Are these 2 definitions of $K$ and $H$ on $U(sl_q(2))$ coherent?

Question

I'm studing $U(sl_q(2))$ and studying how to recover $U(sl(2))$ from $U(sl_q(2))$ I found these two definition for both $H$ and $K$ as formal generators. $$H=\frac{K-K^{-1}}{q-q^{-1}}$$ $$K=q^{H}$$ I'd like to know if they are coherent to each other and how could I derive the first one from the second one if needed? Thank you in advance

Answer 1

I think it's better to think of it as $K = q^h$ where $h$ is the usual generator of $sl_2$. If $V$ is a finite dimensional representation of $sl_2$ then $h$ acts with integer eigenvalues. If $h$ acts on a vector with eigenvalue $n$ then the operator $H = \frac{q^h - q^{-h}}{q-q^{-1}}$ acts on it by the "q-integer" $[n]_q = \frac{q^n - q^{-n}}{q-q^{-1}} = q^n + q^{n-2} + \dots + q^{-n}$. Then it's clear that if we specialize $q$ to $1$ then $H$ just acts by the same thing as $h$.