on a subsemigroup of the universal covering of lie semigroups

Semigroup ForumDOI 10.1007/s00233-014-9597-9

RESEARCH ARTICLE

On a subsemigroup of the universal covering of Liesemigroups

Eyüp Kizil · Jimmie Lawson

Received: 1 October 2010 / Accepted: 22 March 2014© Springer Science+Business Media New York 2014

Abstract Let G be a connected Lie group and S a generating Lie semigroup. Animportant fact is that generating Lie semigroups admit simply connected coveringsemigroups. Denote by ˜S the simply connected universal covering semigroup of S.In connection with the problem of identifying the semigroup �(S) of monotonichomotopy with a certain subsemigroup of the simply connected covering semigroup˜S we consider in this paper the following subsemigroup

˜SL = 〈Exp(L(S))〉 ⊂ ˜S,

where Exp : L(S) → S is the lifting to ˜S of the exponential mapping exp : L(S) → S .We prove that˜SL is also simply connected under the assumption that the Lie semigroupS is right reversible. We further comment how this result should be related to theidentification problem mentioned above.

Keywords Universal covering · Lie semigroups · Monotonic homotopy

Communicated by Joachim Hilgert.

E. KizilInstituto de Ciências Matemáticas e de Computação, Universidade de São Paulo,Cx. Postal 668, São Carlos, SP CEP: 13.560-970, Brazile-mail: [email protected]

J. Lawson (B)Department of Mathematics, Louisiana State University,Baton Rouge, LA 70803, USAe-mail: [email protected]

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1 Introduction

In what follows we assume an elementary familiarity with the basic theory of Liesemigroups and their corresponding subtangent objects in the Lie algebra, the Liewedges, as it may be found, for example, in [2] and [3]. Consistent with [3] we definea Lie semigroup to be a closed subsemigroup of a Lie group that is the smallestclosed subsemigroup generated by the exponential image of its subtangent set, whichof necessity must be a Lie wedge. Let G be a connected Lie group with Lie algebrag = L(G), which we think of as the set of right invariant vector fields defined on G.Let S ⊂ G be a Lie subsemigroup with subtangent Lie wedge L(S) ⊂ g. We assumethroughout that L(S) is Lie generating, that is, the Lie (sub)algebra generated by L(S)

is the whole Lie algebra g. This implies that intS is a dense semigroup ideal in S (cf.Theorem 3.8, [3]).

Lie semigroups admit simply connected covering semigroups, and their basic theoryis laid out in Sect. 3.4 of [3]. Denote by ˜S the simply connected covering semigroup ofS and let p : ˜S → S be the covering map, which is also a semigroup homomorphism. Itfollows that the exponential mapping exp : L(S) → S lifts to a continuous exponentialmapping Exp : L(S) → ˜S, which satisfies

p(Exp(X)) = exp(X) for all X ∈ L(S).

The one-parameter semigroups γ : R+ → ˜S are precisely the maps defined by

γX (t) = Exp(t X) for X ∈ L(S).A semigroup generated by its one-parameter semigroups is said to be infinitesimally

generated. The largest closed subsemigroup of ˜S that is infinitesimally generated isgiven by

˜SL = 〈Exp(L(S))〉,which is obtained by lifting S to ˜S.

In the Lie group setting the universal covering ˜G of a connected Lie group G hasthe universal property that any continuous local homomorphism from a neighborhoodof the identity in G into a topological group H uniquely extends to a continuoushomomorphism from ˜G to H . This universal property of the simply connected coveringdoes not hold in general in the Lie semigroup setting. A semigroup �(S) with theuniversal property does always exist and a construction for it has been given in [4].In some cases it may agree with ˜SL ; indeed no specific example is known where thisis not the case. But in general �(S) is a semigroup mapping homomorphically anddensely into ˜SL , as we observe in the course of this paper.

Our knowledge of the semigroups ˜SL and �(S) is at best fragmentary at this stage,and a major purpose of this paper is to obtain in more detail some structural resultsabout the two semigroups, particularly ˜SL (Sect. 2). We know by definition that ˜S issimply connected, but do not know in general whether ˜SL is, although we expect thisto be the case. We are able, using the results of Sect. 2, to derive our main result inSect. 3, namely that ˜SL is simply connected whenever S is left or right reversible. Weexpect that results along the lines of those found in these two sections should provehelpful in identifying cases where ˜SL and �(S) agree. Also we note that if ˜SL were not

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On a subsemigroup of the universal covering of Lie semigroups

simply connected, then it could not agree with �(S), since we could build a universalcovering semigroup for which ˜SL would not exhibit the universal extension property.

