Mathematica Bohemica, Vol. 145, No. 4, pp. 415-433, 2020
Covariantization of quantized calculi over quantum groups
Seyed Ebrahim Akrami, Shervin Farzi
Received October 29, 2018. Published online December 9, 2019.
Abstract: 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.}
Keywords: Hopf algebra; quantum group; covariant first order differential calculus; quantized calculus; Dirac operator
