We consider in this chapter the relationship, on the infinitesimal level, between multiplicative and additive algebraic structure. This is relevant for the theory of group-valued differential forms, and for the relationship between their exterior derivatives, when formed multiplicatively, and when formed additively. The setting involves a space G equipped with a group structure. In Sections 6.7, 6.8 and 6.9, we assume that G is a manifold (so G is a Lie group); in the other sections, we have more general assumptions that will be explained.

Associative algebras

We begin with some observations in the case where the group G admits some enveloping associative algebra; this means an associative unitary algebra A, and a multiplication preserving injection G → A (so we consider G as a subgroup of the multiplicative monoid of A). The algebra A should be thought of as an auxiliary thing, not necessarily intrinsic to G, and in particular, it does not necessarily have any universal property. Of course, under some foundational assumptions on the category ℰ, there exists a universal such A, the group algebra R(G) of G, but we need to assume that A is a KL algebra, in the sense that the underlying vector space of A is KL, and it is not clear that R(G) has this property, except when G is finite.

We present some examples of enveloping algebras at the end of this chapter.

We shall also consider group bundles, and enveloping KL algebra bundles. Such occur whenever a vector bundle is given; see Example 6.10.1 below.