The semigroup �(S) is constructed as equivalence classes of a subclass of paths inS, called monotonic paths, via an appropriate variant of homotopy for them (see thepapers by Lawson, [4,5]). The relation of being monotonically homotopic betweenmonotonic paths is an equivalence relation and the standard concatenation definesa semigroup structure on the set of equivalence classes. The semigroup �(S) thusobtained is called the semigroup of monotonic homotopy and has connections with ˜Sand ˜SL , which are discussed in Sect. 4. We also briefly allude to absolutely continuousmonotonic paths, which are right invariant trajectories of a differential equation on Sparametrized by controls. We find it appropriate to mention this connection since someresults in [1] in the control setting should be adaptable to the context of semigroupshere.

2 The semigroups ˜S and ˜SL

A Lie semigroup is a closed subsemigroup S of a Lie group G which, as a closedsubsemigroup, is generated by the images of all its one-parameter semigroups

γX : R+ −→ S, γX (t) = exp(t X).

The set of all these one-parameter semigroups may be identified with the set of theircorresponding infinitesimal generators X and hence as a set L(S) in the Lie algebrag. This set is called the subtangent set of S (at the identity) and is known to be a Liewedge (i.e., closed under addition, non-negative scalar multiplication, and invariantunder the adjoint action of the group with Lie algebra L(S) ∩ −L(S)). We say thata Lie semigroup S is generating if the smallest Lie subalgebra in g containing L(S)

is g. In what follows S will denote a generating Lie semigroup. In this setting it is astandard result that the interior of S is a dense open semigroup ideal of S.

The universal covering semigroup ˜S of a generating Lie semigroup S is treatedin [3, Sect. 3.4]. It is first shown that S is path connected, locally path connected,and semi-locally simply connected, and hence has a universal covering space ˜S withcovering map p : ˜S → S. The semigroup structure on S is next lifted to ˜S in such away that p : ˜S → S is a homomorphism.

We recall certain results from [3, Sect. 3.4] that will be used in this paper.

Theorem 2.1 ([3, Theorem 3.14] ) For every generating Lie semigroup S ⊂ G thereexists a locally compact topological semigroup ˜S and a mapping p : ˜S → S with thefollowing properties:

(1) ˜S is path-connected, locally path-connected, and π1(˜S) = {1},(2) p : ˜S → S is a covering and a semigroup homomorphism,(3) For S◦ = int(S), ˜S◦ = p−1(S◦) is a dense semigroup ideal in ˜S.

Proposition 2.2 ([3, Corollary 3.18]) For p : ˜S → S as in the preceding, the funda-mental group π1(S) may be identified with the subgroup p−1(e) of ˜S. It is contained

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in the center of ˜S and the deck transformations of the covering p : ˜S → S are givenby the left translations by elements of π1(S).

Proposition 2.3 ([3, Proposition 3.22]) For X ∈ L(S) we define γX : R+ → S by

γX (t) = exp(t X). Then X → γX , where γX denotes the lift of γX , is a bijectionL(S) → Hom(R+,˜S). Define Exp : L(S) → ˜S by X → γX (1). Then the semigroup

˜SL = 〈Exp(L(S))〉

is a neighborhood of the identity e in ˜S. It is the smallest subsemigroup topologicallygenerated by every neighborhood of e in ˜S. Moreover,

˜S = (π1(S)˜SL). (1)

The following property of ˜S will also be useful for us.

Lemma 2.4 For U open in ˜S and contained in ˜S◦ and s ∈ ˜S, the set sU = {su : u ∈U } is again an open subset of ˜S contained in ˜S◦. A similar statement holds for Us.

