Division Algebras, Lie Algebras, Lie Groups: 7

Division Algebras, Lie Algebras, Lie Groups and Spinors

7. Quaternions: Lie Groups and Algebras

The fact that Q acting on itself from the left and from the right gives rise to two distinct
and commuting copies of Q actions suggests that it would be worth our while to
distinguish the algebras of left actions, right actions, and Q itself, the algebra on which
these adjoint algebras act:

Q: quaternion algebra itself; basis q_m, m=0,1,2,3;
Q_L: adjoint algebra of left actions of Q on itself; basis q_Lm, m=0,1,2,3;
Q_R: adjoint algebra of right actions of Q on itself; basis q_Rm, m=0,1,2,3.

(This notation originates in my book:
Division Algebras: Octonions, Quaternions, Complex Numbers and the Algebraic Design of Physics.)

Why bother with three copies of the same algebra? Because the quaternions are
noncommutative, and there really are three different copies. Using all three makes
it very easy to connect the quaternions to some important Lie groups, and Clifford
algebras and spinors (see the book for a detailed discussion of Clifford algebras).

Any pure imaginary quaternion, A, can be written in the form uθ, where u is a unit
imaginary quaternion (so an element of a 2-sphere; don't forget, the space of imaginary
quaternions is 3-dimensional), and θ = |A|, the positive real magnitude of A.
The element u behaves just like the complex imaginary i when exponentiated, because
u² = -1. Therefore, e^A = exp(uθ) = cos(θ) + u sin(θ). This is also a unit quaternion,
although not a pure imaginary one. In fact, any unit quaternion can be written in this
form. Hence the set {e^A: A linear in q_k, k = 1,2,3} = {U in Q: |U| = 1} = S³ = 3-sphere.

This set is also closed under multiplication, and since it is associative, it is a Lie group,
in this case SU(2) (that is, the "shape" of SU(2) is that of a 3-sphere). The associated
Lie algebra, su(2), has a basis, q_k, k = 1,2,3 (3-dimensional, as is SU(2) itself). That is,
SU(2) is obtained from the elements of su(2) via exponentiation. Note that a Lie algebra
by definition uses the commutator product, under which the set of all elements linear in
q_k, k = 1,2,3, is closed.

However, Lie groups invariably appear in physics as actions on some space, not as some
abstract mathematical object only contextually connected to anything else. In order
to make connection with those ideas we have to start using the adjoint algebras.