I originally posted this on Maths SE, but then I thought it MO might be more fitting.

Let $k$ be a characteristic $0$ field and let $G$ be a linear algebraic group scheme over $k$. Then is it true that $G$ is reductive if and only if the category of representations of $G$, $\mathsf{Rep}(G)$ is semi-simple ?

P/s: $\mathsf{Rep}(\mathcal{G})$ is defined as the functor category $[\mathcal{G}, \mathcal{C}]$, where $\mathcal{G}$ is a groupoid and $\mathcal{C}$ is some category.

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    $\begingroup$ Theorem 22.42 (jmilne.org/math/Books/iag.html) $\endgroup$ – Eoin Dec 14 '19 at 22:44
  • $\begingroup$ @Eoin Would you mind expanding on the relevance of this theorem ? My background in representation theory isn't particularly strong. Thank you. $\endgroup$ – Dat Minh Ha Dec 14 '19 at 23:15
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    $\begingroup$ Compare with 9.54. It answers your question in the affirmative. $\endgroup$ – Eoin Dec 15 '19 at 0:42
  • $\begingroup$ @eoin Thank you. $\endgroup$ – Dat Minh Ha Dec 15 '19 at 0:55

The answer, as Eoin pointed out, is yes. For those who use the pdf version of Milne's book Algebraic groups available on his website, the theorem to look for is 22.138 (apparently there were enumeration changes during the publication process) on p. 424.

screenshot of page 424

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    $\begingroup$ It's good to have this as an answer, but it's probably polite first to ask @Eoin to post it rather than doing so immediately yourself. $\endgroup$ – LSpice Dec 15 '19 at 12:38

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