Cluster realisations of $ı$quantum groups of type AI

Abstract

The $ı$quantum group ${\mathrm{U}}_n^{ı}$ of type ${\text{AI}}_n$ 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 ${\mathrm{U}}_n^{ı}$. Under such a realisation, we give cluster interpretations of some fundamental constructions of ${\mathrm{U}}_n^{ı}$, including braid group symmetries, the coideal structure, and the action of a Coxeter element. Along the way, we study a (rescaled) integral form of ${\mathrm{U}}_n^{ı}$, 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.

40 pages, 9 figures