Seminar Sophus Lie 2 (1992) 213–221

A globality theorem for Lie-wedges that are bounded by a hyperplane-ideal

Dirk Mittenhuber

Abstract

We consider a Lie group G containing a normal subgroup N ⊴ G such that G/N ≅ ℝ, i.e the Lie algebra 𝔫 is a hyperplane ideal in 𝔤. A natural question that arises in this context is the following: Suppose we are given a Lie-wedge W which is contained in a halfspace bounded by 𝔫. Under which conditions is W global in G? We will prove globality for all pointed wedges W such that there exists another wedge W′ ⊆ 𝔫 which is global in N and satisfies (W ∩ 𝔫) ⊇ int_{𝔫}(W′) ∪ {0}. Especially our result applies to the groups and Lorentzian wedges considered by Levichev and Levicheva in this volume. As another application, we solve the globality problem of the Heisenberg-algebra, i.e. we give a complete characterization of all Liewedges that are global in the Heisenberg-group.

1. The globality theorem

Theorem 1.1. Let G be a Lie group, N ⊴ G a normal subgroup such that G/N ≅ ℝ. Let W ⊆ 𝔫 be a Lie wedge which is global in N and X ∈ 𝔤 \ 𝔫 such that the edge H(W) = W ∩ (−W) is ad X -invariant. We define the following set:

W = {W′ ⊆ 𝔫 | W′ global in N and (∃ε > 0)(∀t ∈ [0, ε])e^{t ad X}W ⊆ W′} .

If ⋂ W = W then the Lie wedge W + ℝ^{+}X is global in G.

Proof. The ad X -invariance of H(W) implies that e^{t ad Y} X ∈ X + H(W) for all Y ∈ H(W). Since W is a Lie wedge and H(W) = H(W +ℝ^{+}X) it follows that W +ℝ^{+}X is invariant under the adjoint action of its edge, hence a Lie wedge. Let us denote with α(t) the inner automorphism of G induced by exp(tX), i.e. α(t)(g) = exp(tX)g exp(−tX) and α(t)(exp Y) = exp(Ad(exp(tX)Y) = exp(e^{t ad X}Y). We use the notation ∏_{i=1}^{n} g_{i} = g_{1}g_{2} ⋯ g_{n} and set for t ≥ 0

S_{t} = {(∏_{i=1}^{n} α(t_{i})(g_{i})) exp(tX) | t_{i} ∈ [0, t], g_{i} ∈ exp(W)}  (1)

and S = ⋃_{t≥0} S_{t}. We will prove the following:

(i) S is a semigroup because for s, t ≥ 0 we have S_{t}S_{s} ⊆ S_{t+s}. Indeed, if g = (∏_{i=1}^{n} α(t_{i})(g_{i})) exp(tX) ∈ S_{t} and h = (∏_{j=1}^{m} α(s_{j})(h_{j})) exp(sX) ∈ S_{s} then

gh = (∏_{i=1}^{n} α(t_{i})(g_{i})) exp(tX) (∏_{j=1}^{m} α(s_{j})(h_{j})) exp(sX)

= (∏_{i=1}^{n} α(t_{i})(g_{i})) α(t) (∏_{j=1}^{m} α(s_{j})(h_{j})) exp((s + t)X)

= (∏_{i=1}^{n} α(t_{i})(g_{i})) (∏_{j=1}^{m} α(t + s_{j})(h_{j})) exp((s + t)X) ∈ S_{t+s}.

(ii) L(S) ⊇ W + ℝ^{+}X for obviously exp(W) ∪ exp(ℝ^{+}X) ⊆ S.

(iii) L(S) ⊆ W + ℝ^{+}X. This is the non-trivial part of the proof. Let Y ∈ L(S) then there exist sequences m_{j} ∈ ℝ^{+} and g_{j} ∈ S such that lim_{j} g_{j} = 1 and Y = lim_{j} m_{j} log g_{j}. Now g_{j} = a_{j} exp(t_{j}X) with t_{j} ≥ 0, lim_{j} a_{j} = 1 and lim_{j} t_{j} = 0 because G/N ≅ ℝ. Let an arbitrary W′ ∈ W be given, then there is an ε > 0 such that e^{s ad X}W ⊆ W′ for all s ∈ [0, ε]. Since lim_{j} t_{j} = 0 we have t_{j} < ε for all sufficiently large j. For these j we have

a_{j} = ∏_{i} α(s_{i}) exp(w_{i}) = ∏_{i} exp(e^{s_{i} ad X}w_{i}) ∈ ⟨exp(W′)⟩ .

