Cluster realisations of ıquantum$\imath {\rm quantum}$ groups of type AI

Abstract

AbstractThe group of type is a coideal subalgebra of the quantum group , associated with the symmetric pair . In this paper, we give a cluster realisation of the algebra . Under such a realisation, we give cluster interpretations of some fundamental constructions of , including braid group symmetries, the coideal structure and the action of a Coxeter element. Along the way, we study a (rescaled) integral form of , which is compatible with our cluster realisation. We show that this integral form is invariant under braid group symmetries, and construct the Poincare‐Birkhoff‐Witt (PBW)‐bases for the integral form.