# A particular representation of $SU(2)$ on $\mathbb R^3$

Studying physics I encounter group theory and it has told me that: the matrices that rotates $$\mathbb R^3$$ vectors in the Euclidean space are the representation of $$SU(2)$$. Namely, $$SO(3)$$ matrices are the representation of $$SU(2)$$ on $$\mathbb R^3$$.

After they told me this I searched for the definition on representation and I thought that the map to the identity should also be a representation of $$SU(2)$$ on $$\mathbb R^3$$, namely:

if $$g \in SU(2)$$ and $$f$$ is a map such that $$f(g)=\begin {bmatrix} 1&0&0\\0&1&0\\0&0&1\end {bmatrix}$$ then $$f$$ is a representation of $$SU(2)$$ on $$\mathbb R^3$$.

If this is true it means that there are many representation of $$SU(2)$$ on $$\mathbb R^3$$ and one of these gives us the matrices of $$SO(3)$$. Then what is special in the representation that gives us the matrices of $$SO(3)$$?

I've tried to answer this question by myself but the only thing I came up with is that the representation that gives the $$SO(3)$$ group has the same Lie algebra of $$SU(2)$$. However I'm not able to check if the representation of $$SU(2)$$ on $$\mathbb R^3$$ that gives the $$SO(3)$$ group is the only one with this Lie algebra.

• Would Mathematics be a better home for this question? – Qmechanic May 5 at 9:31
• I don't think because in my opinion there are many possibles properties that makes this representation mathematically special, but the property that makes it special for us is a property that is important for physical reasons – SimoBartz May 5 at 9:36
• I guess it is related to the uniqueness of this representation? – ChoMedit May 5 at 11:54

OP explicitly considers the trivial representation $$SU(2)\to GL(3,\mathbb{R})$$. However, modulo equivalence of representations, the sought-for irreducible representation $$\rho:SU(2)\to GL(3,\mathbb{R})$$ is undoubtedly $$\rho=R\circ\pi$$, where $$\pi:SU(2)\to SO(3)\cong SU(2)/ \mathbb{Z}_2$$ is the covering homomorphism, and $$R:SO(3)\to GL(3,\mathbb{R})$$ is the defining representation for $$SO(3)$$, aka. the vector representation.