Description of Infinite Dimensional Abelian Regular Lie Groups

It is shown that every abelian regular Lie group is a quotient of its Lie algebra via the exponential mapping. This paper is a sequel of [3], see also [4], chapter VIII, where a regular Lie group is defined as a smooth Lie group modeled on convenient vector spaces such that the right logarithmic derivative has a smooth inverse Evol : C(R, g) → C(R, G), the canonical evolution operator, where g is the Lie algebra. We follow the notation and the concepts of this paper closely. Lemma. Let G be an abelian regular Lie group with Lie algebra g. Then the evolution operator is given by Evol(X)(t) := Evol(X)(t) = exp (∫ t 0 X(s)ds ) for X ∈ C(R, g). Proof. Since G is regular it has an exponential mapping exp : g → G which is a smooth group homomorphism, because s 7→ exp(sX) exp(sY ) is a smooth oneparameter group in G with generator X + Y , thus exp(X) exp(Y ) = exp(X + Y ) by uniqueness, [3], 3.6 or [4], 36.7. The Lie algebra g is a convenient vector space with evolution mapping Evolg(X)(t) = ∫ 1 0 X(s)ds, see [3], 5.4, or [4], 38.5. The mapping exp : g → G is a homomorphism of Lie groups and thus intertwines the evolution operators by [3], 5.3 or [4], 38.4, hence the formula. Another proof is by differentiating the right hand side, using [3], 5.10 or [4], 38.2. As consequence we obtain that an abelian Lie group G is regular if and only if an exponential map exists. Furthermore, an exponential map is surjective on a connected abelian Lie group, because exp( ∫ t 0 δc(s)ds) = Evol(δc)(t) = c(t) for any smooth curve c : R → G with c(0) = e. Theorem. Let G be an abelian, connected and regular Lie group, then there is a c-open neighborhood V of zero in g so that exp(V ) is open in G and exp : V → exp(V ) is a diffeomorphism. Moreover, g/ ker(exp) → G is an isomorphism of Lie groups. Proof. Given a connected, abelian and regular Lie group G, we look at the universal covering group G̃ π −→ G, see [4], 27.14, which is also abelian and regular. Any 1991 Mathematics Subject Classification. 22E65, 58B25, 53C05.