# Representations of SL(2) as an algebraic group

I understand the finite-dimensional representations of $\text{SL}(2,\mathbb C)$ as a Lie group and their correspondence with Lie algebra representations of $\mathfrak{sl}(2,\mathbb C)$.

Does anyone know a good reference that explains the analogous theory for the finite dimensional representations of $\text{SL}(2)$ as an algebraic group (over an arbitrary field)? Under what conditions does a similar correspondence hold with the Lie algebra representations?

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For char p the story is much more subtle, even for SL(2). The Frobenius map now gives you unexpected morphisms between the symmetric powers of the standard rep, sending $Sym^k$ to $Sym^{pk}$. The images of these maps in general don't have invariant complements, so the representations are not semisimple, and the situation becomes much more complicated and interesting. Jantzen has much more information about this.