Geodesic
Covariantization of quantized calculi over quantum groups
Mathematica Bohemica, Tome 145 (2020) no. 4, pp. 415-433
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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.
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.
DOI : 10.21136/MB.2019.0142-18
Classification : 58B32, 81Q30
Keywords: Hopf algebra; quantum group; covariant first order differential calculus; quantized calculus; Dirac operator 
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Akrami, Seyed Ebrahim; Farzi, Shervin. Covariantization of quantized calculi over quantum groups. Mathematica Bohemica, Tome 145 (2020) no. 4, pp. 415-433. doi: 10.21136/MB.2019.0142-18

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