How to calculate $X^*(SL_2) = \operatorname{Hom}(SL_2,\mathbb{G_m})$ and $X^*(PSL_2) = \operatorname{Hom}(PSL_2,\mathbb{G_m})$ ? ($SL_2$ and $PSL_2$ are viewed as algebraic groups over a field $K$)
For $SL_2$, I tried to do it with Hopf algebra, which leads to calculate $\operatorname{Hom}(K[X,X^{-1}],K[A,B,C,D]/(AD-BC-1))$, but I have difficulty calculating the group of invertible elements of $K[A,B,C,D]/(AD-BC-1)$.