Journal of Lie Theory Volume 5 (1995) 203–205 ◯^{C} 1995 Heldermann Verlag

A connected complex simple centerfree Lie group whose exponential function is not surjective

Michael Wüstner

Communicated by M. Moskowitz

Abstract. We discuss the example of a complex simple Lie group G, with trivial center, whose exponential map is not surjective and with dim_{ℂ} G = 10.

In [2] I have characterized solvable Lie groups and complex semisimple Lie groups with surjective exponential function. In particular I have proven the following Theorem (See [2, Satz IV.3.27] and [3]):

Theorem 1. A semisimple complex Lie groups has surjective exponential function if and only if it is isomorphic to a finite product of groups PSl(n(i), ℂ).

Independently, in [1] Moskowitz also has proven that finite products of PSl(n(i), ℂ) have surjective exponential map. In the context of this work it is informative to observe that for the exponential map on a connected complex simple Lie group to be surjective it is not enough that it is centerfree. In the following I shall discuss the smallest counterexample known, namely PSP(2, ℂ). Here PSP(2, ℂ) ≅ SP(2, ℂ)/Z(SP(2, ℂ)) is the adjoint Lie group of SP(2, ℂ) of all

matrices ( \begin{matrix}A & B \\ C & D \end{matrix} ) ∈ Sl(4, ℂ) with A, B, C, D ∈ ℂ^{2×2} such that C^{T} A−A^{T} C =

0, D^{T} B − B^{T} D = 0, A^{T} D − C^{T} B = I, where I is the identity of Gl(2, ℂ).

Theorem 2. The complex simple Lie group PSP(2, ℂ) is centerfree and its exponential map is not surjective.

Proof. The center of PSP(2, ℂ) is trivial since it is an adjoint Lie group. We claim that there is an element γ ∈ PSP(2, ℂ), for which there is no x ∈ sp(2, ℂ) such that γ = exp_{PSP(2,ℂ)} x. First, we consider the group SP(2, ℂ).

Let p: SP(2, ℂ) → PSP(2, ℂ) be the quotient map. We claim that there is no g ∈ p^{−1}(γ) which is in the image of exp_{SP(2,ℂ)}. Then we will have finished the proof. We shall prove the claim in the following lemma.

ISSN 0940–2268 / $2.50 ◯^{C} Heldermann Verlag Lemma 3. There is a γ ∈ PSP(2, ℂ) such that no element of p^{−1}(γ) lies in the image of exp_{SP(2,ℂ)}.

Proof. We set g = ( \begin{matrix}1 & 0 & −1 & 0 \\ 0 & −1 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & −1 \end{matrix} ). The proof that g is in SP(2, ℂ) is

straightforward. Now we set γ := p(g).

The elements of the Lie algebra sp(2, ℂ) of SP(2, ℂ) are exactly those

elements ( \begin{matrix}A & B \\ C & D \end{matrix} ) of sl(4, ℂ) with 2 × 2 complex matrtices A, B, C and D,

for which the following conditions hold: A = −D^{T} , B = B^{T} , C = C^{T} .

We assume that there is an x ∈ sp(2, ℂ) such that g = exp_{SP(2,ℂ)} x and consider SP(2, ℂ) as a group of endomorphisms of ℂ^{4}. Since a subspace which is invariant under x must be invariant under g and since g possesses two 2dimensional eigenspaces for two different eigenvalues, respectively, x must possess two 2-dimensional eigenspaces. By the Spectral Mapping Theorem one eigenvalue must be in 2πiℤ and the other must be of the form (2z + 1)πi with z ∈ ℤ. But then the trace of x is equal to (4z_{1} + 2(2z_{2} + 1))πi with z_{1}, z_{2} ∈ ℤ. On the other hand, tr x = 0 because x ∈ sp(2, ℂ) ⊆ sl(4, ℂ). But this implies 2 ∈ 4ℤ and this is impossible. Now we consider PSP(2, ℂ). The center of SP(2, ℂ) is equal to {1, −1}. So, p^{−1}(γ) = {g, −g}. We saw a moment ago that g ∉ im(exp_{SP(2,ℂ)}). But the proof of the assertion −g ∉ im(exp_{SP(2,ℂ)}) is analogous to that for g.

The argument shows that g and −g have no preimage in sl(4, ℂ). But in this case, the center of Sl(4, ℂ) is equal to {I, −I, iI, −iI} and, indeed, ig

has a preimage in sl(4, ℂ), namely ( \begin{matrix}\frac{π}{2}i & 0 & −1 & 0 \\ 0 & −\frac{π}{2}i & 0 & 1 \\ 0 & 0 & \frac{π}{2}i & 0 \\ 0 & 0 & 0 & −\frac{π}{2}i \end{matrix} ). So, of course, we

have not constructed a counterexample to the result in Theorem 1 that PSl(n, ℂ) possesses a surjective exponential function.

References

[1] Moskowitz, M., The surjectivity of the exponential map for certain Lie groups, Ann. Mat. Pura Appl. (4) 166 (1994), 129-143.

[2] Wüstner, M., ,,Beiträge zur Strukturtheorie auflösbarer Lie-Gruppen“, Dissertation, Technische Hochschule Darmstadt, 1995.

[3] Wüstner, M., On the surjectivity of the exponential function of complex algebraic, complex semisimple and complex splittable Lie groups, submitted.

Wüstner 205

Michael Wüstner Fachbereich Mathematik Technische Hochschule Darmstadt Schloßgartenstraße 7 D-64289 Darmstadt wuestner@mathematik.th-darmstadt.de

Received June 20, 1995