A proper generating functional on a Podle\'{s} sphere

We construct a proper generating functional $L$ on a Podle\'{s} sphere and we show that $1$-cocycle arising from $L$ coincides with the one in our previous work. We also show that our 1-cocycle is purely non Gaussian and that the full `group' $C^{\ast}$-algebra of the quantum $SL(2,\mathbb{R})$ is liminal.