1 The spin homomorphism SL 2 ( C ) → SO

is a homomorphism of classical matrix Lie groups. The lefthand group consists of 2 × 2 complex matrices with determinant 1. The righthand group consists of 4× 4 real matrices with determinant 1 which preserve some fixed real quadratic form Q of signature (1, 3). This map is alternately called the spinor map and variations. The image of this map is the identity component of SO1,3(R), denoted SO1,3(R). The kernel is {±I}. Therefore, we obtain an isomorphism PSL2(C) = SL2(C)/± I ' SO1,3(R). This is one of a family of isomorphisms of Lie groups called exceptional isomorphisms. In Section 1.3, we give the spin homomorphism explicitly, although these formulae are unenlightening by themselves. In Section 1.4 we describe O1,3(R) in greater detail as the group of Lorentz transformations. This document describes this homomorphism from three distinct perspectives. The first is very concrete, and constructs, using the language of Minkowski space, Lorentz transformations and Hermitian matrices, an explicit action of SL2(C) on R preserving Q (Section 1.5). The second is geometric, and describes the zero locus of Q as a twisted form of P × P; it then inherits an action from P × P (Section 1.6). The third is the most general and develops the basic theory of Clifford algebras and Clifford groups in order to define the Spin group, before specializing to our case, in which SL2(C) ' Spin1,3(R) (Sections 1.7 to 1.9). This last perspective is the most challenging, but it also leads to other exceptional isomorphisms in Lie theory (which is beyond the scope of this note).