On the ${\mathbb Z}_3$-Graded Structures

Abstract

After introducing some ${\mathbb Z}_3$-graded structures, we first give the definition of a ${\mathbb Z}_3$-graded quantum space and show that the algebra of functions on it, denoted by ${\cal O}(\widetilde{\mathbb C}_q^{1|1|1})$, has a ${\mathbb Z}_3$-graded Hopf algebra structure. Later, we obtain a new ${\mathbb Z}_3$-graded quantum group, denoted by $\widetilde{\rm GL}_q(1|1)$, and show that the algebra of functions on this group is a ${\mathbb Z}_3$-graded Hopf algebra. Finally, we construct two non-commutative differential calculi on the algebra ${\cal O}(\widetilde{\mathbb C}_q^{1|1})$ which are left covariant with respect to the ${\mathbb Z}_3$-graded Hopf algebra ${\cal O}(\widetilde{\rm GL}_q(1|1))$.