Applying the formula for the Campbell-Hausdorff-multiplication ∗ we obtain

m_{j} log g_{j} = m_{j} log(a_{j} exp(t_{j}X)) = m_{j}(log a_{j} ∗ t_{j}X)
= \underbrace{m_{j} log(a_{j})}_{∈𝔫} +m_{j}t_{j}X + m_{j}r_{j}  (2)

with r_{j} ∈ 𝔫 and ∥r_{j}∥ ≤ |t_{j}| ∥log a_{j}∥ for a suitable norm on 𝔤. Now the existence of Y = lim m_{j} log g_{j} implies that m_{j}t_{j} ≥ 0 converges, hence is bounded. Thus ∥m_{j}r_{j}∥ ≤ |m_{j}t_{j}| ∥log a_{j}∥ → 0 since a_{j} → 1, and we conclude that m_{j} log a_{j} also converges. Now the globality of W′ in N yields lim_{j} m_{j} log a_{j} ∈ W′ which implies Y ∈ W′ + ℝ^{+}X. Since W′ ∈ W was arbitrary, it follows that

L(S) ⊆ ⋂_{W′∈W}(W′ + ℝ^{+}X) = W + ℝ^{+}X,  (3)

which proves (iii).

From (i)–(iii) we deduce the globality of W + ℝ^{+}X in G, because we have proved the existence of a semigroup S with the prescribed tangent wedge.

Remark 1.2. The sets S_{t} defined in the previous proof are bigger than they ought to be. Indeed, it would be sufficient to take

S̃_{t} = {α(t_{1})(g_{1}) ⋯ α(t_{n})(g_{n}) exp(tX) | 0 ≤ t_{1} ≤ t_{2} ≤ … ≤ t_{n} ≤ t, g_{i} ∈ exp(W)} in order to obtain a semigroup via S̃ = ⋃_{t≥0} S̃_{t}. But this does not change the proof of (iii). The main difference is that with our choice, the sets T_{t} = S_{t} exp(−tX) ⊆ N are subsemigroups of N whereas the corresponding sets T̃_{t} = S̃_{t} ⊆ N need not be subsemigroups of N. If N is abelian then obviously S_{t} = S̃_{t} because we may rearrange the factors α(t_{i}) exp(w_{i}) such that t_{i} ≤ t_{i+1}, but in the non-abelian case S_{t} \ S̃_{t} may be non-empty. The assumption that ⋂ W = W may not be weakened which will be shown by a counter example.

2. A counterexample

The following counterexample shows, that we cannot weaken the assumption that ⋂ W = W. We take N to be the Heisenberg group and G = N⋊ℝ the four-dimensional oscillator group where ℝ acts on N by rotations around the center Z(N). So we may identify G with the set ℂ × ℝ × ℝ where the multiplication is given by

(v, s, r)(w, t, r′) = (v + e^{ir}w, s + t + \frac{1}{2}ℑ(\overline{v}e^{ir}w), r + r′) .  (4)

We may identify 𝔤 with the set ℂ × ℝ × ℝ and let P = (1, 0, 0), Q = (i, 0, 0), Z = (0, 1, 0), R = (0, 0, 1), then {P, Q, Z} is a standard base for the Heisenberg algebra 𝔫, i.e. [P, Q] = Z, Z is central. Now we consider the following wedge:

W = ℝ^{+}(Z − P) + ℝ^{+}(Z + P).  (5)

Then W is global in N because W +(−W) is an abelian subalgebra of 𝔫, but every wedge in W wedge has central elements in its interior, hence generates the whole group N (see [1, V.4.10]), so W = {𝔫} and ⋂ W = 𝔫 ≠ W. The non-globality of W + ℝ^{+}R in G follows from the following proposition:

Proposition 2.1. Let us denote with S_{r} the slice ⟨exp(W + ℝ^{+}R)⟩ ∩ ℂ × ℝ × {r}, then for every r > 0 the set Z(N)(0, 0, 2r) = {0} × ℝ × {2r} is contained in S_{2r}, hence Z(N) ⊆ \overline{⟨exp W + ℝ^{+}R⟩}.

