Download Seminar in Topology and Actions of Groups. Topological Groups

Seminar in Topology and Actions of Groups.
Topological Groups, Semi-Direct Products
and Relative Minimality
Meny Shlossberg
May, 2005
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Abstract
In this lecture we shall prove some theorems about topological groups.
We shall generalize the classical Heisenberg group using the concept of
semi-direct products and prove a theorem regarding relatively minimal
subgroups.
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Topological Groups
Definition 1.1. A topological group is a set G which carries a group
structure and a topology and satisfies the following two axioms:
(i) The mapping (x, y) 7→ xy of G × G into G is continuous.
(ii) The mapping x 7→ x−1 of G into G is continuous.
A group structure and a topology on a set G are
said to be compatible if they satisfy (i) and (ii).
Example 1. 1.The discrete topology on a group G is compatible with the
group structure. A topological group whose topology is discrete is called a
discrete group.
2. The trivial topology on G is compatible with the group structure of G.
3. Every normed space has an additive group structure which is compatible
with the topology induced by the norm.
4. The group GLn (R) with operation of matrices multiplication is compati2
ble with the topology induced from Rn .
5. Let G be a topological group and let H be a subgroup of G .
The subgroup H with the induced topology has a structure of a topological
group.
Lemma 1.1. Let X be a topological space. If the diagonal ∆ of X × X is
closed then X is Hausdorff .
Proof. If x 6= y then (x, y) ∈ X × X is not in the diagonal ∆, hence there is
a nbd V of x and a nbd W of y such that (V × W ) ∩ ∆ = ∅, which means
that V ∩ W = ∅
Theorem 1.2. A topological group G is Hausdorff iff the set {e} is closed.
Proof. Clearly, if G is Hausdorff, then {e} is closed.
Conversely, if {e} is closed, then the diagonal ∆ of G × G is closed,
because it is the inverse image of {e} under the continuous mapping (x, y) 7→
xy −1 we finish the proof using lemma 1.1
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Now we will prove some propositions regarding topological subgroups.
Theorem 1.3. The closure H of a subgroup H of a
topological group G is a subgroup of G.
If H is a normal subgroup of G then so is H .
Proof. Let a, b ∈ H.
The mapping f : (x, y) 7→ xy −1 is continuous on G × G and transforms
H × H into H, hence, f (H × H) = f (H × H) j f (H × H) = H, and we get
ab−1 ∈ H
Theorem 1.4. Every open subgroup of a topological group is closed.
Proof. Let H Be an open subgroup in a topological group G. H = G −
U
x6∈H xH .We know that each coset from the union is open,because of the
fact that H is open, and so is the union,hence H is closed.
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Semi-Direct Products
Theorem 2.1. Let N and L be two groups, e1 and e2 their respective identity elements.Suppose that we are given a homeomorphism y 7→ σy of L into
Aut(N ).Then:
(i) On the product set S = N ×L, the internal law of composition (x1 , y1 )(x2 , y2 ) =
(x1 σy1 (x2 ), y1 y2 ).
defines a group structure, for which j1 : x 7→ (x, e2 ) is an isomorphism of N
onto a normal subgroup of S,j2 : y 7→ (e1 , y) is an isomorphism of L onto
a subgroup of S, and pr2 : S → L is an epimorphism whose kernel is j1 (N )
and which is such that pr2 ◦ j2 = IdAut(L) .
(ii)Let f : N → G, g : L → G be two homomorphisms into a group G such
that
f (σy (x) = g(y)f (x)g(y −1 )∀x ∈ N, y ∈ L.
Then there is a unique homomorphism h : S → L such that f = h ◦ j1
and g = h ◦ j2
Proof. The associativity of the operation follows from the facts that y 7→ σy
is a homomorphism and that σy ∈ Aut(N ). Clearly, (e1 , e2 ) is the identity
element of s, and finally (x, y)(σy−1 (x−1 ), y −1 ) = (σy−1 (x−1 ), y −1 ) = (e1 , e2 )
so that (x, y) has an inverse in S.The other assertions of (i) are clear.
