On the homomorphism of the Lorentz algebra representation (1/2, 0)
I was reading this answer and I don't quite understand how the $\rho$ homomorphism works. The generators of the two copies of $\mathfrak{su}(2)$ in $\mathfrak{su}(2)\oplus\mathfrak{su}(2)$ are given by $N_i^+ = \frac{1}{2}(J_i+\mathrm{i}K_i)$ , $N_i^- = \frac{1}{2}(J_i-\mathrm{i}K_i)$ respectively. The $J_i$'s are the generators corresponding to rotations in $SO^+(1,3)$ and $K_i$'s are the generators corresponding to boosts in $SO^+(1,3)$. This is the excerpt where it is defined.
"You are given the $(1/2,0)$ representation $\rho : \mathfrak{su}(2)\oplus\mathfrak{su}(2)\to\mathfrak{gl}(\mathbb{C}^2)$. 656
Since $\rho$, as a representation, is a <PERSON> algebra homomorphism, you know that $\rho(N_i^-) = 0$ implies $\rho(J_i) = \mathrm{i}\rho(K_i)$. Here, all matrices $N_i^-,J_i,K_i,0$ matrices are two-dimensional matrices on $\mathbb{C}^2$. You know that $\rho(N_i^-) = 0$ as two-dimensional matrices because of how the $(s_1,s_2)$ representation is defined: Take the individual representations $\rho^+ : \mathfrak{su}(2)\to\mathfrak{gl}(\mathbb{C}^{2s_1+1})$ and $\rho^- : \mathfrak{su}(2)\to\mathfrak{gl}(\mathbb{C}^{2s_2+1})$ and define the total representation map by $$ \rho : \mathfrak{su}(2)\oplus\mathfrak{su}(2)\to\mathfrak{gl}(\mathbb{C}^{2s_1+1}\otimes\mathbb{C}^{2s_2+1}), h\mapsto \rho^+(h)\otimes 1 + 1 \otimes \rho^-(h)$$ where I really mean the tensor product of vector spaces with $\otimes$. For $s_1 = 1/2,s_2 = 0$, this is a two-dimensional representation where $\rho^-$ is identically zero - and the zero is the two-dimensional zero matrix in the two-by-two matrices $\mathfrak{gl}(\mathbb{C}^2)$."
1) If $h = N_i ^+$, $\rho ( N_i ^+) = \rho ^+ (N_i^+)\otimes1 + 1 \otimes \rho^-(N_i^+)$ , why is $\rho^-$ defined on $N_i^+$?
My guess of how it ends: $\rho (N_i^+) = (\sigma_i /2)\otimes1 + 1 \otimes {0} = \sigma_i /2$
2)If $h = N_i ^-$, $\rho ( N_i ^-) = \rho ^+ (N_i^-)\otimes1 + 1 \otimes \rho^-(N_i^-)$ , same, why is $\rho^+$ defined on $N_i^-$?
I guess $\rho^+(N_i^-) = 0$ so that $\rho (N_i^-) = 0\otimes1 + 1 \otimes {0} = 0$ , but I am not sure why. Thanks in advance. 364