From supersymmetry to quantum commutativity

Let (H, R) be a quasitriangular Hopf algebra acting on an algebra A. We study a concept of A being quantum commutative with respect to (H, R). Superalgebras which are graded commutative (called sometimes commutative superalgebras) are shown to be examples of such an action. There is an analogous notion of quantum commutativity for comodule algebra. The quantum plane C_q[x, y] is an example, both under the coaction of quantum 2 × 2 matrices, and also in a more novel way at q a root of unity. If H is a cocommutative finite dimensional Hopf algebra and (D(H), R) its Drinfeld double we show that H is quantum commutative with respect to (D(H), R). We discuss further examples of such actions and coactions, and show that the category _{A # H}Mod resembles (for such actions) the category of modules over commutative rings.