Proof. Let r > 0 be given. Since S_{r}(0, 0, r′ − r) ⊆ S_{r′} for r ≤ r′ we only have to consider small r > 0, so we may assume w.l.o.g. that r < π/2, i.e. sin(2r) > 0. With respect to the ℂ-factor we will steer along the following path in order to reach all points in {0} × ℝ × {2r}:

[figure: r; r; s; s cos r; s; s cos r; ℂ]

For s ≥ 0 and z ≥ 0 we compute:

S_{2r} ∋ (0, z, 0)(s, s, 0)(0, 0, r)(−2s cos r, 2s cos r, 0)(0, 0, r)(s, s, 0)
= (s, s + z, 0)(0, 0, r)(−2s cos r, 2s cos r, 0)(se^{ir}, s, r)
= (s, s + z, 0)(0, 0, r) (−2s cos r + se^{ir}, 2s cos r + s + ℑ(−s cos rse^{ir}), r)
= (s, s + z, 0)(0, 0, r) (\underbrace{−s cos r + is sin r}_{−se^{−ir}}, s + 2s cos r − \frac{s^{2}}{2} sin(2r), r)

= (s, s + z, 0) (−s, s + 2s cos r − \frac{s^{2}}{2} sin(2r), 2r)
= (0, z + 2s + 2s cos r − \frac{s^{2}}{2} sin(2r) + \frac{1}{2}ℑ(−s^{2}), 2r)
= (0, z + 2s(1 + cos r) − s^{2} \frac{sin(2r)}{2}, 2r) .

Now lim_{s→∞} z + 2s(1 + cos r) − s^{2} \frac{sin(2r)}{2} = −∞ thus \{0\} × ℝ × \{2r\} ⊆ S_{2r} for any 0 < r < π/2 which proves our claim.

As an observation, we still prove that our example is in some sense as bad as possible.

Proposition 2.2. For any r > 0 we have S_{r} = N × {r}.

Proof. First we will prove pr_{ℂ}(S_{2r}) = ℂ: We have for 0 ≤ t ≤ r and s ≥ 0:

(0, 0, t)(s, s, 0)(0, 0, r − t) ∈ S_{r} and (0, 0, t)(−s, s, 0)(0, 0, r) ∈ S_{r}.

Hence pr_{ℂ}(S_{r}) ⊇ C_{r}^{1} ∪ C_{r}^{2} with

C_{r}^{1} = {v | arg(v) ∈ [0, r]} and C_{r}^{2} = −C_{r}^{1} = {v | arg(v) ∈ [π, π + r]} .

Thus pr_{ℂ}(S_{2r}) ⊇ (C_{r}^{1} + e^{ir}C_{r}^{2}) ∪ (C_{r}^{2} + e^{ir}C_{r}^{1}). Now

C_{r}^{2} + e^{ir}C_{r}^{1} = {arg(v) ∈ [π, π + r]} + {arg(v) ∈ [r, 2r]} = {arg(v) ∈ [r, π + r]},

i.e. a halfspace E_{+}, and because of C_{r}^{2} = −C_{r}^{1} we have C_{r}^{1} + e^{ir}C_{r}^{2} = −E_{+} which proves pr_{ℂ}(S_{2r}) ⊇ ℂ. From the previous proposition we know that {0}×ℝ×{r} ⊆ S_{r} so that we may deduce S_{3r} = H × {r} by right-multiplication with appropriate elements of {0} × ℝ × {r}.

This counterexample is minimal with respect to the dimension of G, because if the hyperplane ideal 𝔫 has dimension ≤ 2 then it is either abelian or the twodimensional non-abelian algebra. In both cases every wedge W ⊆ 𝔫 is global in N, (see [1, V.4.13] for the non-abelian case). So if we take a pointed wedge W, then ⋂ W = W and if the edge H(W) ≠ {0} then W must be a halfspace. Since H(W) is ad X -invariant and 𝔫 \ H(W) is disconnected, W must be ad X -invariant itself which implies W ∈ W. Thus there is no counterexample of lower dimension. It is also possible to construct a nilpotent counterexample, just take 𝔫⋊_{D}ℝ with 𝔫 the Heisenberg-algebra, W as in the former example and D ∈ Der(𝔫) is

D = ( \begin{matrix}0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} )

with respect to the base {P, Q, Z}.

