Serre–Lusztig Relations for $$\imath $$Quantum Groups

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