# Application of Weyl's Theorem

Recall that Weyl's theorem says that any finite dimensional representation of semi simple Lie algebra is completely reducible. I'm trying some examples to understand this theorem properly.

Since $\bf sl_2(\bf C)$ acts on $V=\bf C^2$ by matrix multiplication, $V$ is an $\bf sl_2$-module.

Decompose $V \otimes V$ as $\bf sl_2$-module.

Since I've just learned about Weyl's theorem so please don't use any high machinery. Any idea or reference is appreciated.

• It decomposes as the symmetric square and the exterior square. The exterior square is trivial and the symmetric square is irreducible. – Qiaochu Yuan Feb 20 '16 at 17:35