Reference Request: $H^1(\mathfrak g, V)=0$ for semisimple Lie algebra $\mathfrak g$ and $\mathfrak g$-module $V$

Question

I read the following theorem in the lecture note of Victor Kac. Let $\mathfrak g$ be a finite-dimensional semisimple Lie algebra over an algebraically closed field of characteristic zero.

Theorem(Vanishing Theorem) If $V$ is a finite-dimensional $\mathfrak g$-module, then $H^1(\mathfrak g, V)=0$

I would like to request a reference for this theorem. In particular, I seek an introductory textbook which covers this theorem.

Answer 1

A different reference is:

Hilton Stammbach "A course in Homological Algebra", Chap VII, Proposition 5.6 and 6.1.

Moreover, I have to mention that all the Chapter VII is an introduction to cohomology of Lie Algebras and that section 5 and 6 analyze the special case of semisimple Lie algebras.

Answer 2

This is theorem 3.3 from A. L. Onischik and E. B. Vinberg's Lie Groups and Lie Algebras III:

Theorem: The following properties of a finite-dimensional Lie algebra $\mathfrak g$ are equivalent:

1. $\mathfrak g$ is semisimple;
2. $H^1(\mathfrak g,V)=\{0\}$ for any finite-dimensional $\mathfrak g$-module $V$;
3. any finite-dimensional linear representation of $\mathfrak g$ is completely reducible.

Answer 3

This theorem appears as Exercise $7.8.4$ in Weibel's An Introduction to Homological Algebra in the chapter on Lie algebra homology and cohomology. It is written as follows

If $\mathfrak{g}$ is a finite-dimensional Lie algebra over a field of characteristic $0$, show that $\mathfrak{g}$ is semisimple iff $H^1(\mathfrak{g}, M) = 0$ for all finite-dimensional $\mathfrak{g}$-modules $M$.

So it doesn't explicitly cover the theorem as you request, but the chapter does provides enough information for the reader to construct the proof themselves, otherwise it wouldn't appear as an exercise!