Proof Let x ∈ U . Note sx ∈ ˜S◦ since ˜S◦ is an semigroup ideal of ˜S. Pick an openset V ⊆ ˜S◦ containing sx such that p : V → p(V ) is a homeomorphism onto anopen subset of S◦; this is possible since p is a covering map. Pick W open containingx such that sW ⊆ V , and W ⊆ U , and p : W → p(W ) is a homeomorphism ontop(W ). Let λs denote left translation by s on ˜S and λp(s) left translation by p(s) on S.From the fact p is a homomorphism of semigroups we have the following commutingdiagram:

Wλs−−−−→ V

⏐

⏐

�

p⏐

⏐

�

p

p(W )λp(s)−−−−→ p(V )

Because λp(s) is an open map on G, hence on S◦, we conclude that p−1λp(s) p(W ) isopen. From the commutativity of the diagram and the fact that p maps V bijectivelyonto p(V ), we conclude that λs(W ) = p−1λp(s) p(W ) is an open neighborhood of sxcontained in λs(U ). We conclude that λs(U ) is open. That sU ⊆ ˜S◦ follows from thefact ˜S◦ is a semigroup ideal in ˜S. ��Corollary 2.5 The set ˜S◦

L is an open path-connected ideal of ˜SL .

Proof By the preceding lemma s˜S◦L and ˜S◦

Ls are open subsets of the semigroup ˜SL forall s ∈ ˜SL , and hence are contained in ˜S◦

L . Thus ˜S◦L is an ideal.

We note first that ˜SL is connected since it is the closure of the connected set⋃∞

n=1(Exp(L(S))n . We then use the fact that in a connected topological semigroup Twith identity any ideal I is connected. Indeed for each x ∈ I , the ideal Ix := T xT isthe continuous image of T × T , hence connected, and I = ⋃

x∈I Ix . For z ∈ I fixed,each Ix meets Iz since Ix Iz ⊆ Ix ∩ Iz , and hence the union

⋃

x∈I Ix = I is connected.

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Finally since ˜S◦L is an open subset, the restriction of p to ˜S◦

L into S◦ is a localhomeomorphism. It follows that ˜S◦

L is locally path connected. It is a standard resultthat a connected locally path connected space is path connected. ��Corollary 2.6 The semigroup ˜SL is path connected.

Proof In light of the previous corollary it is only necessary to see that each element of˜S lies on an arc that intersects ˜S◦

L . Since some neighborhood of e in ˜S lies entirely in ˜SL

and since p is a local homeomorphism, it follows from the local path connectednessof S (Theorem 2.1) that there is a path connected neighborhood N of e contained in˜SL . Since ˜S◦ is dense, there is a nonempty open set W contained in N ∩˜S◦; it followsthat W ⊆ ˜S◦

L . There thus exists a continuous curve β : [0, 1] → N such that β(0) = eand β(1) ∈ ˜S◦

L . Then for any s ∈ ˜SL , the arc γ (t) = sβ(t) stretches from s to ˜S◦L .

��Lemma 2.7 For U open in S containing the identity e,

〈S◦ ∩ U 〉sg = S◦,

where 〈A〉sg denotes the subsemigroup generated by A. Similarly, for U open in ˜Scontaining e, the identity of ˜S, 〈˜S◦

L ∩ U 〉sg is dense in ˜SL .

Proof Let T = 〈S◦ ∩ U 〉sg . For X ∈ L(S), for all sufficiently small t it will be thecase that exp(t X) ∈ U . Thus the semigroup 〈U 〉sg will contain exp(R+ X) for eachX ∈ L(S), and thus must be dense in S. Since S◦ is dense in S, it follows that S◦ ∩ Uis dense in U . It follows that T is dense in 〈U 〉sg and hence in S.

Let s ∈ S◦. Pick W = W −1 open in G containing e such that W ∩ S ⊆ U andsW ⊆ S◦. Then V = W ∩S◦ is nonempty and open and sV −1 ⊆ sW ⊂ S◦. By densityof T there exists r = sv−1 ∈ T ∩ sV −1. Then s = sv−1v ∈ T (S◦ ∩ U ) ⊆ T 2 ⊆ T .

Let U be open in ˜S containing e. The fact that ˜SL is a neighborhood of e in ˜S allowsus to assume without loss of generality that U = ˜SL ∩U . In a manner analogous to thatused in the first paragraph, one shows that Exp(R+

L(S)) is contained in 〈U 〉sg , andhence that this semigroup is thus dense in ˜SL . The density of ˜S◦ implies that U ∩ ˜S◦is dense in U , and we note that the intersection is also open, hence contained in ˜S◦

L .We now complete the proof as in the first paragraph. ��

The next lemma is a variant of the fact that ˜S = π1(S)˜SL .

Lemma 2.8 In ˜S, ˜S◦ = π1(S)˜S◦L .

