Journal of Lie Theory Volume 13 (2003) 307–309 \begin{matrix}c \\ ◯ \end{matrix} 2003 Heldermann Verlag

A Connected Lie Group Equals the Square of the Exponential Image

Michael Wüstner

Communicated by J. Hilgert

Abstract. It is shown that in every connnected real Lie group every element is the product of two elements in the exponential image.

More than one hundred years ago, F. Engel and E. Study considered the question of the surjectivity of the exponential function of Lie groups ([2],[3], compare also the remarks in [14]). Interestingly, a general solution of this problem is not found yet though there exist equivalent criteria for the surjectivity of special classes of Lie groups (compare [12]). M. Moskowitz and R. Sacksteder first formulated the fact that every real connected Lie group is equal to (exp 𝔤)^{2} ([7]) and provided a proof in the final version of their paper using control theory like methods. Using partly well-known decomposition results on Lie groups, we shall provide a short proof for this result.

In the whole article, G denotes a real connected Lie group. A subgroup K ⊆ G is called compactly embedded if Ad(K) is relatively compact in GL(𝔤). It is shown in [10] that there exist maximal compactly embedded subgroups and that all maximal compactly embedded subgroups are closed and connected. The last property ensures that all maximal compactly embedded subgroups are conjugate to each other. Moreover, it is well-known that every compactly embedded connected subgroup K has surjective exponential function: Indeed, K = K′Z_{0}, where Z_{0} denotes the identity component of the center. Moreover, K′ is compact, hence has surjective exponential function. This implies the surjectivity of the exponential function of K.

First, let us assume that G is semisimple. The Iwasawa decomposition says that every semisimple Lie group can be written as KAM where K is maximal compactly embedded, A is abelian connected and consists of Ad-semisimple elements with spec ad(a) ⊆ ℝ for every a ∈ 𝔞, M is nilpotent connected, and AM is solvable ([4]). By Dixmier’s and Saito’s Theorem ([1],[9]), S := AM = exp(𝔞 + 𝔪). Thus, G = exp 𝔨 exp 𝔰.

Now we recall the Levi decomposition: If G is a real connected Lie group, then for every maximal semisimple subgroup (called Levi factor) L we have G = L Rad(G) and L ∩ Rad(G) discrete. All Levi factors are conjugate to each other.

ISSN 0949–5932 / $2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag This follows directly from the Levi decomposition of real finite-dimensional Lie algebras (see e.g. Korollar II.4.8 of [5]).

A Cartan subalgebra 𝔥 ⊂ 𝔤 of a real finite dimensional Lie algebra 𝔤 is a nilpotent subalgebra which equals its own normalizer 𝔫_{𝔤}(𝔥). Cartan subgroups are defined in various ways which in case of connected Lie groups are all equivalent (compare [8]). We will use the following definition: H ⊆ G is a Cartan subgroup if 𝔥 := L(H) is a Cartan subalgebra and for all g ∈ H and all x ∈ 𝔥 we have Ad(g) ad(x)_{s} = ad(x)_{s} Ad(g) where ad(x)_{s} is the semisimple part of the Jordan decomposition of ad(x).

Theorem 1.9 (ii) of [13] states that for all Cartan subgroups H the intersection H ∩ Rad(G) is connected. Since H ∩ Rad(G) is nilpotent, we have H ∩ Rad(G) = exp(𝔥 ∩ rad(𝔤)). Theorem 1.11 of [13] says that for each Cartan subgroup H of G there exists a Levi factor L such that H ∩ L is a Cartan subgroup of L, H = (H ∩ L)(H ∩ Rad(G)), and H ∩ Rad(G) ⊆ Z_{G}(L). This leads to the following:

Lemma 1. If 𝔥 is a Cartan subalgebra of a real finite-dimensional Lie algebra 𝔤 and 𝔫 the nilradical of 𝔤, then rad(𝔤) = (𝔥∩rad(𝔤))+𝔫. If H is a Cartan subgroup of a real Lie group G and N the nilradical, then Rad(G) = (H ∩ Rad(G))N.

Proof. We choose 𝔩 such that 𝔥 = (𝔥 ∩ 𝔩) ⊕ (𝔥 ∩ rad(𝔤)) (compare Theorem 1.8 of [13]). We observe that 𝔤′ ⊆ 𝔩 + 𝔫. Moreover, 𝔤 = 𝔥 + 𝔤′ = (𝔥 ∩ rad(𝔤)) + 𝔫 + 𝔩, hence rad(𝔤) = (𝔥 ∩ rad(𝔤)) + 𝔫. The proof for Lie groups works similarly.

