Cluster realisations of $\imath$quantum groups of type AI

The $\imath$quantum group ${\mathrm{U}^\imath_{n}}$ of type $\textrm{AI}_n$ is a coideal subalgebra of the quantum group $U_q(\mathfrak{sl}_{n+1})$, associated with the symmetric pair $(\mathfrak{sl}_{n+1},\mathfrak{so}_{n+1})$. In this paper, we give a cluster realisation of the algebra ${\mathrm{U}^\imath_{n}}$. Under such a realisation, we give cluster interpretations of some fundamental constructions of ${\mathrm{U}^\imath_{n}}$, including braid group symmetries, the coideal structure, and the action of a Coxeter element. Along the way, we study a (rescaled) integral form of ${\mathrm{U}^\imath_{n}}$, which is compatible with our cluster realisation. We show that this integral form is invariant under braid group symmetries, and construct PBW-bases for the integral form.