Serre-Lusztig relations for $\mathrm{ı}$quantum groups

Let $(\mathbf{U},{\mathbf{U}}^{\mathbf{ı}})$ be a quantum symmetric pair of Kac-Moody type. The $\mathrm{ı}$quantum groups ${\mathbf{U}}^{\mathbf{ı}}$ and the universal $\mathrm{ı}$quantum groups ${\tilde{\mathbf{U}}}^{\mathrm{ı}}$ can be viewed as a generalization of quantum groups and Drinfeld doubles $\tilde{\mathbf{U}}$. In this paper we formulate and establish Serre-Lusztig relations for $\mathrm{ı}$quantum groups in terms of $\mathrm{ı}$divided powers, which are an $\mathrm{ı}$-analog of Lusztig’s higher order Serre relations for quantum groups. This has applications to braid group symmetries on $\mathrm{ı}$quantum groups.