\varrho({\rm coh}(\mathbb{X}))$, where $\varrho$ is an involution of ${\rm coh}(\mathbb{X})$. This $\imath$Hall algebra is used to realize the quasi-split $\imath$quantum loop algebra, which is a generalization of the $\imath$quantum group arising from the quantum symmetric pair of quasi-split affine type ADE in its Drinfeld type presentation. |
| title | $\imath$Hall algebras of weighted projective lines and quantum symmetric pairs III: quasi-split type |
| topic | Quantum Algebra Representation Theory |
| url | https://arxiv.org/abs/2411.13078 |