Chapter VIII: Conformal group and conformal completion

Abstract

Having defined the conformal Lie algebra co(T), the next step in our program is obviously the definition of a suitable group Co(T) belonging to this Lie algebra which will be called the conformal group. This is not an abstract group, but a group coming in a particular realization, namely as a group of birational maps of a vector space, corresponding to the realization of the conformal Lie algebra as a Lie algebra of polynomial vector fields. Following Koecher ([Koe69a,b]) one may define the conformal group to be the group of birational maps of V which preserve the conformal Lie algebra, and this definition even makes sense over quite general base fields. Since we are working over the base field of real numbers, we can avoid the assumption of rationality and consider just locally defined smooth maps which preserve the conformal Lie algebra. It follows automatically that they are birational. Thus we start with a pseudogroup of diffeomorphisms (which is a useful concept in differential geometry, cf. [Ko72] and end up with a group. We show that this group is a Lie group whose Lie algebra is the conformal Lie algebra.