Proof Let s ∈ ˜S◦. Pick U ⊆ ˜SL open in ˜S containing e such that p : U → p(U )

is a homeomorphism onto p(U ), an open neighborhood of e in S. By Lemma 2.7there exist s1, . . . , sn ∈ p(U ) ∩ S◦ such that s = p(s) = s1 · · · sn . Pick si ∈ Usuch that p(si ) = si for 1 ≤ i ≤ n and let u = s1 · · · sn . Since si ∈ S◦ for each i ,si ∈ U ∩ ˜S◦ ⊆ ˜S◦

L for each i . Hence u ∈ (s1 · · · sn−1)(U ∩ ˜S◦), an open set of ˜S byLemma 2.4, and a subset of ˜SL since it consists of products of members of U . Thusu ∈ ˜S◦

L .We note that p(u) = s = p(s). Thus s is the image of u under a deck transformation,

i.e., s = hu for some h ∈ π1(S). ��

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Proposition 2.9 For all g, h ∈ π1(S), g˜S◦L∩h˜S◦

L �= ∅and for all x ∈ ˜S, x˜S◦L∩˜S◦

L �= ∅.

Proof Set I = {

g ∈ π1(S) : g˜S◦L ∩ ˜S◦

L �= ∅}

. We first claim that I is a subgroup ofπ1(S). Clearly e ∈ I . If g˜S◦

L∩˜S◦L �= ∅, then multiplying by g−1 yields˜S◦

L∩g−1˜S◦

L �= ∅,so I is closed under inversion. If g, h ∈ I , then gs1, hs2 ∈ ˜S◦

L for some s1, s2 ∈ ˜S◦L .

Since g, h are central ghs1s2 = gs1hs2 ∈ ˜S◦L˜S◦

L ⊆ ˜S◦L , where the last inclusion and

the fact s1s2 ∈ ˜S◦L follow from Corollary 2.5.

We next note that

g˜S◦L ∩ h˜S◦

L �= ∅ ⇐⇒ h−1g˜S◦L ∩ ˜S◦

L �= ∅⇐⇒ h−1g ∈ I

⇐⇒ hI = gI.

Since by Lemma 2.8 ˜S◦ is covered by the collection of open sets {k˜S◦L : k ∈ π1(S)},

connectedness (Corollary 2.5) implies that any two of the open sets can be joinedby a chain of connecting open sets. Specifically, given g, h ∈ π1(S) there existx1, x2, . . . , xn ∈ π1(S) such that

g˜S◦L ∩ x1˜S

◦L �= ∅, x1˜S

◦L ∩ x2˜S

◦L �= ∅ , . . . , xn˜S

◦L ∩ h˜S◦

L �= ∅.

We conclude thatgI = x1 I = · · · = xn I = hI,

and hence g˜S◦L ∩ h˜S◦

L �= ∅.For x ∈ ˜S, we pick z ∈ ˜S◦

L and note xz ∈ ˜S◦ since ˜S◦ is an ideal. By Lemma 2.8xz = hs = sh for some h ∈ π1(S) and s ∈ ˜S◦

L . Since by the preceding h˜S◦L ∩ ˜S◦

L =h˜S◦

L ∩ e˜S◦L �= ∅, there exist s1, s2 ∈ ˜S◦

L such that hs1 = s2. Then xzs1 ∈ x˜S◦L since

z, s1 ∈ ˜S◦L and xzs1 = shs1 = ss2 ∈ ˜S◦

L . Thus x˜S◦L ∩ ˜S◦

L �= ∅. ��Analogous arguments show alternatively that ˜S◦

L x ∩ ˜S◦L �= ∅ for every x ∈ ˜S.

3 Reversibility and simple connectivity

In the study of ˜S and ˜SL there is an important distinction between the case where thetwo are equal and the case in which they are different. The first case occurs whenthe inclusion map from H(S) = S ∩ S−1, the group of units of S, into S induces anisomorphism between the fundamental groups and the second occurs when this is notthe case. As an example of the latter, consider the following.

Example Let C = {(x, y) ∈ R2 : 0 ≤ x, −x ≤ y ≤ x}, an additive cone in the right

half-plane, an elementary example of a Lie semigroup. The image S of C in the infinitehorizontal cylinder R×R/Z = R×S1 is a Lie semigroup in which H(S) is a singletonwhile S eventually wraps around the whole cylinder and hence has π1(S) = Z (see[3, Section 1.6]). The covering semigroup ˜S in this case may be identified with theunion of the infinite family of cones {C + (0, n) : n ∈ Z} endowed with the restrictedaddition of R

2. The semigroup ˜SL , the union of the one-parameter semigroups of ˜S,is just the original cone C . ��

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Of course the universal covering semigroup˜S of a Lie semigroup S is always simplyconnected, since the universal covering of a path connected, locally path connected,and semi-locally simply connected space always is. A natural question is whether ˜SL issimply connected; this makes sense only in the case the two are distinct. In this sectionwe give a positive solution to this question in the case S is right (or left) reversible.

