Division Algebras, Lie Algebras, Lie Groups: 8

Division Algebras, Lie Algebras, Lie Groups and Spinors

8. Quaternions: Lie Groups, Clifford Algebras and Spinors

Let A be a pure imaginary quaternion as on the previous page, so U = exp(A) is an element
of SU(2) = 3-sphere. Suppose this SU(2) acts on some space, M. Well, does U act on M from
the left or right? It can do either, and it matters. M has a copy of Q in it, and it's this copy
that receives the action of U.

Given A = A^kq_k, sum k=1,2,3,
define A_L = A^kq_Lk, and U_L = exp(A_L);
define A_R = A^kq_Rk, and U_R = exp(A_R).

So U_L[M] = UM, and U_R[M] = MU, and both of these are SU(2) actions, but they're distinct
SU(2)'s, and the left action SU(2) commutes with the right action SU(2).

By the way, using U_L and U_R we can construct a copy of SO(3), the automorphism group of
Q. In particular, if X is in Q, then U_LU^-1_R[X] = UXU^-1 leaves the real part of X alone and
performs an SO(3) rotation on the imaginary 3-dimensional part (the reader should see
this action is obviously a Q automorphism).

Q is also a Clifford algebra, and an integral part of Clifford algebra theory. However, Clifford
algebras also act on some space (which are called spinor spaces), so we should once again
specify the direction of action. Let CL(p,q) be the Clifford algebra of the real pseudo-orthogonal
space with signature p(+), q(-) (see the book). Then Q_L is isomorphic to CL(0,2), a 1-vector
basis being {q_L1, q_L2}, and the sole 2-vector basis element: q_L3 = q_L1q_L2. Likewise Q_R is
isomorphic to CL(0,2).

What if we allow elements of both Q_L and Q_R? We'll denote by Q_A the algebra of combined
left/right actions of Q on itself. This algebra is isomorphic to R(4) (hence also to CL(3,1) and
CL(2,2)). But this is a path down which I haven't the patience at present to trod. Time for
octonions.