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

Let $$(\mathbf{U}, \mathbf{U}^\imath )$$ be a quantum symmetric pair of Kac–Moody type. The $$\imath $$ quantum groups $$\mathbf{U}^\imath $$ and the universal $$\imath $$ quantum groups $$\widetilde{\mathbf{U}}^\imath $$ can be viewed as a generalization of quantum groups and Drinfeld doubles $$\widetilde{\mathbf{U}}$$ . 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 Lusztig’s higher order Serre relations for quantum groups. This has applications to braid group symmetries on $$\imath $$ quantum groups.