Bicovariant differential calculi for finite global quotients

Let (M,G) be a finite global quotient, that is, a finite set M with an action by a finite group G. In this note, we classify all bicovariant first order differential calculi (FODCs) over the weak Hopf algebra k(G⋉M) ≃ k[G⋉M ], where G⋉M is the action groupoid associated to (M,G), and k[G ⋉M ] is the groupoid algebra of G ⋉ M . Specifically, we prove a necessary and sufficient condition for a FODC over k(G ⋉ M) to be bicovariant and then show that the isomorphism classes of bicovariant FODCs over k(G⋉M) are in one-to-one correspondence with subsets of a certain quotient space.