Formulae of $\imath$-divided powers in ${\mathbf U}_q(\mathfrak{sl}_2)$

Abstract: The existence of the $\imath$-canonical basis (also known as the $\imath$-divided powers) for the coideal subalgebra of the quantum $\mathfrak{sl}_2$ were established by Bao and Wang, with conjectural explicit formulae. In this paper we prove the conjectured formulae of these $\imath$-divided powers. This is achieved by first establishing closed formulae of the $\imath$-divided powers in basis for quantum $\mathfrak{sl}_2$ and then formulae for the $\imath$-canonical basis in terms of Lusztig's divided powers in each finite-dimensional simple module of quantum $\mathfrak{sl}_2$. These formulae exhibit integrality and positivity properties.