3. Applications of the globality Theorem

The following corollary ([1, VI.5.2]) enables us to prove the globality in many cases:

Corollary 3.1. Suppose W_{2} ⊆ 𝔤 is global in G and W_{1} ⊆ W_{2} satisfies

(i) W_{1} ∩ H(W_{2}) ⊆ H(W_{1}).

(ii) The analytical subgroup ⟨exp H(W_{1})⟩ is closed in G.

Then W_{1} is also global in G.

Remark 3.2. If W_{2} is pointed, then these conditions are trivially satisfied. We will now give a sufficient condition for the equality ⋂ W = W.

Definition 3.3. Let V be a vector space and W_{1}, W ⊆ V two wedges. We say that W_{1} is surrounding W (in V ), if W \ H(W) ⊆ int_{V} (W_{1}) or equivalently W ⊆ int_{V} (W_{1}) ∪ H(W).

Proposition 3.4. Assume W ⊆ 𝔫 is pointed and there exists a pointed, global wedge W_{1} ⊆ 𝔫 surrounding W in 𝔫. Then W = ⋂ W.

Proof. If v ∈ 𝔫\W then there exists a wedge W_{2} surrounding W with v ∉ W_{2}. Therefore W′ := W_{1} ∩ W_{2} is global and since W′ surrounds W, we can find an ε > 0 such that e^{[0,ε] ad X} ⊆ W′ no matter what the action of ad X on 𝔫 is. Thus W′ ∈ W and v ∉ ⋂ W which proves the assertion.

So we may state

Corollary 3.5. Let G be a Lie group containing a normal subgroup N such that G/N ≅ ℝ. Let W ⊆ 𝔫 be a pointed wedge such that there exists a global wedge W_{1} surrounding W in 𝔫. Then for every X ∈ 𝔤 \ 𝔫 the Lie wedge W + ℝ^{+}X is global in G.

Proof. Since W is pointed, contained in W_{1}, it is global in N and its edge H(W) = {0} is trivially ad X -invariant. Proposition 3.4 yields W = ⋂ W, so we may apply Theorem 1.1.

The next question which arises is: Suppose we are given a Lie wedge bounded by the hyperplane ideal 𝔫, i.e. W ⊆ 𝔫 + ℝ^{+}X for an X ∈ 𝔤 \ 𝔫. When are these wedges global? In view of corollary 3.1 one should try to find a suitable wedge W′ ⊆ 𝔫 satisfying the assumptions of Theorem 1.1 such that W′ + ℝ^{+}X ⊇ W. If we assume that W is pointed then the intersection W ∩ 𝔫 will play an important role.

4. A Lemma on pointed wedges

If (X, d) is a metric space, we may endow the set C(X) of compact subsets with the so-called Vietoris topology which is defined through the metric

d(A, B) = max (max{d(a, B) | a ∈ A}, max{d(b, A) | b ∈ B}) .

where d(a, B) = min{d(a, b) | b ∈ B} is the distance of a from B.

Proposition 4.1. Assume X is a metric space and (K_{n})_{n∈ℕ} ⊆ X a decreasing sequence of compact sets. Let K = ⋂_{n∈ℕ} K_{n}, then K_{n} → K in the Vietoris topology.

Proposition 4.2. Suppose f: X → X is a continous function and K_{n} → K in the Vietoris topology. Then f(K_{n}) → f(K) in the Vietoris topology.

Lemma 4.3. Let V be a vector space, C ⊆ V a pointed, (closed, convex) cone, H a supporting hyperplane of C and the cone C̃ ⊆ H is surrounding C ∩ H in H, i.e. int_{H}(C̃) ⊇ (C ∩ H) \ {0}. Then there exists an x ∈ V \ H such that C ⊆ C̃ + ℝ^{+}x.

