$\imath$Hall algebras of weighted projective lines and quantum symmetric pairs
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866916088765218816 |
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| author | Lu, Ming Ruan, Shiquan |
| author_facet | Lu, Ming Ruan, Shiquan |
| contents | The $\imath$Hall algebra of a weighted projective line is defined to be the semi-derived Ringel-Hall algebra of the category of $1$-periodic complexes of coherent sheaves on the weighted projective line over a finite field. We show that this Hall algebra provides a realization of the $\imath$quantum loop algebra, which is a generalization of the $\imath$quantum group arising from the quantum symmetric pair of split affine type ADE in its Drinfeld type presentation. The $\imath$Hall algebra of the $\imath$quiver algebra of split affine type A was known earlier to realize the same algebra in its Serre presentation. We then establish a derived equivalence which induces an isomorphism of these two $\imath$Hall algebras, explaining the isomorphism of the $\imath$quantum group of split affine type A under the two presentations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2110_02575 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | $\imath$Hall algebras of weighted projective lines and quantum symmetric pairs Lu, Ming Ruan, Shiquan Quantum Algebra Combinatorics Representation Theory The $\imath$Hall algebra of a weighted projective line is defined to be the semi-derived Ringel-Hall algebra of the category of $1$-periodic complexes of coherent sheaves on the weighted projective line over a finite field. We show that this Hall algebra provides a realization of the $\imath$quantum loop algebra, which is a generalization of the $\imath$quantum group arising from the quantum symmetric pair of split affine type ADE in its Drinfeld type presentation. The $\imath$Hall algebra of the $\imath$quiver algebra of split affine type A was known earlier to realize the same algebra in its Serre presentation. We then establish a derived equivalence which induces an isomorphism of these two $\imath$Hall algebras, explaining the isomorphism of the $\imath$quantum group of split affine type A under the two presentations. |
| title | $\imath$Hall algebras of weighted projective lines and quantum symmetric pairs |
| topic | Quantum Algebra Combinatorics Representation Theory |
| url | https://arxiv.org/abs/2110.02575 |