2. — Lie groups.

 

with kernel the fourth roots of unity. Since these two groups have the same dimension it follows that (2.1) is a local isomorphism or that

The Klein representation can also be looked at from the point of view of Lie groups. The group 8L(4:, 0), of complex 4x4 matrices of deter­minant 1, acts on P3(G) by projective transformations, and hence it acts on the space of lines i.e. the quadric Qt. But the group of projective trans­formations in P5 keeping Qi fixed is just the complex orthogonal group 0(6, G) modulo {± 1} and so we get a homomorphism

 

 

This is one of the coincidences that happen in low dimensions between the various classical groups.

If we start from the group Spin (6, 0) then the isomorphism (2.2) im­plies that Spin (6, G) has a representation on Oi. This is one of the half-spin representations. The other half-spin representation is on the dual G4.

These complex Lie groups have many real forms, and the local iso­morphism (2.1) leads in particular to the following local isomorphisms of real Lie groups:

 

 

Case (ii) arises from Minkowski space and is the one mainly studied in the Penrose theory. Case (iii) arises from Euclidean space and is the one that concerns us. The action of 8L(2, H) on P3(C) preserves the fibration (1.1) and induces the conformal group action on 84. In other words the conformal group of 84 acts naturally on the fibration (1.1).

we see that the fibration (1.1), expressed in terms of compact Lie groups becomes

 

Taking maximal compact subgroups of (iii) we get the local isomorphism

 

 

Thus 8p(2) is the group of automorphisms of the fibration (1.1) which pre­serves in addition the natural metrics on P3(G) and on 8*. Since 8p(2) acts transitively on P3(G) we can express P3(G) as a homogeneous (coset) space:

 

 

 

Since

 

 

with fibre 8p(l)/U(l) = 82. This description shows clearly that, starting from 8* we have essentially two different choices to produce P3(G) depending on which 8p( 1) factor we pick. These two choices switch when we reverse orientation on S4 and lead to dualizing P3(G).

The fibration (2.3) is also related to the basic $p(l)-instanton on $4. As explained in Chapter II the basic instanton can be thought of as the quaternion line-bundle over P^H) with its connection induced from the fixed space H2. An equivalent description is to say that the principal $p(l)-bundle of the instanton is the fibration

 

 

The connection can now be described in terms of horizontals to the fibres

 

 

If we give all spaces their natural metrics, inherited from the bi-invariant metric of the compact group 8p(2), we can choose the orthogonals to the fibres as horizontals. This gives a connection admitting 8p(2) ~ S0{5) as symmetry group. Moreover this is the unique choice that has this property because any invariant choice of horizontals must, at each point, be invariant under the action of the isotropy group 8p( 1), which acts by inequivalent representations in the vertical (fibre) and horizontal directions (vertically we have the adjoint representation of 8p( 1) and horizontally we have the C2 representation of 8p(l) ^ 8U(2)). Since the instanton connection admits 80(5) as symmetry group it follows that our connection must be the instan­ton or anti-instanton, depending on which 8p( 1) factor we choose on the left hand side of (2.4).

Thus P3(0) is naturally the quotient of the principal bundle of the /8y(l)-instanton by the action of U{1). Equivalently P3(0) can be obtained from the instanton, considered as H = G2 bundle over $4, by replacing each fibre by the corresponding projective space P1(0). This way of obtain­ing P3(G) from $4 can be generalized to other 4-manifolds as explained in [3].