Seminar Sophus Lie 3 (1993) 35–46

New sufficient criteria for the globality of Lie wedges

Dirk Mittenhuber^{∗}

Abstract We consider a simply connected Lie group G with Lie algebra 𝔤 and Lie semigroups S, T ⊆ G with tangent wedges L(S), resp. L(T). We give some sufficient conditions for W := L(S) + L(T) to be a global Lie wedge. The first theorem applies to the situation where G = NH is a semidirect decomposition and W is adapted to this decomposition in the sense of L(S) ⊆ 𝔫 and L(T) ⊆ 𝔥 while the second theorem applies to the case where the semigroup generated by S and T admits a product decomposition ⟨S ∪ T ⟩ = ST. As an application, we prove that in a threedimensional Lie algebra every Lie wedge that lies in the intersection of two distinct halfspace-semialgebras is global in the corresponding simply connected Lie group.

1. Introduction

First we recall some definitions and notations concerning Lie wedges, Lie semigroups and orders. In what follows, G denotes an arbitrary Lie group and 𝔤 its Lie algebra. We consider only real Lie algebras.

Definition 1.1. Suppose S ⊆ G is a subsemigroup, then we define the tangent wedge L(S) by X ∈ L(S) if and only if there exist sequences (s_{j})_{j∈ℕ} ⊆ S and m_{j} ∈ ℝ^{+} such that

lim_{j} s_{j} = 1, and X = lim_{j} m_{j} log s_{j},

where log denotes the logarithm which is well-defined on a sufficiently small neighborhood of 1. A subsemigroup S ⊆ G is a Lie semigroup iff it is reconstructible from its tangent wedge, i.e.

S = \overline{⟨exp L(S)⟩}.

Remark 1.2. Four our purposes the former definition turns out to be more convenient than the equivalent (see [1])

L(S) =^{def} \{X ∈ 𝔤 | exp ℝ^{+}X ⊆ \overline{S}\} .

∗supported by a DFG-grant

In any case it’s worth to note that L(S) = L(\overline{S}). Therefore we will be able to restrict most of our considerations to dense subsemigroups of Lie semigroups.

Definition 1.3. Let 𝔤 be a Lie algebra, W ⊆ 𝔤 is a Lie wedge, if it satisfies

\overline{W} = W, W + W = W, ℝ^{+}W = W and e^{ad H(W)}W = W,

where H(W) =^{def} W ∩ −W is called the edge of W and ad is the adjoint representation. Let G be a Lie group with Lie algebra 𝔤. A Lie wedge W ⊆ 𝔤 is global in G, if there exists a subsemigroup S ⊆ G with W = L(S). A Lie wedge is global, if it is global in the simply connected Lie group corresponding to 𝔤.

The starting point of the Lie theory of semigroups was the observation that for any subsemigroup S ⊆ G the tangent wedge L(S) ⊆ 𝔤 is a Lie wedge. Conversely, if W ⊆ 𝔤 is a Lie wedge, then S(W) =^{def} \overline{⟨exp_{G} W ⟩} is the smallest Lie semigroup with L(S(W)) ⊇ W, but the inclusion may be proper. Equality holds if and only if W is global in G. The so-called halfspace semialgebras are of special importance, because they are maximal global Lie wedges:

Definition 1.4. A halfspace-semialgebra in a Lie algebra 𝔤 is a halfspace such that its bounding hyperplane is a subalgebra. An intersection of halfspacesemialgebras is called an intersection-semialgebra.

Finally every semigroup gives rise to a left-invariant partial order on the whole group:

Definition 1.5. Let S ⊆ G be a subsemigroup, then we denote with ≤_{S} the left invariant order on G induced by S:

x ≤_{S} y iff xS ∋ y.

For x, y ∈ G the order interval [x, y]_{S} is defined as [x, y]_{S} = xS ∩ yS^{−1}.

2. A globality theorem for wedges that are adapted to a semidirect decomposition

Lemma 2.1. Suppose G is a Lie Group, N a closed normal subgroup, H a closed subgroup satisfying N ∩ H = \{1\} and NH = \overline{NH}. Assume g_{j} = n_{j}h_{j} with n_{j} ∈ N, h_{j} ∈ H and lim_{j} g_{j} = 1. Then h_{j} converges to 1 and so does n_{j}.

