Braid group action and quasi-split affine $\imath$quantum groups I
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| Main Authors: | , , |
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| Format: | Preprint |
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2022
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| _version_ | 1866911247626141696 |
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| author | Lu, Ming Wang, Weiqiang Zhang, Weinan |
| author_facet | Lu, Ming Wang, Weiqiang Zhang, Weinan |
| contents | This is the first of two papers on quasi-split affine quantum symmetric pairs $\big(\widetilde{\mathbf U}(\widehat{\mathfrak g}), \widetilde{\mathbf U}^\imath \big)$, focusing on the real rank one case, i.e., $\mathfrak{g}= \mathfrak{sl}_3$ equipped with a diagram involution. We construct explicitly a relative braid group action of type $A_2^{(2)}$ on the affine $\imath$quantum group $\widetilde{\mathbf U}^\imath$. Real and imaginary root vectors for $\widetilde{\mathbf U}^\imath$ are constructed, and a Drinfeld type presentation of $\widetilde{\mathbf U}^\imath$ is then established. This provides a new basic ingredient for the Drinfeld type presentation of higher rank quasi-split affine $\imath$quantum groups in the sequel. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2203_11286 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Braid group action and quasi-split affine $\imath$quantum groups I Lu, Ming Wang, Weiqiang Zhang, Weinan Representation Theory Quantum Algebra 17B37 This is the first of two papers on quasi-split affine quantum symmetric pairs $\big(\widetilde{\mathbf U}(\widehat{\mathfrak g}), \widetilde{\mathbf U}^\imath \big)$, focusing on the real rank one case, i.e., $\mathfrak{g}= \mathfrak{sl}_3$ equipped with a diagram involution. We construct explicitly a relative braid group action of type $A_2^{(2)}$ on the affine $\imath$quantum group $\widetilde{\mathbf U}^\imath$. Real and imaginary root vectors for $\widetilde{\mathbf U}^\imath$ are constructed, and a Drinfeld type presentation of $\widetilde{\mathbf U}^\imath$ is then established. This provides a new basic ingredient for the Drinfeld type presentation of higher rank quasi-split affine $\imath$quantum groups in the sequel. |
| title | Braid group action and quasi-split affine $\imath$quantum groups I |
| topic | Representation Theory Quantum Algebra 17B37 |
| url | https://arxiv.org/abs/2203.11286 |