Actions of compact and discrete quantum groups on operator systems

We introduce the notion of an action of a discrete or compact quantum group on an operator system, and study equivariant operator system injectivity. We then prove a duality result that relates equivariant injectivity with dual injectivity on associated crossed products. As an application, we give a description of the equivariant injective envelope of the reduced crossed product built from an action of a discrete quantum group on an operator system.