Definition 3.1 A subsemigroup S is said to be right reversible (resp. left reversible)if Sx ∩ Sy �= ∅ for all x, y ∈ S (resp. x S ∩ yS �= ∅). We say that S is reversible if itis both right and left reversible.

It should be noted that Proposition 2.9 does not imply that ˜SL is a right reversiblesubsemigroup of ˜S in the sense of Definition 3.1. Since the semigroup ˜SL is obtainedfrom S through the lifting process it is more natural to assume right reversibility onS and show this implies right reversibility of ˜SL . This is what we do in the followingTheorem 3.2, which in part makes use of Proposition 2.9.

Theorem 3.2 With respect to the three semigroups S, ˜SL , and ˜S, any one of them isright reversible if and only if they all are.

Proof (1) We assume that S right reversible and show ˜SL is. Let u, v ∈ ˜SL and letx = p(u) and y = p(v). Then there exist s1, s2 ∈ S such that s1x = s2 y. Replacings1 by ts1 and s2 by ts2 for some t ∈ S◦ if necessary, we see we may assume thats1, s2 ∈ S◦. Using the fact that small neighborhoods of e are contained in ˜SL and maponto small neighborhoods of e in S, which in turn generate all elements of S◦ (Lemma2.7), we see that we may pick u1, . . . , un ∈ ˜SL such that p(u1) · · · p(un) = s1. Settings1 = u1 · · · un we have s1 ∈ ˜SL and p(s1) = s1. Similarly we pick s2 ∈ ˜SL such thatp(s2) = s2. Then p(s1u) = p(s2v), so there exists h ∈ π1S such that hs1u = s2v.Since by Proposition 2.9 ˜SL h ∩ ˜SL �= ∅, there exists r ∈ ˜SL such that rh ∈ ˜SL . Thenrhs1u = r s2v, rhs1u ∈ ˜SLu, and r s2v ∈ ˜SLv, which completes the proof.

(2) We assume that ˜SL is right reversible and let u, v ∈ ˜S. Pick z ∈ ˜S◦ and notezu, zv ∈ ˜S◦. By Lemma 2.8 we may write zu = u1h1 and zv = u2h2 for u1, u2 ∈ ˜SL

and h1, h2 ∈ π1(S). By hypothesis there exist t1, t2 ∈ ˜SL such that t1u1 = t2u2. Usingthe centrality of π1(S), we obtain

(t1h2z)u = t1h2u1h1 = t1u1h2h1 = t2u2h1h2 = t2h1u2h2 = (t2h1z)v.

Thus ˜Su ∩ ˜Sv �= ∅, so ˜S is right reversible.(3) One sees easily that the surjective homomorphic image of a right reversible

semigroup is again right reversible, so ˜S right reversible implies that S is. ��Let S be a semigroup with identity. We recall that the L- quasiorder on S is defined

by s ≤L t if s ∈ St and note that this relation is invariant under right translations.If s ∈ Ss1 ∩ Ss2, then s ≤L s1, s2. Hence if S is right reversible, the quasiorder isdown-directed or filtered, and conversely (see [3, Sect. 3.6]).

A well known fact concerning the reversibility property states that compact subsetsof groups can be translated into reversible subsemigroups (cf. Lemma 3.37, [3]). In[6] it is proved that the translation can be done by an element of the subsemigroup.

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Lemma 3.3 Let S be right reversible. Given a compact set K ⊂ ˜S, there exists anelement s ∈ ˜SL such that sK ⊂ ˜SL .

Proof By Proposition 2.9 there exists for each x ∈ K an element sx ∈ ˜S◦L such that

sx x ∈ ˜S◦L . By continuity there exists Ux an open set containing x such that sxUx ⊆ ˜S◦

L .By compactness there exist finitely many of the open sets, say {Ui }1≤i≤n , that coverK , and we have for the corresponding si that siUi ⊆ ˜S◦

L .By Theorem 3.2 ˜SL is right reversible, hence has a filtered L-quasiorder. Thus we

may find s ∈ ˜SL such that s ≤L si for 1 ≤ i ≤ n. Then for each i , there exists ti ∈ ˜SL

such that s = ti si . We then have

sK ⊆n

⋃

i=1

sUi =n

⋃

i=1

ti siUi ⊆n

⋃

i=1

ti˜S◦L ⊆ ˜S◦

L .