Proof. We consider the factor group G/N. The quotient map π: G → G/N is continuous, hence π|_{H}: H → π(H) ⊆ G/N is a surjective, continuous homomorphism onto a locally compact group. Since H ∩ N = {1} we know that π|_{H} is injective, hence bijective. Now the open mapping theorem applies and yields a continuous homomorphism ψ: π(H) → H which inverts π|_{H}. The continuity of π and g_{j} → 1 in G imply π(g_{j}) = π(h_{j}) → 1 in G/N. Therefore h_{j} = ψ(π(h_{j})) → 1 by the continuity of ψ. Thus h_{j} → 1 in G which immediately yields n_{j} = g_{j}h_{j}^{−1} → 1 in G.

Theorem 2.2. Suppose G is a Lie group, N = N̄ ⊴ G a normal subgroup and H = H̄ ≤ G a Lie subgroup such that N ∩ H = {1} and NH ⊆ G is closed. Let \overline{S} ⊆ N and \overline{T} ⊆ H be Lie semigroups with int_{H}(T) ≠ Ø and tangent wedges L(S) ⊆ 𝔫, L(T) ⊆ 𝔥. We define the following family of wedges:

W = {W′ ⊆ 𝔫 | W′ global in N , (∃t_{0} ∈ int_{H} T)(∀t ∈ [1, t_{0}]_{T}) Ad(t) L(S) ⊆ W′} .

Then the following holds: If ⋂ W = L(S) then the Lie wedge L(S) + L(T) is global in G.

Proof. Let t ∈ T, then we denote with α(t) the inner automorphism of G induced by t ∈ T, i.e. α(t)(g) = tgt^{−1} and we let

E_{t} = {α(t_{1})(s_{1}) … α(t_{n})(s_{n})s | t_{i} ∈ [1, t]_{T} , s_{i} ∈ exp L(S)}

and E = ⋃_{t∈T} E_{t}. We will prove the following:

(i) E is a semigroup because for t, t′ ∈ T we have E_{t}E_{t′} ⊆ E_{tt′}. Indeed, if g = (∏_{i=1}^{n} α(t_{i})(s_{i}))t ∈ E_{t} and g′ = (∏_{j=1}^{m} α(t′_{j})(s′_{j}))t′ ∈ E_{t′} then

gg′ = (∏_{i=1}^{n} α(t_{i})(s_{i})) t (∏_{j=1}^{m} α(t′_{j})(s′_{j})) t′

= (∏_{i=1}^{n} α(t_{i})(s_{i})) α(t) (∏_{j=1}^{m} α(t′_{j})(s′_{j})) tt′

= (∏_{i=1}^{n} α(t_{i})(s_{i})) (∏_{j=1}^{m} α(tt′_{j})(s′_{j})) tt′ ∈ E_{tt′}.

(ii) L(E) ⊇ L(S) + L(T) for obviously exp L(S) ∪ exp L(T) ⊆ E. (iii) L(E) ⊆ L(S) + L(T). This is the non-trivial part of the proof. Let Y ∈ L(E) then there exist sequences m_{j} ∈ ℝ^{+} and q_{j} ∈ E such that lim_{j} q_{j} = 1 and Y = lim_{j} m_{j} log(q_{j}). Now q_{j} = n_{j}t_{j} with n_{j} ∈ N, t_{j} ∈ T and lim_{j} n_{j} = 1 = lim_{j} t_{j}. Applying the Campbell-Hausdorff-formula we obtain

m_{j} log q_{j} = m_{j} log n_{j}t_{j} = m_{j}(log n_{j} ∗ log t_{j}) = \underbrace{m_{j} log n_{j}}_{∈𝔫} + \underbrace{m_{j} log t_{j}}_{∈𝔥} + \underbrace{m_{j}r_{j}}_{∈𝔫} .