Proof. Let α ∈ H^{⊥} ∩ C^{∗} and pick an arbitrary x_{0} with α(x_{0}) = 1. Pick ω ∈ int(C^{∗}) and let K = C ∩ ω^{−1}(1), i.e. K is compact and C = ℝ^{+}K. We will construct an invertible linear map ψ : V → V with the following properties:

(i) ψ|_{H} = id_{H},

(ii) ψ(K) ⊆ C̃ + ℝ^{+}x_{0}.

Since ℝ^{+}K = C we may then conclude that ψ(C) = ψ(ℝ^{+}K) ⊆ C̃ + ℝ^{+}x_{0}, hence

C ⊆ ψ^{−1}(C̃) + ℝ^{+}ψ^{−1}(x_{0}) = C̃ + ℝ^{+}x

with x = ψ^{−1}(x_{0}) because of (i). Now we construct the map ψ: We denote the projection along x_{0} onto H with pr_{H}, i.e. pr_{H}(v) = v − α(v)x_{0} since α(x_{0}) = 1. Now K is compact and so is pr_{H}(K). We denote with K_{r} the slices K ∩ α^{−1}([0, r]) which are also compact for any r > 0. Since K_{r} → K_{0} = K ∩ H ⊆ int(C̃) for r → 0 in the Vietoris topology, there is an r_{0} > 0 such that pr_{H}(K_{r_{0}}) ⊆ C̃. Since int_{H}(C̃) ≠ Ø and pr_{H}(K) is compact, we can find some w ∈ int_{H}(C̃) such that

r_{0}w + pr_{H}(K) ⊆ C̃.

Now let ψ(v) = pr_{H}(v) + α(v)w + α(v)x_{0}, then ψ|_{H} = id_{H}, ψ(x_{0} − w) = −w + w + x_{0} = x_{0}, hence ψ is surjective and therefore invertible. If v ∈ K then either α(v) ≤ r_{0} which implies

ψ(v) = \underbrace{pr_{H}(v)}_{∈_{C}} + \underbrace{α(v)w}_{∈_{C}} +α(v)x_{0} ∈ C̃ + ℝ^{+}x_{0}

or α(v) ≥ r_{0} and therefore

ψ(v) = \underbrace{pr_{H}(v) + r_{0}w}_{∈_{C}} + \underbrace{(α(v) − r_{0})w}_{∈_{C}} +α(v)x_{0} ∈ C̃ + ℝ^{+}x_{0},

thus ψ(K) ⊆ C̃ + ℝ^{+}x_{0}.

5. Globality of wedges that are bounded by a hyperplane ideal

Now we have all the tools to prove

Theorem 5.1. Let G be a Lie group containing a normal subgroup N with G/N ≅ ℝ. Suppose W ⊆ 𝔤 is a pointed wedge such that

(i) 𝔫 is a supporting hyperplane of W,

(ii) There exists a pointed wedge W_{1} ⊆ 𝔫 such that W_{1} is surrounding W ∩ 𝔫 in 𝔫 and W_{1} is global in N.

Then W is global in G.

Proof. Since W_{1} is surrounding W ∩ 𝔫 in 𝔫, we may choose a wedge W_{2} such that W_{1} is surounding W_{2} and W_{2} is surrounding W ∩ 𝔫 in 𝔫. According to Lemma 4.3 we can find an X_{0} ∈ 𝔤 \ 𝔫 such that W ⊆ W_{2} + ℝ^{+}X_{0}, the latter being global in G in view of Corollary 3.5. Now Corollary 3.1 yields the globality of W.

Remark 5.2. In view of the counterexample, it is clear that we cannot drop the assumption that there is a global wedge W′ surrounding W ∩ 𝔫. Nevertheless this condition is not necessary for globality, because if we consider the invariant wedge W in the oscillator-algebra, then this wedge is naturally global, but there is no pointed global wedge W′ surrounding W ∩ 𝔫 = ℝ^{+}Z.

