Quantum torsors

We present a quantum analogue of the “parallelogram” axioms for torsors (J. ReineAngew. Math. 160 (1929) 199). A quantum torsor is an algebra equipped with a triple coproduct and with an algebra endomorphism playing the role of the square of the antipode in a Hopf algebra. When the ground ring is a field or a ring of formal series, we prove that any quantum torsor is equipped with a natural structure of bicomodule-algebra. In this framework, we reformulate the rules for the composition of torsors and we define an invariant group associated with any Hopf algebra. We then discuss the relation of our approach with the theory of Hopf–Galois extensions and with the theory of Hopf–Galois systems (a quantum analogue of the “groupoid” formulation of torsor axioms). Finally, we give a general categorical reformulation for faithfully flat Hopf–Galois extensions over a ring.