In Peskin & Schroeder section $3.2$ they begin by telling us that they want to construct spin $1/2$ representations of the Lorentz algebra. One way to do that, they say, is to first find a representation of the Dirac algebra from which one can construct the Lorentz algebra via commutator. Then they say that the smallest possible representation of the Dirac Algebra are $4 \times 4$ matrices, implying that the smallest possible (complex) dimension of the representation space is $4$. But then in problem $3.1$ when we are asked to find all the representations of the Lorentz algebra we find there are $0$ and $2$ dimensional representations as well (the $(0,1/2)$, and $(1/2,0)$ representations for instance). So how are the above two statements compatible? Aren't the $(0,1/2)$, and $(1/2,0)$ also spin $1/2$ representations of the Lorentz algebra as well?

In Peskin & Schroeder section $3.2$ they begin by telling us that they want to construct spin $1/2$ representations of the Lorentz Lie algebra. One way to do that, they say, is to first find a representation of the Dirac algebra from which one can construct the Lorentz Lie algebra via commutator. Then they say that the smallest possible representation of the Dirac Algebra are $4 \times 4$ matrices, implying that the smallest possible (complex) dimension of the representation space is $4$. But then in problem $3.1$ when we are asked to find all the representations of the Lorentz Lie algebra we find there are $0$ and $2$ dimensional representations as well (the $(0,1/2)$, and $(1/2,0)$ representations for instance). So how are the above two statements compatible? Aren't the $(0,1/2)$, and $(1/2,0)$ also spin $1/2$ representations of the Lorentz Lie algebra as well?