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Let $V$ be a finite dimensional $k$-vector space which is a simple $\mathfrak{sl}_2$-module. Here $k$ is a field of characteristic $0$ and we let $\bar{k}$ denote its algebraic closure.

I was wondering is $V \otimes \bar{k}$ still a simple $\mathfrak{sl}_2$-module? Any comments would be appreciated. Thank you.

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