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Let $\mathfrak{g}$ be a basic classical simple Lie super algebra. I want to prove that every finite dimensional module over $\mathfrak{g}$ has a highest weight vector.

My feeling is, since $e_i$'s are rising operators it will kill a non-zero vector and this will give us a highest weight vector and may be we need to use Lie's theorem.

But I am unable to connect these things to get a perfect answer. If some one can tell me clearly what is happening here, that would help me a lot. Thank you.

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    $\begingroup$ There are books out there that handle this stuff. This is hardly a research level question. $\endgroup$
    – user130903
    Commented Nov 6, 2018 at 8:02

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This is contained in section 1.5.3 of the book "Dualities and representations of Lie superalgebras" by Cheng and Wang. Chapter 1 of their book happens to be available for free on the AMS bookstore website: https://bookstore.ams.org/gsm-144.

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