��With this lemma at hand we now prove our main result that ˜SL is a simply connected

semigroup following the same technique used in [6].

Theorem 3.4 Suppose the Lie semigroup S is right reversible. Then the subsemigroup˜SL of the simply connected covering semigroup ˜S is simply connected as well.

Proof Let [γ ] ∈ π1(˜SL , x0) where γ : [0, 1] → ˜SL is such that γ (0) = γ (1) = x0for some x0 ∈ ˜SL . (The choice of x0 does not matter since ˜SL is path connected byCorollary 2.6.) Since ˜S is simply connected it follows that the loop γ is homotopic in˜S to the point map to x0. Denote by H the homotopy between these two maps. Thenthe image K = Im(H) is a compact set in ˜S. Hence, there exists by Lemma 3.3 anelement g ∈ ˜SL such that g(Im(H)) ⊂ ˜SL . Observe that the mapping F(s, t) = gH(s, t) defines a homotopy in ˜SL between the loop gγ and the constant map cgx0 .

Let β(t) = gt be a path in ˜SL that connects e to g and set α(t) = Rx0(gt ) = gt x0,its (right) translation starting at x0 and ending at gx0. Let α− be the curve obtainedfrom α by reversing t . Then,

α− ∗ gγ ∗ α � α− ∗ cgx0 ∗ α � cx0

in ˜SL since (s, t) → F(s, t) is a deformation of gγ to gx0 in ˜SL . On the other hand,

t −→ α− ∣

∣[0,t] ∗ α(t)γ ∗ α∣

∣[0,t]

defines a continuous deformation of α− ∗ gγ ∗ α to γ . Hence, we get that γ � cx0 in˜SL , completing the proof. ��Remark According to Ruppert’s Theorem (Proposition 3.45, [3]) a subsemigroupS ⊂ G is (left and right) reversible:

(i) if G is a nilpotent Lie group, or

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(ii) if G is a connected Lie group with compact Lie algebra such that intS is not empty.Combining Theorems 3.2 and 3.4 with this information establishes the existenceof numerous examples in which ˜SL is a right (resp. left) reversible and simplyconnected subsemigroup of ˜S.On the other hand, [3, Proposition 3.45]) states that

(iii) A subsemigroup S ⊂ G with non-empty interior of a connected solvable Liegroup G is mid-reversible, which means G = S−1SS−1.

The following example shows that a subsemigroup of even a completely solvable Liegroup G may not be reversible, and hence Rupert’s theorem cannot be extended tocompletely solvable, much less solvable, Lie groups. Nonetheless, despite the absenceof reversibility, ˜SL remains simply connected in this example.

Example Let

G ={(

x y0 1

)

: x, y ∈ R, x > 0

}

be the nonabelian completely solvable two-dimensional connected Lie group on theright half plane and let S ⊂ G be the triangle semigroup

S ={(

x y0 1

)

∈ G : 0 < x ≤ 1, 0 ≤ y ≤ x

}

.

Then S is closed in G, the interior of S is dense in S, and S = exp(L(S)) since eachelement of S lies on a one-parameter subsemigroup of G, namely the ray through thepoint emanating from (1, 0) and stretching to the y-axis. Thus, in particular, S is a Liesemigroup with generating Lie wedge. Elementary calculations show that S is rightreversible, hence G = S−1S, but not left reversible. It follows that the subsemigroupS = S × S−1 of the completely solvable Lie group G = G × G is a Lie subsemigroupwith generating Lie wedge and dense interior. Thus we have a subsemigroup S withdense interior in a connected solvable Lie group G which is, by the item (iii), mid-reversible. Since S−1 is anti-isomorphic to S, hence not right reversible, we concludethat S is neither left nor right reversible.

Since S and hence S × S−1 are contractible, hence simply connected, it followsthat S = ˜S = ˜SL , so trivially the latter is simply connected. Hence the reversibilityassumption is not necessary in order for the semigroup ˜SL to be simply connected.Indeed, we conjecture that this may always be the case. ��

4 Monotonic paths and homotopy

Let G be a connected Lie group and S be a subsemigroup of G such that the identitye is in the closure of the interior of S. Denote by Lg : G → G, h → gh, and byRg : G → G, h → hg, the left and right translation maps on G, respectively.