Since L(H) ∩ L(N) = {0} the sequence m_{j} log t_{j} converges and since t_{j} ∈ T, t_{j} → 1 it follows that lim m_{j} log t_{j} ∈ L(T). Since ∥r_{j}∥ ≤ c ∥log t_{j}∥ ∥log n_{j}∥ for a suitable norm and positive constant c, we conclude lim_{j} r_{j} = 0 for ∥log n_{j}∥ → 0 and m_{j} ∥log s_{j}∥ is bounded. Thus lim_{j} m_{j} log n_{j} also exists. Now let W′ ∈ W be given, then there is a t_{0} ∈ int_{H}(T) such that Ad(t) L(S) ⊆ W′ for all t ∈ [1, t_{0}]_{T}. But t_{0} ∈ int_{H}(T) implies the existence of an open, symmetric U = U^{−1} such that Ut_{0} ∩ H ⊆ T. Therefore H ∩ U = H ∩ U^{−1} ⊆ t_{0}T^{−1}. Since t_{j} → 1, we may conclude t_{j} ≤_{T} t_{0} for all sufficiently large j. Hence n_{j} ∈ ⟨exp W′⟩ and the globality of W′ in N implies lim m_{j} log n_{j} ∈ W′. This proves Y ∈ W′ + L(T) for all W′ ∈ W, hence

L(E) ⊆ ⋂_{W′∈W} W′ + L(T) = L(S) + L(T),

by our assumption on ⋂ W which proves (iii).

From (i)–(iii) we deduce that L(S) + L(T) = L(E) is the tangent wedge of a subsemigroup E ⊆ G, therefore it is a Lie wedge and global in G.

This is a slightly more general version of Theorem 1.1 in [2]. There was an additional assumption imposed on L(S) and L(T) in order to ensure that L(S) + L(T) is a Lie wedge, but this assumption may be dropped, because the latter follows automatically from L(E) = L(S) + L(T). In fact the assumption made in [2] is necessary (but not sufficient) for ⋂ W = L(S) to hold. Investigating the condition ⋂ W = L(S), one obtains by the same arguments as in [2] the following corollary:

Corollary 2.3. If the ideal 𝔫 is almost-abelian and L(S), L(T) are pointed wedges, then ⋂ W = L(S) always holds, i.e. L(S) + L(T) is global in G.

3. A globality Theorem for a sum of pointed wedges

Proposition 3.1. Let G be a topological group and S, T ⊆ G. Let t_{j} ∈ T and s_{j} ∈ S be arbitrary sequences, then the following are equivalent:

(i) s_{j}t_{j} → 1 iff s_{j} → 1 and t_{j} → 1.

(ii) (∀V)(∃U)S ∩ UT^{−1} ⊆ V .

(iii) (∀V)(∃U)S^{−1}U ∩ T ⊆ V .

Proof. We will prove: (i)⇒(ii)⇒(iii)⇒(i). Suppose (i) holds, so s_{j}t_{j} → 1 implies s_{j} → 1 and t_{j} → 1. Now we assume that there is a V such that for any U we have S ∩UT^{−1} ⊈ V . Then we take a sequence U_{j} of 1-neighborhoods converging to 1. According to our assumption we find an element s_{j} ∈ (S ∩ U_{j}T^{−1}) \ V and an t_{j} ∈ T with s_{j} ∈ U_{j}t_{j}^{−1}. Hence s_{j}t_{j} ∈ U_{j} which implies s_{j}t_{j} → 1 but s_{j} ↛ 1 since s_{j} ∈ V^{C}, a contradiction to (i). Thus (ii) must hold. Next we suppose that (ii) holds. Let V be given then we choose V′ = V^{′−1} with V′V′ ⊆ V . Since (ii) holds, we find a U = U^{−1} ⊆ V′ such that S ∩ UT^{−1} ⊆ V′. Hence S^{−1} ∩ T U ⊆ V′ which proves (iii) since

S^{−1}U ∩ T ⊆ (S^{−1} ∩ T U)U ⊆ V′U ⊆ V′V′ ⊆ V.

Finally we assume that (iii) holds and that s_{j}t_{j} → 1. We will prove t_{j} → 1, i.e. (∀V)(∃j_{0})(∀j ≥ j_{0})s_{j} ∈ V . Let V be given, then we can find U such that S^{−1}U ∩ T ⊆ V . Choose j_{0} such that s_{j}t_{j} ∈ U for all j ≥ j_{0}. Then we have t_{j} ∈ s_{j}^{−1}U, thus t_{j} ∈ T^{−1}U ∩ S ⊆ V for all j ≥ j_{0} proving (i).

Definition 3.2. Let S, T ⊆ G be Lie semigroups. We say that S and T are well-separated, if one of the equivalent conditions in Proposition 3.1 holds. Let W_{1}, W_{2} ⊆ V be two wedges in a vector space. We call W_{1} and W_{2} wellseparated, if W_{1} ∩ −W_{2} = {0}.