Thus, with the above notation we obtain G = KS(H ∩ Rad(G))N = K(H ∩Rad(G))SN = exp 𝔨 exp(𝔥∩rad(𝔤)) exp 𝔰 exp 𝔫. Since [𝔨, 𝔥∩rad(𝔤)] = {0}, we get exp 𝔨 exp(𝔥 ∩ rad(𝔤)) = exp(𝔨 + (𝔥 ∩ rad(𝔤))). Moreover, SN is a solvable connected Lie group and spec ad_{𝔤}(s) ⊆ ℝ for each s ∈ 𝔰 by Proposition 3.7 of [11]. Again by Dixmier’s and Saito’s Theorem we get SN = exp(𝔰 + 𝔫) because by Proposition 3.7 of [11], every element x ∈ 𝔰 + 𝔫 satisfies spec ad(x) ⊆ ℝ.

Definition 2. An element x of a real finite-dimensional Lie algebra is called exp-regular if spec ad x ∩ 2πiℤ = {0}. The set of all exp-regular elements of 𝔤 is denoted by reg exp.

Taken together, we have proved the following:

Theorem 3. If G is a real connected Lie group, then G = (exp 𝔤)^{2}. Moreover, for every element g ∈ G there is an exp-regular element x and an element y in 𝔤 with g = exp x exp y.

Theorem 4. If G is a connected real Lie group and for every g ∈ G there is an exp-regular x ∈ 𝔤 and a y ∈ 𝔤 with g = exp x exp y, then there are exp-regular u, w ∈ 𝔤 with g = exp u exp w.

Proof. Since reg exp is dense in 𝔤 (for example by Lemma 2 of [6]), the image exp(reg exp) is dense in exp 𝔤. Since exp is regular at each x ∈ reg exp, the set exp(reg exp) is open in G, hence also in exp 𝔤. The openess of exp(reg exp) implies that there is a symmetric 1-neighborhood U such that exp x · U ⊆ exp(reg exp). The density implies that exp y · U ∩ exp reg exp ≠ Ø. So there is a u ∈ U such that (exp x)u^{−1} and u exp y are in exp(reg exp). This implies the assertion.

References

[1] Dixmier, J., L’application exponentielle dans les groupes de Lie résolubles, Bull. Soc. Math. France 85(1957), 113–121.

[2] Engel, F., Die Erzeugung der endlichen Transformationen einer projectiven Gruppe durch infinitesimale Transformationen der Gruppe. (Erste Mittheilung), Leipziger Berichte 44 (1892), 279–291.

[3] —, Die Erzeugung der endlichen Transformationen einer projectiven Gruppe durch die infinitesimalen Transformationen der Gruppe. (Zweite Mittheilung, mit Beiträgen von E. Study), Leipziger Berichte 45 (1893), 659–696.

[4] Helgason, S., “Differential geometry, Lie groups, and symmetric spaces”, Academic Press, New York, 1978.

[5] Hilgert, J., und K.-H. Neeb, ,,Lie-Gruppen und Lie-Algebren“, Vieweg, Braunschweig, 1991.

[6] Hofmann, K. H., A memo on the exponential function and regular points, Arch. Math. 59 (1992), 24–37.

[7] Moskowitz, M. and R. Sacksteder, The exponential map and differential equations on real Lie groups, J. of Lie Theory 13 (2003), 291–306.

[8]  Neeb, K.-H., Weakly exponential Lie groups, J. Alg. 179 (1996), 331–361.

[9] Saito, M., Sur certains groupes de Lie résolubles, Sci. Papers College Gen. Ed. Univ. Tokyo 7 (1957), 1–11 and 157–168.

[10] Wüstner, M., Product decompositions of solvable Lie groups, Manusc. Math. 91 (1996), 179–194.

[11] —, On the surjectivity of the exponential function of solvable Lie groups, Math. Nachr. 192 (1998), 255–266.

[12] —, “Lie groups with surjective exponential function”, Habilitationsschrift, TU Darmstadt 2000 und Shaker-Verlag, Aachen, 2001.

[13] —, A generalization of the Jordan decomposition, to appear in Forum Math..

[14] —, Historical remarks on the surjectivity of the exponential function, Hist. Math. 29 (2002).

Michael Wüstner Fachbereich Mathematik Technische Universität Darmstadt Schloßgartenstr. 7 64289 Darmstadt, Germany asterix.w@gmx.de

Received July 25, 2002 and in final form August 9, 2002