Journal of Lie Theory Volume 9 (1999) 487–489 C ◯1999 Heldermann Verlag

Description of infinite dimensional abelian regular Lie groups

Peter Michor and Josef Teichmann

Communicated by W. A. F. Ruppert

Abstract. It is shown that every abelian regular Lie group is a quotient of its Lie algebra via the exponential mapping. Key words: Regular Lie groups, infinite dimensional Lie groups 1991 Mathematics subject classification: 22E65, 58B25, 53C05

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^{r}(X)(t) = exp(∫_{0}^{t} 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 ↦ 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 Evol_{𝔤}(X)(t) = ∫_{0}^{t} 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 a 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

ISSN 0949–5932 / $2.50 ^{_{C} ◯} Heldermann Verlag on a connected abelian Lie group, because exp(∫_{0}^{t} δ^{r}c(s)ds) = Evol(δ^{r}c)(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.

J.T. was supported by ‘Fonds zur Förderung der wissenschaftlichen Forschung, Projekt P 10037 PHY’.

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 tangent Lie algebra homomorphism from a simply connected Lie group to a regular Lie group can be uniquely integrated to a Lie group homomorphism by [5] or [3], 7.3 or [4], 40.3. Consequently, there exists a homomorphism Φ : G̃ → 𝔤 with Φ′ = id_{𝔤}. Since G̃ is regular there is a map from 𝔤 to G̃ extending id, which has to be the inverse of Φ and which is a fortiori the exponential map ẽxp of G̃, so Φ is an isomorphism of Lie groups. The universal covering projection π intertwines ẽxp and exp, so the result follows. The quotient 𝔤/ ker(exp) is a Lie group since there are natural chart maps and the quotient space is a Hausdorff space by the Hausdorff property on G.

Remarks. Given a convenient vector space E and a subgroup Z, it is not obvious how to determine simple conditions to ensure that E/Z is a Hausdorff space, because c^{∞}E is not a topological vector space in general (see [4], Chapter I): An additive subgroup Z of E is called “discrete” if there is a c^{∞} -open zero neighborhood V with V ∩ (Z + V) = {0} and for any x ∉ Z there is a c^{∞} -open zero neighborhood U so that (x + Z + U) ∩ (Z + U) = Ø. The above kernel of exp naturally has this property, consequently any regular connected abelian Lie group is a convenient vector space modulo a “discrete” subgroup.

Let E be a Fréchet space, then a subgroup is “discrete” if and only if there is an open zero neighborhood V with V ∩ (Z + V) = {0}, because c^{∞}E = E. This leads immediately to a generalization of a result of Galanis ([2]), who proved that every abelian Fréchet-Lie group which admits an exponential map being a local diffeomorphism around zero is a projective limit of Banach Lie groups. With the above theorem one can easily write down this limit in general.

With the above methods it is necessary to assume regularity: Otherwise one obtains as image of Φ a dense arcwise connected subgroup of the convenient vector space 𝔤, which does not allow any conclusion in contradiction to the finite dimensional case. Note that the closed subgroup of integer-valued functions in L^{2}([0, 1], ℝ) is arcwise connected but not a Lie subgroup (see [1]) so that Yamabe’s theorem is already wrong on the level of infinite dimensional Hilbert spaces.

References

[1] Chen, Su-Ching, and R. W. Yoh, The Category of Generalized Lie Groups, Trans. Amer. Math. Soc. 199 (1974), 281–294.

[2] Galanis, G., Projective Limits of Banach Lie Groups, Periodica Mathematica Hungarica 32 (1996), 179–191.

[3] Kriegl, A., and P. W. Michor, Regular infinite dimensional Lie groups, J. Lie Theory 7 (1997), 61–99.

[4] —, “The Convenient Setting of Global Analysis,” Amer. Math. Soc., Mathematical Surveys and Monographs 53, 1997.

[5] Pestov, V., Regular Lie groups and a theorem of Lie-Palais, J. Lie Theory 5 (1995), 173–178.

Institut für Mathematik Universität Wien Strudlhofgasse 4 A-1090 Wien Austria; and: Erwin Schrödinger Institut für Mathematische Physik Boltzmanngasse, A-1090 Wien Austria peter.michor@esi.ac.at Institut für Mathematik, Universität Wien Strudlhofgasse 4 A-1090 Wien Austria jteichma@radon.mat.univie.ac.at

Received July 21, 1998