Hopf algebra; quantum group; covariant first order differential calculus; quantized calculus; Dirac operator
We introduce a method for construction of a covariant differential calculus over a Hopf algebra $A$ from a quantized calculus $da=[D,a]$, $a\in A$, where $D$ is a candidate for a Dirac operator for $A$. We recover the method of construction of a bicovariant differential calculus given by T. Brzeziński and S. Majid created from a central element of the dual Hopf algebra $A^\circ $. We apply this method to the Dirac operator for the quantum $\rm SL(2)$ given by S. Majid. We find that the differential calculus obtained by our method is the standard bicovariant 4D-calculus. We also apply this method to the Dirac operator for the quantum $\rm SL(2)$ given by P. N. Bibikov and P. P. Kulish and show that the resulted differential calculus is $8$-dimensional.