First Order Differential Calculus on the Jordanian Twisted \(\star \)-Hopf Supersymmetry Algebra

A Jordanian deformation of the \(N=2\) supersymmetry algebra and its first-order noncommutative differential calculus is obtained from the q-deformed Hopf supersymmetry algebra via a singular limit of a linear transformation. We show that the Jordanian \(N=2\) supersymmetry algebra carries a Hopf superalgebra structure. It is enhanced to a twisted Hopf star superalgebra by four inequivalent families of \(\star \)-structures. It is demonstrated that these star operations induce four types of \(\star \)-involutions on the differential one-forms and partial derivatives. We introduce an appropriate Jordanian super-Hopf algebra that includes two even and two odd generators equipped with four different types of \(\star \)-structures and its corresponding supergroup on the Hopf \(N=2\) supersymmetry algebra. It is shown that the noncommutative differential calculus over the Jordanian \(N=2\) supersymmetry algebra is left-covariant with respect to the Jordanian supergroup. The \(\star \)-structures of the supergroup are \(\star \)-preserving on the supersymmetry algebra only for zero values of their parameters.