We first recall some definitions from [5]. The first definition is given there for leftmonotonic paths in the case that S is a local semigroup, and we adapt the definitionto right monotonic paths for a global semigroup S.

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Definition 4.1 A continuous function α : [0, 1] → S ⊂ G is said to be a rightmonotonic path in S or simply a monotonic path if α(0) = e and

α(t2) ∈ Sα(t1) (2)

whenever t1 < t2.

A basic example of a monotonic path is a one-parameter semigroup α(t) = exp(t X)

for S a Lie semigroup and X ∈ L(S).Denote by Mon(S) the set of monotonic paths. The set Mon(S) becomes a semi-

group with the binary operation defined as follows:

(α ∗ β)(s) ={

α(2s), s ∈ [0, 1/2]β(2s − 1) · α(1), s ∈ [1/2, 1] .

Definition 4.2 A monotonic homotopy between monotonic paths α and β with α(1) =β(1) is a continuous function H : [0, 1] × [0, 1] → S ⊂ G satisfying

(1) H(s, 0) = α(s) and H(s, 1) = β(s) for 0 ≤ s ≤ 1,(2) H(1, t) = α(1) = β(1) for each t ∈ [0, 1], and(3) γt (s) = H(s, t) is a monotonic path for each t ∈ [0, 1].

We say that two monotonic paths α and β are monotonically homotopic (and writeα �m β) if there exists a monotonic homotopy between them. It follows that therelation of being monotonically homotopic is an equivalence relation and moreover,the standard concatenation defines a semigroup structure on the set of equivalenceclasses. We denote this semigroup by �(S) = Mon(S)/ �m .

The following theorem is a special case of the main result of [4][Theorem 7.2].

Theorem 4.3 Let S be a Lie semigroup and f : S ∩ U → T be a continuous localhomomorphism, f (uv) = f (u) f (v) whenever u, v, uv ∈ S ∩ U, from a neighbor-hood of the identity in S into a topological monoid T . Then f extends uniquely to ahomomorphism f : �(S) → T such that f ([α]m) = f (α(1)) if α([0, 1]) ⊆ U.

We make a connection with our earlier material.

Proposition 4.4 Let S be a Lie semigroup. Then there exists a homomorphism σ :�(S) → ˜SL sending a monotonic homotopy class [α]m to its homotopy class [α].Furthermore, the image of σ is dense in ˜SL .

Proof We have seen previously that there exists an open neighborhood ˜U of e in ˜SL

that maps homeomorphically under p onto an open neighborhood U of e in S. Let j :U → ˜U be the inverse map, which is a homeomorphism and a local homomorphism.Since S is locally path-connected, we may choose V open such that e ∈ V ⊆ U , V ispath connected, and any two paths in V from e to the same endpoint are homotopicin U . Set ˜V = j (V ). For the map j , we may take any point x ∈ V a path α from eto x in V and then j (x) will be the homotopy class [α] in ˜S. By Theorem 4.3 thereexists a unique homomorphism σ : �(S) → ˜SL such that σ([α]m) = j (α(1)) if

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α([0, 1]) ⊆ V . But in the latter case we can take for j (α(1)) the homotopy class [α]of α, by definition of j . The map sending a monotonic homotopy class [α]m to itshomotopy class [α] is easily seen to be a homomorphism from �(S) into ˜S, and wehave just seen that it agrees with σ on monotonic paths in V . Hence by uniqueness ofσ , the two must agree.

Since each one-parameter semigroup in S is a monotonic path, the image of σ isa semigroup containing Exp(L(W )), and hence by definition of ˜SL must be dense in˜SL . ��

It is conjectured in [4] that for a Lie semigroup S, �(S) = ˜SL , or more precisely inlight of Proposition 4.4 that the map [α]m to [α] is an isomorphism. This breaks downinto two parts. The question of the surjectivity of σ is equivalent to the condition ofthe following problem.

Problem 1 In a Lie semigroup S, is every path in S starting from e homotopic to amonotonic path? A related more specific question is whether every element of S lieson a monotonic path.

Problem 2 Is the map σ of Proposition 4.4 injective?

Our results in this paper have some bearing on injectivity questions. For example, inthe case that ˜SL is simply connected, it follows that the map σ must be an isomorphismif it is a covering projection. This covering projection property might be possible toestablish in the case that σ is a finite-to-one map.