6. The globality problem of the Heisenberg-algebra

Let us consider the Heisenberg-algebra 𝔤 with the standard base {P, Q, Z} and brackets [P, Q] = Z, [P, Z] = [Q, Z] = 0. We identify it with the Heisenberg group G via the Campbell-Hausdorff-multiplication. From [1] we know, that the hyperplane-subalgebras of 𝔤 play an important role. A half-space which is bounded by a subalgebra is called a halfspace-semialgebra. The following is an immediate consequence of [1, V.5.41]:

Corollary 6.1. If W is Lie generating and not contained in a halfspacesemialgebra, then W is controllable.

The hyperplane subaglebras of 𝔤 may be described as follows:

Proposition 6.2. A hyperplane 𝔥 ⊆ 𝔤 is a subalgebra iff 𝔷 ⊆ 𝔥. Any proper subalgebra is contained in a hyperplane subalgebra.

Now we are ready to solve the globality problem of the Heisenberg-algebra.

Theorem 6.3. Suppose W ⊆ 𝔤 is a Lie-wedge, then the following are equivalent:

(i) W is global in G.

(ii) W is contained in a halfspace-semialgebra.

(iii) W^{∗} ∩ 𝔷^{⊥} ≠ {0}.

Proof. The equivalence of (ii) and (iii) follows from the classification of hyperplane subalgebras in 𝔤. For proving (i)⇒(ii), we consider a global wedge W. If W is Lie generating then Corollary 6.1 proves (ii). If W is not Lie generating, then it is contained in a proper subalgebra. Hence (ii) holds in view of Proposition 6.2. The crucial part of the proof is (iii)⇒(i). We have to consider the different cases of dim H(W) seperately:

1. dim H(W) = 2. In this case W is either a hyperplane-subalgebra or a halfspace-semialgebra, hence global.

2. dim H(W) = 1. The invariance of W under e^{ad H(W)} implies

[H(W), W] ⊆ H(W) ∩ 𝔷,

thus either H(W) = 𝔷 or W ⊆ H(W) + 𝔷. In the first case, W satisfies the conditions of corollary 3.1, and in the second case W is contained in an abelian subalgebra, so W is global in any case.

3. dim H(W) = 0. Let ω ∈ W^{∗} ∩ 𝔷^{⊥}, then 𝔫 := ker ω is an abelian subalgebra and G/N ≃ ℝ. Take an arbitrary pointed W′ surrounding W in 𝔤, then W′ ∩ 𝔫 is pointed, surrounds W ∩ 𝔫 in 𝔫 and is global, since 𝔫 is abelian. Therefore Theorem 5.1 applies, proving the globality of W.

Thus everything is proved.

7. Some remarks on Lorentzian Lie groups

In [2] Levichev and Levicheva consider the Lorentzian manifold structure on a Lie group G obtained by choosing a Lorentzian wedge W ⊆ 𝔤. For a closed semigroup S ⊆ G we may define the partial order ≤_{S} on G by x ≤_{S} y iff xS ∋ y. The notion of future-distinguishability is then equivalent to the condition that S is pointed, i.e. S ∩ S^{−1} = {1}. Hence the Lorentzian manifold G is futuredistinguishing iff the Lie wedge W is global in G.

It may happen that the Lorentzian manifold G is not geodesically complete, i.e. if we denote Exp the exponential function of the affine connection on G induced by the choice of a bilinear form on 𝔤, then Exp_{1} is not defined on all of 𝔤. Nevertheless the Lie semigroup generated by a Lorentzian cone is always welldefined, so the future-distinguishability-property is independent of geodesically completeness. Thus it may happen that the future of an event is well-behaved although some geodesics are not infinitely extendable.

As a final remark, we mention that Theorem 5.1 applies to the cases considered in [2], because there the hyperplane ideal 𝔫 is either abelian or almostabelian, and therefore every wedge W′ ⊆ 𝔫 is global in the corresponding group (cf. [1, V.4.13]).

References

[1] Hilgert, J., K. H. Hofmann and J. D. Lawson, “Lie groups, Convex Cones and Semigroups,” Oxford University Press, 1989.

Mittenhuber 221

[2] Levichev A., and V. Levicheva, Distinguishability Condition and the Future Subsemigroup, Seminar Sophus Lie 2 1992, 205–212.

Fachbereich Mathematik TH Darmstadt Schloßgartenstr. 7 D-6100 Darmstadt Received October 20, 1992