In 3+1 Minkowski spacetime, we can use the fact that the Lie algebra of the Lorentz group decomposes (I am omitting some details here to keep it short) into $su(2) \oplus su(2)$ and then use the fact that spinors are the fundamental representations of $su(2)$ to start building representations of the Lorentz group$^\dagger$ of the form $(j_+, j_-)$ where $j_\pm$ label representations of $su(2)$.
My question is: since there are systems that include spinors and live in 2+1 or 1+1 dimensions being studied at the moment, how do spinors come into play? How do we get them as representations of the (universal cover of the) Lorentz group in 2+1 and 1+1 dimensions? Can their Lie algebras be decomposed in similar ways or maybe there is a different definition of spinors in these dimensions (i.e. not as the fundamental representation of $su(2)$)?
$^\dagger$or rather its universal cover since we're interested in projective reps of the Lorentz group