There is another useful technique for attacking injectivity. Suppose that for eachpoint x ∈ S there is a distinguished monotonic path αx from e to x such that thedistinguished paths vary continuously with x . Then it is possible to show that anymonotonic path to x is monotonically homotopic to the distinguished one and henceall monotonic paths to x are monotonically homotopic. Thus �(S) may be identifiedwith the subsemigroup of S of elements lying on monotonic curves. As a simpleexample consider the case of Lie semigroups in which every element lies on a uniqueone-parameter semigroup, for example, the triangle semigroup S and the productsemigroup S = S × S−1 of Sect. 3. Then one may take the initial segments of the one-parameter semigroups as the continuous family of monotonic paths. In these examplesS = �(S) = ˜S = ˜SL . A similar, slightly more sophisticated version of this approachyields the same result for the important class of Lie semigroups called Ol’shanskiisemigroups [4, Sect. 9].

Similar considerations for guaranteeing the injectivity of σ apply to the case thatthe continuous families of monotonic paths can be constructed in ˜SL rather than S.

There are close connections between the preceding material and aspects of con-trol theory involving control systems of right-invariant vector fields on a Lie group,particularly the study of universal reachable sets and universal control systems as itappears, for example, in [1].

For a Lie semigroup S in a Lie group G, let u : [0, 1] → L(S) be integrable.Consider the differential equation on S

dg

dt= d Rg(u(t)) (3)

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E. Kizil, J. Lawson

with initial point g(0) = e, the identity. In a more compact notation, we could writethe Eq. in (3) as:

·g = u(t)g, g(0) = e.

Standard existence and uniqueness theorems on solutions of differential equationson Lie groups guarantee that there exists a unique (absolutely continuous) solutionα : [0, 1] → G of (3) such that α(0) = e. We call such a solution a right invarianttrajectory of S. Since the members of L(S) may be viewed as right invariant vectorfields on G, solutions with other initial points are obtained by right translations ofsolutions beginning at e.

Remark Right invariant trajectories are examples of right monotonic paths; indeedthey are absolutely continuous monotonic paths. The first observation follows from thefact that the solutions for constant steering functions u(·) are just one-parameter semi-groups of S (which are basic examples of monotonic paths), solutions for the piece-wise constant functions are concatenations of initial segments of the one-parametersemigroups, and from the density of the piecewise constant steering functions in themeasurable ones, the solution trajectories for general measurable controls are uniformlimits of continuous monotonic functions, which are again continuous monotonic func-tions. Being solutions of Eq. (3), they are absolutely continuous. Conversely given anabsolutely continuous monotonic path, one may right translate its derivatives back tothe identity to obtain a control equation for which it is the trajectory.

The constructions of the universal reachable set from e for the family of vector fieldsfrom L(S) bear close resemblance to the construction of �(S). Another motivationfor the line of study of this paper is the development of tools to understand betterthe connections between these constructions, a development that we hope to pursuefurther in future work. We refer to the last section of [5] for more details concerningthe connections between geometric control and the semigroup �(S).

Acknowledgments Research partially supported by Fapesp Grant No. 2012/20818-5.

References

1. Colonius, F., Kizil, E., San, L.A.B.: Martin, Covering space for monotonic homotopy of trajectories ofcontrol systems. J. Differ. Equ. 216, 324–353 (2005)

2. Hilgert, J., Hofmann, K.H., Lawson, J.D.: Lie Groups, Convex Cones, and Semigroups. Oxford Press,Oxford (1989)

3. Hilgert, J., Neeb, K.H.: Lie Semigroups and their Applications. Lecture Notes in Mathematics, vol.1552 Springer-Verlag, Berlin (1993)

4. Lawson, J.: Free local semigroup constructions. Monatsh. Math. 121, 309–333 (1996)5. Lawson, J.: Universal objects in lie semigroup theory, In: Hilgert, J., Lawson, J.D., Neeb, K.-H., Vinberg,

E. B. (eds), Positivity in Lie Theory, (De Gruyter Expositions in Mathematics), vol. 20, pp. 121–132.De Gruyter, Berlin, (1995)

6. San Martin, L.A.B., Santana, A.J.: The homotopy type of lie semigroups in semi-simple lie groups.Monatsh. Math. 136, 151–173 (2002)

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on a subsemigroup of the universal covering of lie semigroups

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