Remark 3.3. If the Lie semigroups S and T are well-separated, then so are their tangent wedges L(S) and L(T). The converse is not true. It suffices to choose W_{1} pointed and controllable in order to obtain a counterexample.

Proposition 3.4. Let W_{1}, W_{2} be pointed and well-separated wedges, then their sum W_{1} + W_{2} is also a pointed wedge. In addition, we can find surrounding wedges W̃_{1} and W̃_{2} which are still well-separated.

Theorem 3.5. Let \overline{S} and \overline{T} ⊆ G be well-separated Lie semigroups with pointed tangent wedges L(S) and L(T). We further assume that ⟨S ∪ T ⟩ = ST, then L(S) + L(T) is global in G.

Proof. Let g ∈ ⟨S ∪T ⟩, then we may write g = st with s ∈ S and t ∈ T. If we have a sequence g_{j} = s_{j}t_{j} → 1 and Y = lim_{j} m_{j} log g_{j}, then the well-separateness of S and T implies t_{j} → 1 and s_{j} → 1. Let W_{S} and W_{T} be well-separated wedges surrounding L(S), resp. L(T). Then

m_{j} log(s_{j}t_{j}) = \underbrace{m_{j} log s_{j}}_{∈W_{S}} + \underbrace{m_{j} log t_{j}}_{∈W_{T}} +m_{j}r_{j}

Since W_{S} and W_{T} are well-separated, W_{S} + W_{T} is pointed, so its dual has nonempty interior. Let ω ∈ int((W_{S} + W_{T})^{∗}), then there exists a constant c > 0 such that ω(w) ≥ c ∥w∥ for all w ∈ W_{S} + W_{T}. Thus

ω(m_{j} log g_{j}) = m_{j}ω(log s_{j}) + m_{j}ω(log t_{j}) + m_{j}ω(r_{j})
≥ m_{j}(c ∥log s_{j}∥ + c ∥log t_{j}∥ − M ∥log s_{j}∥ ∥log t_{j}∥)

with some positive constant M > 0. But the function

f(x, y; c, M) = cx + cy − Mxy = M (\frac{c^{2}}{M^{2}} − (x − \frac{c}{M} ) (y − \frac{c}{M} ))

is positive for all (x, y) ∈ [0, \frac{2c}{M} ] × [0, \frac{2c}{M} ]. So we may conclude ω(Y) ≥ 0. Since

ω ∈ int((W_{S} + W_{T})^{∗}) was arbitrary, we obtain Y ∈ ( int((W_{S} + W_{T})^{∗}))^{∗} = W_{S} + W_{T}. Since this holds for arbitrary W_{S} and W_{T} (close enough to L(S), resp. L(T)), we conclude

Y ∈ ⋂_{W_{S},W_{T}} W_{S} + W_{T} = L(S) + L(T),

proving our claim.

Remark 3.6. A problem always occurring in this kind of proof is getting rid of the term m_{j}r_{j}. This can easily be done if S ⊆ Z(G) because then r_{j} ≡ 0. Another idea applies if S is already an invariant subsemigroup, i.e. Ad(G) L(S) = L(S). In that case we may conclude log s_{j} ∈ L(S) for s_{j} sufficiently close to 1. Therefore r_{j} ∈ Ad(G) log s_{j} ⊆ L(S), implying m_{j} log g_{j} ∈ L(S) + W_{T}. So we may drop the assumption that L(T) is pointed.

It may be a difficult task to prove the well-separateness of two Lie semigroups S and T. We will need the following characterization of globality from [3, II.11; IV.1.]:

Theorem 3.7. A Lie generating wedge W ⊆ 𝔤 is global in G if and only if the analytic subgroup H corresponding to H(W) = W ∩ −W is closed in G and there exists a function f ∈ C^{∞}(G) such that

⟨df(g), dλ_{g}(1)X⟩ > 0 for all X ∈ W \ H(W), g ∈ G,

i.e. df(g) ∘ dλ_{g}(1) ∈ algintW^{∗} = int_{W^{∗}−W^{∗}} W^{∗}. The function f is called W - monotone, it is constant on all cosets gH.

