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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 $K[A,B,C,D]/(AD-BC-1)$.

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    "The algebraic group $PSL_2$" doesn't actually exist. See http://mathoverflow.net/questions/16145/what-is-the-difference-between-psl-2-and-pgl-2.2012-01-27

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$SL_2$ is a simple algebraic group, and so admits no non-trivial characters.

$PSL_2$ is a quotient of $SL_2$, and so any character of $PSL_2$ would also be a character of $SL_2$; thus $PSL_2$ also admits no non-trivial characters.