Topics in Representation Theory: Spin Groups 1 Spin Groups

One of the most important aspects of Clifford algebras is that they can be used to explicitly construct groups called Spin(n) which are non-trivial double covers of the orthogonal groups SO(n). For a more exhaustive discussion of this than the simplified one that will follow, there are many possible sources to read, but two in particular that may be useful are [2] Chapters I.2 and I.6, and [1] Chapter I.6. There are several equivalent possible ways to go about defining the Spin(n) groups as groups of invertible elements in the Clifford algebra. 1. One can define Spin(n) in terms of invertible elements g ̃ of Ceven(n) that leave the space V = Rn invariant under conjugation g̃V g̃−1 ⊂ V 2. One can show that, for v, w ∈ V, v → wvw−1 is reflection in the hyperplane perpendicular to w. Then Pin(n) is defined as the group generated by such reflections with ||w||2 = 1. Spin(n) is the subgroup of Pin(n) of even elements. Any rotation can be implemented as an even number of reflections (Cartan-Dieudonné) theorem. 3. One can define the Lie algebra of Spin(n) in terms of quadratic elements of the Clifford algebra. This is what we will do here. The Lie algebra of SO(n) consists of n by n antisymmetric real matrices. A basis for these is given by Lij = Eij − Eji for i < j. The Lij generate rotations in the i − j plane. They satisfy the commutation relations [Lij, Lkl] = δilLkj − δikLlj + δjlLik − δjkLil The generators of ei of the Clifford algebra C(n) satisfy the relations eiej + ejei = −2δij and one can use these to show that the 12eiej satisfy the same commutation relations as the Lij.