I'm going through the representation theory of $\mathfrak{so}(1,3)$, building Dirac/Weyl spinors and vectors, and I'm a bit confused on the mathematical definitions involved. We have $\mathfrak{so}(1,3)\cong \mathfrak{su}(2)\oplus \mathfrak{su}(2)$ as algebras. I'm good with the formalities of this isomorphism: if $J^i$ and $K^i$ are our elements of the six-dimensional algebra $\mathfrak{so}(1,3)$, we can always write any linear combination of $J^i$ and $K^i$ as $a_i J_+^i+b_jJ_-^j$ where $J_+$ satisfies its own $su(2)$ algebra, $J_-$ does too, and all $J_+$ and $J_-$ commute.  To build out a representation of $\mathfrak{so}(1,3)$, I see the notation $\left(n_1,n_2\right)$ used. So $\left(\frac{1}{2},0\right)$ can denote the left-handed Weyl spinors, $\left(0,\frac{1}{2}\right)$ the right-handed ones, $\left(\frac{1}{2},\frac{1}{2}\right)$ vectors, and $\left(\frac{1}{2},0\right)\oplus\left(0,\frac{1}{2}\right)$ Dirac spinors. I'm a bit confused about what this notation really means.  It can't be the tensor product: $\frac{1}{2}\otimes \frac{1}{2}\cong 1\oplus 0$ is still just a representation of the three-dimensional Lie algebra $su(2)$, with group action (for example) $J^i=(\frac{1}{2}\sigma^i)\otimes 1+ 1\otimes (\frac{1}{2}\sigma^i)$.   Is the following a good definition for $(n_1,n_2)$?  Take $V_1$ to be the vector space of the spin $n_1$ representation, and $V_2$ the vector space corresponding to $n_2$. Then $(n_1,n_2)$ is the representation of $su(2)\oplus su(2)$ which acts on the vector space $V_1\otimes V_2$, with action $(A,B)(v_1\otimes v_2)=(Av_1)\otimes(B v_2)$ (where $(A,B)\in su(2)\oplus su(2)$ and their actions on $V_i$ are determined by the spin $n_i$ representation).  I think this is correct, I just got very confused with the contrast between addition of angular momentum: in this case the vector space is still the tensor product, but the group action is different.
EDIT:  The vector space for the $(\frac{1}{2},0)$ Representation is $\mathbb{C}^2$ as mentioned by Qmechanic in the comments to his answer below! The vector spaces for the other representations remain unanswered.  The definition of a representation is a map (a homomorphism) to the space of linear operators over a vector space. My question is: What are the corresponding vector spaces for the   $(0,0)$ Representation $(\frac{1}{2},0)$ Representation  $(0,\frac{1}{2})$ Representation $(\frac{1}{2},0) \oplus (0,\frac{1}{2}) $ Representation $(\frac{1}{2},\frac{1}{2})$ Representation infinite dimensional Representation?
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