In view of the preceding theorem and the globality result for subwedges of global wedges in [1, VI.5.2] the following is an interesting application:

Proposition 3.8. Suppose W ⊆ 𝔤 is a global Lie wedge and 𝔥 = W ∩ −W its edge. Suppose \overline{S} and \overline{T} are Lie semigroups such that

(i) L(S) ⊆ W, (ii) L(S) ∩ 𝔥 = {0}, (iii) L(T) ⊆ 𝔥, Then \overline{S} and \overline{T} are well-separated.

Proof. We have to prove: If s_{j}t_{j} → 1, then s_{j} → 1 and thus t_{j} → 1. Since W is global, there exists a W -positive function f: G → ℝ satisfying f(1) = 0, f(gT) = f(g) and df(g) ∘ dλ_{g}(1) ∈ algintW^{∗} ⊆ int L(S)^{∗}. If s_{j}t_{j} → 1, then f(s_{j}) = f(s_{j}t_{j}) → 0 thus s_{j} → 1, because f is strictly L(S)-positive, proving our claim.

Corollary 3.9. Suppose W ⊆ 𝔤 is a global Lie wedge with edge 𝔥 = W ∩−W. Suppose \overline{S} and \overline{T} are pointed Lie semigroups with the following properties:

(i) L(S) ⊆ W, (ii) L(S) ∩ 𝔥 = {0}, (iii) L(T) ⊆ 𝔥, (iv) ⟨S ∪ T ⟩ = ST. Then L(S) + L(T) is global in G.

Proof. According to Proposition 3.8 the Lie semigroups \overline{S} and \overline{T} are wellseparated, hence we may apply Theorem 3.5.

4. Some preliminaries on almost abelian groups

Since we are interested in having nice product decompositions of the form ⟨S ∪ T ⟩ = ST, we investigate at first semigroups in the two-dimensional nonabelian Lie group because we will use these semigroups as building blocks for larger semigroups. In addition the geometric interpretation of these results is of its own interest. At first we need some information about almost abelian Lie algebras:

Definition 4.1. A Lie algebra 𝔤 is called almost abelian if there is a linear form α: 𝔤 → ℝ such that the bracket is given by

[X, Y] = α(X)Y − α(Y)X.

If α ≠ 0 then 𝔤 is called truly almost abelian.

Proposition 4.2. For every n ≥ 2 there exists up to isomorphism a unique truly almost abelian Lie algebra 𝔤 with dim 𝔤 = n and a unique Lie group G with Lie algebra 𝔤. The group is G ≅ ℝ^{n−1}⋊ℝ_{+}^{∗} with group multiplication

(x, s)(y, t) = (x + sy, st) ,

the Lie algebra is 𝔤 ≅ ℝ^{n−1}⋊ℝ with bracket [(x, s), (y, t)] = (sy − tx, 0) and the exponential function is given by

exp(x, s) = (\frac{e^{s} − 1}{s}x, e^{s}) = (f(s)x, e^{s}).

Theorem 4.3. Let 𝔤 be a Lie algebra, then the following are equivalent:

(i) 𝔤 is almost abelian. (ii) Every hyperplane is a subalgebra. (iii) Every wedge is a Lie wedge. (iii) Every wedge is a Lie semialgebra. (iv) Every Lie wedge is global.

(v) For every Lie wedge W the following equality holds:

\overline{⟨exp W ⟩} = exp W.

If W is a polyhedral wedge in an almost abelian Lie algebra, then the semigroup exp W possesses nice product decompositions. This can be verified by some elementary computations, but it is best elucidated by the following geometric considerations on the two-dimensional non-abelian Lie group: The group of motions of the real line is the unique two-dimensional non-abelian Lie group. It may be represented as G = ℝ⋊ℝ_{+}^{∗} with group operation

(b, a)(b′, a′) = (b + ab′, aa′).

The identity element is 1 = (0, 1) and the group may be visualised as the upper half plane in ℝ^{2}. The unique normal subgroup N = comm(G) corresponds to the horizontal line {(b, 1) | b ∈ ℝ} This is a convenient representation because of the geometric significance of fundamental objects such as one-parameter subgroups and left and right cosets:

• If g = (b, a) ∈ G \ {1}, then the one-parameter subgroup containing g is just the line through g and 1. The one-parameter subgroups are the lines containing 1.

