On divided powers quantum supergroups

In the nineties Lusztig studied divided powers forms of quantum groups and the Hopf algebras obtained at a root of unity. His interest comes from representation theory of algebraic groups iin positive characteristic, as they also have a Frobenius map whose kernel is the so called small quantum group.They differ from that studied by de Concini, Kac and Procesi, where the small quantum grooup appears as a quotient. In this talk we will recall Lusztig's construction of these quantum groups, and a generalization to the context of quantum supergroups obatined in collaboration with C. Vay.