In Majid's book (A Quantum Groups Primer) (pg11), the antipode for the Hopf algebra $SL_q(2)$ is defined as $Sd=a$, $Sa=d$, $Sb=-qb$, $Sc=-q^{-1}c$.
However, in Kassel's book (Quantum Groups) (pg 84), there is an additional factor of $\det_q^{-1}$, i.e. $S(a)=\det_q^{-1} d$,
$S(b)=\det_q^{-1} (-qb)$,
$S(c)=\det_q^{-1} (-q^{-1}c)$,
$S(d)=\det_q^{-1} a$
($\det_q^{-1}=ad-q^{-1}bc$)
May I ask, how do we reconcile these two different presentations of the antipode?
Sincere thanks for any help!