Cluster realisations of $\mathrm{ı}$quantum groups of type AI

The $\mathrm{ı}$quantum group ${\mathrm{U}}_n^{\mathrm{ı}}$ of type ${\text{AI}}_n$ is a coideal subalgebra of the quantum group $U_q({\mathfrak{s}\mathfrak{l}}_{n+1})$, associated with the symmetric pair $({\mathfrak{s}\mathfrak{l}}_{n+1},{\mathfrak{s}\mathfrak{o}}_{n+1})$. In this paper, we give a cluster realisation of the algebra ${\mathrm{U}}_n^{\mathrm{ı}}$. Under such a realisation, we give cluster interpretations of some fundamental constructions of ${\mathrm{U}}_n^{\mathrm{ı}}$, including braid group symmetries, the coideal structure, and the action of a Coxeter element. Along the way, we study a (rescaled) integral form of ${\mathrm{U}}_n^{\mathrm{ı}}$, 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.