Differential calculi on the Z3-graded Hopf algebra F((C)over-tilde1|1|1q)

Abstract

Considering a Z(3)-graded 3-dimensional space, we develop Z(3)-graded differential calculi on the algebra of functions defined on this space, which is a Z(3)-graded Hopf algebra. We demonstrate that these differential calculi admit a *-operation. Additionally, we discover a new Z(3)-graded upper triangular quantum group and explicitly define its Z(3)-graded Hopf algebraic structure. We also show in detail that this quantum group is the symmetry group of one of these calculi.