# When can you build up all representations from the fundamental and antifundamental ones?

Under what conditions can you determine all representations of a Lie algebra from the fundamental and antifundamental ones using just the tensor product, direct sum and Clebsch-Gordan decomposition? I think this is true for $\mathfrak{su}(2)$ and $\mathfrak{sl}(2,\mathbb{C})$, or at least that's what physics books lead me to believe!

Is it true in general for semisimple Lie algebras?

## 1 Answer

Theorem: Let $G$ be a compact Lie group and let $V$ be a faithful (finite-dimensional, continuous, complex) representation of $G$. Then every (finite-dimensional, continuous, complex) irreducible representation of $G$ is a subrepresentation of a tensor product of copies of $V$ and $V^{\ast}$.

Proof. This follows from Stone-Weierstrass and the orthogonality relations for matrix coefficients. See Proof 3 in this blog post.

This gives the desired result for Lie algebras of simply-connected compact Lie groups (provided that "fundamental representation" implies "faithful representation of the corresponding simply-connected compact Lie group"; I am not actually sure what this term means).

• Yes - that's exactly what I mean; many thanks! I'll certainly have a good read of the blog post. It looks excellent! Do you have a recommendation as to the best reference to read more about this in? Cheers. – Edward Hughes Feb 21 '13 at 20:58
• Nope. I've mostly been learning my Lie theory in the wilderness. One mild warning: there is an obvious representation of $\mathfrak{so}(n)$ which is not a faithful representation of the corresponding simply-connected compact Lie group when $n \ge 3$ (namely $\text{Spin}(n)$), so the hypotheses of this result don't apply. Indeed, you can't get spin representations from it. – Qiaochu Yuan Feb 21 '13 at 21:01