Invariants for the $SU(2)$ representation The quantities $\delta_{ij}a_ib_j$ and $\epsilon_{ijk}a_ib_jc_k$ are invariant under the transformation of the $j=1$ (fundamental) representation of $SO(3)$. What would be the analogous expressions for the $j=1/2$ $SU(2)$ representation?
Attempt: The canonical definition of these groups are $$SO(3) = \left\{O \in GL(3, \mathbb{R}) : O^TO = 1, \det O = 1 \right\}\,\,\,\,\\SU(2) = \left\{U \in GL(2,\mathbb{C}) : U^{\dagger}U = 1, \det U = 1\right\}$$
Since the invariant in the rep $j=1$ in $SO(3)$, $\epsilon_{ijk}a_ib_jc_k$, only makes sense for $3D$ vectors, I am not really sure how to generalize this to $2D$. Can I say the invariant is the same in both cases, with the realization that I have simply a zero entry for the third component. (i.e just write $(a, b)$ in $2D$ is equivalent to $(a,b,0)$  in $3D$. 
Thanks.
 A: The crucial difference between $\mathrm{SO(}N\mathrm{)}$ and $\mathrm{SU(}N\mathrm{)}$ is that for $\mathrm{SU(}N\mathrm{)}$ one can raise and lower indices, while for $\mathrm{SO(}N\mathrm{)}$ this is irrelevant. See also these notes for more information.
For $\mathrm{SU(}N\mathrm{)}$ we have:
$$
U^\dagger U = 1
$$
which in component form is:
$$
U^j{}_i U_j{}^k = \delta^k{}_i \;\;\; \text{and} \;\;\; U_i{}^j U^k{}_j = \delta^k{}_i
$$
Thus:
$$
\varphi_i \varphi^i \to \varphi'_i \varphi'^i = U_i{}^j U^i{}_k \varphi_j \varphi^k = \delta^j{}_k \varphi_j \varphi^k=\varphi_i \varphi^i 
$$
and:
$$
\delta^{i}{}_{j} \to \delta'^{i}{}_{j} = U^i{}_k U_j{}^l \delta^k{}_l = U^i{}_k U_j{}^k = \delta^i{}_j
$$
Furthermore:
\begin{equation}
\begin{aligned}
\varepsilon^{i_1 i_2 \dots i_N} \to \varepsilon'^{i_1 i_2 \dots i_N} & = U^{i_1}{}_{j_1}  U^{i_2}{}_{j_2} \cdots  U^{i_N}{}_{j_N} \varepsilon^{j_1 j_2 \dots j_N} \\&
= \mathrm{det}(U)\varepsilon^{i_1 i_2 \dots i_N} \\&
= \varepsilon^{i_1 i_2 \dots i_N} 
\end{aligned}
\end{equation}
and using a similar argument one can show that:
\begin{equation}
\varepsilon_{i_1 i_2 \dots i_N} \to \varepsilon'_{i_1 i_2 \dots i_N} = \varepsilon_{i_1 i_2 \dots i_N}
\end{equation}
