Complex special linear group

The complex special linear group in two dimensions, denoted by $S L (2, C)$ , is a fundamental group in abstract algebra and linear algebra. It consists of all 2x2 complex matrices with a determinant of 1.
It is defined as

S L (2, C) = {(\begin{matrix} a & b \\ c & d \end{matrix}) \in C^{2 \times 2} ∣ a d - b c = 1}

It is the spin group corresponding to the connected component of the identity in the Lorentz group, $S O^{+} (1, 3)$ .
There is an analogy between $S U (2) \to S O (3)$ and $S L (2, C) \to S O^{+} (1, 3)$ , and their representations. See @baez1994gauge page 182.

If we add translations, by means of the semidirect product, to this group I think we obtain the inhomogeneous special linear group:

C^{2} ⋊ S L (2, C) .

It turns out that spinor fields and their tensor products form all the physically relevant unitary representations of the inhomogeneous $S L (2, C)$ . Finally, when considering quantum theories of the various fields, we have the well-known spin-statistics theorem: in a reasonable quantum field theory, spinor fields and their odd tensor products describe fermions, while scalar fields and even tensor products of spinor fields describe bosons. Taken from here (calibre).