[figure: v_{2}; v_{1}; g_{1}L_{2}; L_{1}g_{2}; 1; L_{2}; L_{1}; g_{1}; g_{1}g_{2}; g_{2}]

Figure 1: Geometric construction of g_{1}g_{2}

• If L is a one-parameter subgroup and g ∈ G \ L, then the left coset gL is the line parallel to L containing g.

• If L is a one-parameter subgroup and g ∈ G \ L, then the right coset Lg is the line that joins g and v_{0} where v_{0} is the point of intersection of L with the horizontal line a = 0. If L is horizontal, i.e. there is no point of intersection, then Lg is the horizontal line containing g.

• If g_{1}, g_{2} are elements of G that do not lie on the the same one-parameter subgroup, i.e. g_{i} ∈ L_{i} with L_{1} ≠ L_{2}, then one can obtain their product g_{1}g_{2} by taking the intersection of cosets (i.e. lines):

g_{1}g_{2} = g_{1}L_{2} ∩ L_{1}g_{2},

Figure 1 shows how the product g_{1}g_{2} is obtained geometrically.

A one parameter semigroup is a ray emanating from 1 and the semigroup that is generated by two such rays is the convex region inbetween. But there is a structural difference depending on whether the two rays are of the same type or not. Let us denote these ray-semigroups with S_{1}, S_{2} and the semigroup generated by them with S = ⟨S_{1} ∪ S_{2}⟩, then the following cases may occur:

(i) One of them is horizontal, w.l.o.g. S_{1} is horizontal, then S_{1}S_{2} = S_{2}S_{1} = S simply because S_{1} is invariant under all inner automorphisms of G.

(ii) None of them is horizontal, either both rays meet the horizontal axis y = 0 or none of them does. One can assume the latter w.l.o.g. because otherwise it suffices to replace S_{i} by its inverse S_{i}^{−1}. Applying an inner automorphism one can further assume that S_{1} is the vertical ray and that S_{2} is located as in figure 2. Now one can easily prove S_{1}S_{2} = S_{2}S_{1} = S. Figure 2 shows how the factors g_{i}, h_{i} ∈ S_{i} are determined for an arbitrary g ∈ S such that g_{1}g_{2} = g = h_{2}h_{1}.

[figure: S_{1}; S_{2}; 1; h_{1}; g_{1}; g_{2}; h_{2}; g; S; v_{1}; v_{2}]

Figure 2: Geometric construction of the factors such that g_{1}g_{2} = g = h_{2}h_{1}.

(iii) One of them, let’s say S_{2} meets the horizontal axis and the other one, S_{1} does not. Again applying an inner automorphism of G, one may assume w.l.o.g. that the configuration is as in figures 3,4. We have S = S_{1}S_{2}, but there is a large “gap” between S_{1}S_{2} and S_{2}S_{1}, one can only reach the points between S_{1} and the parallel ray through the intersection of S_{2} with the horizontal axis, the points on this parallel ray are not in S_{2}S_{1}. But nevertheless it’s clear that S = N_{12}S_{2}S_{1} where N_{12} = S_{1}S_{2} ∩ N denotes the horizontal ray that is the intersection of S with the normal subgroup N.

Thus one can prove by repeated applications of the preceding arguments:

Proposition 4.4. Suppose 𝔤 is an almost abelian Lie algebra, S_{i} = exp ℝ^{+}X_{i} with α(X_{i}) > 0, T_{j} = exp ℝ^{+}Y_{j} with α(Y_{j}) < 0 and N_{k} = exp ℝ^{+}Z_{k} with α(Z_{k}) = 0, i = 1 … n, j = 1 … m, k = 1 … l, then

⟨ ⋃_{i=1}^{n} S_{i} ∪ ⋃_{j=1}^{m} T_{j} ∪ ⋃_{k=1}^{l} N_{k}⟩ = N_{1} ⋯ N_{k}S_{1} ⋯ S_{n}T_{1} ⋯ T_{m}
= N_{1} ⋯ N_{k}N_{11}N_{12} ⋯ N_{nm}T_{1} ⋯ T_{m}S_{1} ⋯ S_{n},

with N_{ij} = N ∩ S_{i}T_{j}, where N denotes the maximal normal subgroup of G, N = comm(G).

