Braid group action and quasi-split affine \(\imath\)quantum groups. I. (English) Zbl 07772658
Summary: This is the first of our 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 sequels.
MSC:
| 17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
Keywords:
affine \(\imath\)quantum groups; braid group action; quantum symmetric pairs; Drinfeld presentationReferences:
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