Left covariant differential calculi on ${\widetilde{\mathrm{G}\mathrm{L}}}_q(2)$

In this work, we introduce the ${\mathbb{Z}}_3$-graded differential algebra, denoted by $\mathrm{\Omega }({\widetilde{\mathrm{G}\mathrm{L}}}_q(2))$, treated as the ${\mathbb Z}_3$-gradedquantumdeRhamcomplexof${\mathbb Z}_3$-gradedquantumgroup$\widetilde{\rm GL}_q(2)$. In this sense, we construct left-covariant differential calculi on the ${\mathbb{Z}}_3$-graded quantum group ${\widetilde{\mathrm{G}\mathrm{L}}}_q(2)$.