$\imath$Hall algebra of the projective line and $q$-Onsager algebra
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866916276069203968 |
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| author | Lu, Ming Ruan, Shiquan Wang, Weiqiang |
| author_facet | Lu, Ming Ruan, Shiquan Wang, Weiqiang |
| contents | The $\imath$Hall algebra of the projective line is by definition the twisted semi-derived Ringel-Hall algebra of the category of $1$-periodic complexes of coherent sheaves on the projective line. This $\imath$Hall algebra is shown to realize the universal $q$-Onsager algebra (i.e., $\imath$quantum group of split affine $A_1$ type) in its Drinfeld type presentation. The $\imath$Hall algebra of the Kronecker quiver was known earlier to realize the same algebra in its Serre type presentation. We then establish a derived equivalence which induces an isomorphism of these two $\imath$Hall algebras, explaining the isomorphism of the $q$-Onsager algebra under the two presentations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2010_00646 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | $\imath$Hall algebra of the projective line and $q$-Onsager algebra Lu, Ming Ruan, Shiquan Wang, Weiqiang Representation Theory Quantum Algebra The $\imath$Hall algebra of the projective line is by definition the twisted semi-derived Ringel-Hall algebra of the category of $1$-periodic complexes of coherent sheaves on the projective line. This $\imath$Hall algebra is shown to realize the universal $q$-Onsager algebra (i.e., $\imath$quantum group of split affine $A_1$ type) in its Drinfeld type presentation. The $\imath$Hall algebra of the Kronecker quiver was known earlier to realize the same algebra in its Serre type presentation. We then establish a derived equivalence which induces an isomorphism of these two $\imath$Hall algebras, explaining the isomorphism of the $q$-Onsager algebra under the two presentations. |
| title | $\imath$Hall algebra of the projective line and $q$-Onsager algebra |
| topic | Representation Theory Quantum Algebra |
| url | https://arxiv.org/abs/2010.00646 |