Formulae of ı-divided powers in Uq(sl2)

The existence of the ı-canonical basis (also known as the ı-divided powers) for the coideal subalgebra of the quantum sl2 were established by Bao and Wang, with conjectural explicit formulae. In this paper we prove the conjectured formulae of these ı-divided powers. This is achieved by first establishing closed formulae of the ı-divided powers in basis for quantum sl2 and then formulae for the ı-canonical basis in terms of Lusztig's divided powers in each finite-dimensional simple module of quantum sl2. These formulae exhibit integrality and positivity properties.