5. Applications to three-dimensional groups

Now we are ready to prove the globality result stated at the beginning:

[figure: S_{1}; S_{2}; 1; v_{2}]

Figure 3: The set S_{1}S_{2} regarded as union of left-translates of S_{2}

[figure: v_{2}; S_{1}; S_{2}; S \ S_{2}S_{1}; 1; N_{12}]

Figure 4: The set S_{2}S_{1} visualised as union of left-translates of S_{1}.

Proposition 5.1. Suppose 𝔤 is a three-dimensional Lie algebra and W ⊆ 𝔤 is a pointed Lie wedge which lies in an intersection-semialgebra, i.e. W ⊆ 𝔥_{1}^{+} ∩ 𝔥_{2}^{+}, where 𝔥_{i}^{+} = ω_{i}^{−1}(ℝ^{+}), 𝔥_{i} = ker ω_{i} is a hyperplane subalgebra and 𝔥_{1} ≠ 𝔥_{2}, i.e. dim 𝔥_{1}∩𝔥_{2} = 1. Then W is global in the simply-connected Lie group corresponding to 𝔤.

Proof. Let 𝔥 = 𝔥_{1} ∩ 𝔥_{2}. If 𝔥 ∩ W = {0}, we may apply [1, VI.5.2] in order to prove the globality of W. Now suppose W ∩ 𝔥 ≠ {0}. Since W is pointed, 𝔥\W ≠ Ø and we can find an α ∈ W^{∗}\𝔥^{⊥}. Pick 0 ≠ X ∈ 𝔥∩W, 0 ≠ Y_{i} ∈ ker α∩𝔥_{i} and define T = exp ℝ^{+}X, S_{i} = exp ℝ^{+}Y_{i}. The Lie subgroups H_{i} corresponding to 𝔥_{i} are almost abelian, so we may apply proposition 4.4 in order to obtain a product decomposition of ⟨S_{i} ∪ T ⟩ = N_{i}S_{i}T. Either N_{i} = {1} or N_{i} = T S_{i} ∩ comm(H_{i}), in any case L(N_{i}) ∩ 𝔥 = {0}. If we let

S = ⟨S_{1} ∪ N_{1} ∪ S_{2} ∪ N_{2}⟩ ,

then L(S) = ℝ^{+}Y_{1}+ℝ^{+}Y_{2}+L(N_{1})+L(N_{2}), because the right-hand side is a global Lie wedge by [1, VI.5.2]. In addition L(T) = ℝ^{+}X and W ⊆ L(S) + L(T). By the choice of S_{i} and N_{i}, we immediately obtain

⟨S ∪ T ⟩ = ST,

because of T N_{i}S_{i} = N_{i}S_{i}T. Thus corollary 3.9 proves the globality of L(S)+L(T) and hence of W.

As an interesting example we consider the Lie algebra 𝔤 = sl(2, ℝ), and the Lie group Sl(2, ℝ). We denote G the simply connected Lie group corresponding to 𝔤 and π: G → Sl(2) the quotient homomorphism. The semigroup of all 2 × 2matrices in Sl(2) with nonnegative entries is denoted Sl(2)^{+} and its tangent wedge P^{+}. Then in view of [1, V.4.25], globality in Sl(2, ℝ) is now completely characterized:

Theorem 5.2. A Lie wedge W ⊆ sl(2) is global in Sl(2) iff W is a subalgebra or a conjugate of W is contained in P^{+}.

Proof. The if-part is proved in [1, V.4.25] and the only-if-part follows immediately from the preceding proposition and the fact that ⟨exp_{G} P^{+}⟩ ∩ ker π = {1}, the latter being a consequence of [1, V.4.24, V.4.40].

Obviously the same statement holds for P Sl(2, ℝ) = G/Z(G). The characterization of globality in the simply connected group G is still an open research problem.

References

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

[2] Mittenhuber, D., A globality Theorem for Lie wedges that are bounded by a hyperplane ideal, Seminar Sophus Lie 2:2 1992.

[3] Neeb, K.-H., The duality between subsemigroups of Lie groups and monotone functions, Transactions of the AMS, to appear.

Fachbereich Mathematik Arbeitsgruppe Funktionalanalysis TH Darmstadt Schloßgartenstr. 7 D-6100 Darmstadt mittenhuber@mathematik.th-darmstadt.de

Received March 15, 1993 and in final form April 28, 1993