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Let $(bf U, bf U^imath)$ be a quantum symmetric pair of Kac-Moody type. The $imath$quantum groups $bf U^imath$ and the universal $imath$quantum groups $widetilde{bf U}^imath$ can be viewed as a generalization of quantum groups and Drinfeld doubles $widetilde{bf U}$. In this paper we formulate and establish Serre-Lusztig relations for $imath$quantum groups in terms of $imath$divided powers, which are an $imath$-analog of Lusztigs higher order Serre relations for quantum groups. This has applications to braid group symmetries on $imath$quantum groups.
The $imath$Serre relations and the corresponding Serre-Lusztig relations are formulated for arbitrary $imath$quantum groups arising from quantum symmetric pairs of Kac-Moody type. This generalizes the main results in [CLW18, CLW20].
let $widetilde{bf U}^imath$ be a quasi-split universal $imath$quantum group associated to a quantum symmetric pair $(widetilde{bf U}, widetilde{bf U}^imath)$ of Kac-Moody type with a diagram involution $tau$. We establish the Serre-Lusztig relations
Let $(bf U, bf U^imath)$ be a quasi-split quantum symmetric pair of arbitrary Kac-Moody type, where quasi-split means the corresponding Satake diagram contains no black node. We give a presentation of the $imath$quantum group $bf U^imath$ with explic
Expanding the classic works of Kazhdan-Lusztig and Deodhar, we establish bar involutions and canonical (i.e., quasi-parabolic KL) bases on quasi-permutation modules over the type B Hecke algebra, where the bases are parameterized by cosets of (possib
$imath$quantum groups are generalizations of quantum groups which appear as coideal subalgebras of quantum groups in the theory of quantum symmetric pairs. In this paper, we define the notion of classical weight modules over an $imath$quantum group,