Noncommutative instantons come in families

A construction of instantons in the context of noncommutative geometry, in particular SU(2) instantons on a noncommutative 4 sphere, has been recently reported. Firstly, a noncommutative principal fibration A(S(theta)(4)) hooked right arrow A(S(theta)(7)) which 'quantizes' the classical SU(2)-Hopf fibration over S(4), has been constructed in [11] on the toric noncommutative four-sphere S(theta)(4). The generators of A(S(theta)(4)) are the entries of a projection p which describes the basic instanton on A(S(theta)(4)). That is, p gives a projective module of finite type p[A(S(theta)(4))] 4 and a connection del = p o d on it which has a self- dual curvature and charge 1, in some appropriate sense; this is the basic instanton. In [12] infinitesimal instantons - 'the tangent space to the moduli space' were constructed using infinitesimal conformal transformations, that is elements in a quantized enveloping algebra U(theta)(so(5, 1)). In [10] we looked at a global construction and obtain generic charge 1 instantons by 'quantizing' the action of the Lie groups SL(2, H) and SO(2) on the basic instanton which enter the classical construction [1]. We review all this here.