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.

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

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