1. — Quaternions.

In B2 it is well known that many formulae are simpler when written in terms of complex nnmbers. It was the discovery of Hamilton that for Bi one can introduce a non-commutative extension of the complex nnmbers called quaternions, and it was Hamilton's hope that quaternions would turn out to be the natural tool for an algebraic description of the physical world. Although this was an exaggerated hope there is some merit in Hamilton's point of view as we shall see shortly. For the present however, we simply use quaternions as a convenient algebraic formalism that simpli­fies notation. This is particularly relevant to the group $77(2) which as we shall see can be identified with the group of quaternions of unit norm, in the same way that 77(1) is the complex numbers of unit norm.

We begin by briefly recalling the definition and elementary properties of quaternions. Just as the complex numbers 0 are formed from the real numbers B by adjoining a symbol i with i2 = ~ 1, so the quaternions H (in honour of Hamilton) are formed from B by adjoining three symbols i, j, k satisfying the identities:

 

Thus a general quaternion x is of the form

 

 

 

 

 

where real numbers. The conjugate quaternion x is

defined by

 

and conjugation is an anti-involution, i.e. (xy) — yx. In virtue of the identity (1.1), one finds

 

 

This quantity is denoted |a;|2 and is zero only for x = 0. If x #0 it has a unique inverse ar1 given by

 

 

The quaternions x with norm 1, i.e. \x\ = 1, form therefore a multiplicative group which is geometrically the 3-sphere

 

 

In analogy again with the complex numbers we refer to the component xx in (1.2) as the real part of x and the remainder x - - xx as the imaginary part.

If we identify i with the usual complex number we can regard the complex numbers 0 as contained in H (taking xs= xt — 0). Moreover every quaternion x as in (1.2) has a unique expression.

 

 

This identifies H with O2. Now consider the quaternion multiplication x ->xg where g = gx ~f <72h with gx, g2e G. Computing we find

 

 

so that the vector (zx, z2) is multiplied on the right by the 2x2 complex matrix

 

 

Thus, if we wish, we can identify the algebra of quaternions as a sub-algebra of the 2x2 complex matrices in which i, j, h are the matrices

 

In particular the group Sp(l) of quaternions of unit norm gets identified

 

with $17(2) and its Lie algebra can be viewed as the pure imaginary qua­ternions with natural basis i, j, h.

We now identify It* with II via (1.2) and an $E7(2)-potential will be given by functions An{x) whose values are imaginary quaternions. It will be convenient if we go further and write

 

 

so that A(x) is a differential form with values in Im (H). Finally we shall consider the quaternion differential

 

 

and its conjugate

 

 

just as in complex variable theory one uses dz — dx -f- idy and dz = dx~ i dy. If f(x) is any function of the quaternion variable x with quaternion values the expression

 

 

will represent an SI7(2)-potential. Here f(x)dx is computed formally, written as ^a/idx» with a^eH and then A„ = Im (a„). Note that, before taking the imaginary part we have a potential for the group II* of all non­zero quaternions which is SU(2) times a scale factor.

We shall also write the curvature F as an exterior 2-form

 

Then F can be computed from A by

 

 

 

 

where

 

 

and

 

When A is written in the quaternionic form (1.3) we get a similar formula

 

 

Taking imaginary parts commutes with formation of curvature because of our remark above concerning the larger group H*.

The use of the quaternion differentials dx and dx is also convenient in connection with the study of self-duality. To see this let us compute dx/\dx. We get

 

 

The coefficients of i, j, k in this expression are precisely a basis for the self- dual 2-forms, i.e. 2-forms co with *eo = co. Hence dx/\dx is a 2-form, with values in the Lie algebra of 8U(2), which is self-dual. A similar computation shows that dx/\dx is anti-self-dual.