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On the other hand ,since (x, y) = (x, e2 )(e1 , y), a homomorphism h which
satisfies the conditions of (ii) must satisfy h(x, y) = f (x)g(y), hence is unique
if it exists,moreover it is immediate from equation 2.1 that
f (x1 σy1 (x2 )g(y1 y2 ) = f (x1 )g(y1 )f (x2 )g(y1−1 )g(y1 )g(y2 )
which shows that (x, y) 7→ (x, y −1 ) is indeed the desired homomorphism.
Corollary 2.2. The homomorphism h defined in (ii) of the previous proposition is injective iff f and g are injective and f (N ) ∩ g(L) = {e} and h is
surjective iff f (N )g(L) = G .
Definition 2.1. The group S defined in Proposition 2.1 is said to be the
external semi-direct productof N and L (relative to σ).We denote it by
S =N hL .
Theorem 2.3. Let G be a group ,L, N subgroups of G N C G. then
LN = N L is a subgroup of G if in addition N ∩ L = {e} and N L = G we
receive a homomorphism y 7→ σy of L into Aut(N ) where σy (x) = yxy −1 ,thus
the external semi-direct product S = N h L is well define and. S ∼
= G.
Proof. Take f, g from Proposition 2.1to be the homomorphisms f : N → G
and g : L → G such that f (x) = x ,g(y) = y. In this case the conditions of
Corollary 2.2 hold and h(x, y) = xy is isomorphism from S to G.
In this case say that G is the semi-direct product of N and L.
Definition 2.2. An isomorphism f of a topological group G
onto a topological group G0 is a mapping of G onto G0
which is simultaneously an isomorphism of groups and a homeomorphism.
The next proposition is an immediate consequence of the definitions and
of the properties of the product topology.
Theorem 2.4. Let L, N be two topological groups, let y 7→ σy of L into
Aut(N ) of the (non-topological) group structure of N ; and suppose that the
mapping (x, y) 7→ σy (x) of N × L into N is continuous then:
(i) On the external semi-direct product S of N and L the product of the
topologies of N and L is compatible with the group structure;
(ii) Let f : N → G and g : L → G be two continuous homomorphisms into
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a topological group G satisfying the equation 2.1 ; then the homomorphism
(x, y) → f (x)g(y) of S into G is continuous.
Remark 2.1. We can also define the external topological semi-direct
productof N and L in terms of actions of groups .The mapping π : L × N →
N defined by π(y, x) = σy (x) is a continuous action of L on N , we can say
also that L operates continuously on N through automorphisms and all the
following conditions hold:
(1) π(eL , x) = x ∀x ∈ N .
(2) π is continuous.
(3) π(t, π(s, x)) = π(ts, 7→ x) ∀x ∈ N and s, t ∈ L.
(4) π(t, xy) = π(t, x)π(t, y).
(5) π(t, eN ) = eN ∀t ∈ L.
Example 2. (i)∀y ∈ L let σy be the identity element of Aut(N ) then we retrive to ½
the
¾ (x, y)(a, b) = (xσy (a), yb) = (xa, yb).
µ notion¶of¯ direct product
¯
a b ¯
0 6= a, b ∈ R ∼
(ii)F =
= R h R∗ .
c d ¯µ
¶
a b
7→ (b, a) and the action π : R∗ × R → R is
The isomorphism is
c d
∗
defined by ∀c
Rπ(c, x) = cx.
 ∈ R , x ∈
¾
½ 1 x a ¯
¯
(iii) H =  0 1 y  ¯¯x, y, a ∈ R ∼
= (R × R) h R.
0 0 1
The group H is known
 as Heisenberg
 Group .
1 x a
The isomorphism is  0 1 y  7→ (a, y, x) and the action π : R × (R ×
0 0 1
R) → (R × R) is defined by ∀x, y, a ∈ R
π(x, (a, y)) = (a + xy, y).
Definition 2.3. Let E, F, A be abelian groups.A map w : E × F → A is
said to be biadditive if the induced mappings
: F → A, wf : E → A, wx (f ) := w(x, f ) =: wf (x)
are homomorphisms for all x ∈ E and f ∈ F .
Definition 2.4. Let E, F and a be Hausdorff abelian topological groups
and w : E × F → A be a continuous biadditive mapping.Denote by H(w) =
(A × E) h F the semidirect product (say,generalized Heisenberg group induced by w of F and the group A × E .The group operation is defined
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like this :for a pair u1 = (a1 , x1 , f1 ), u2 = (a2 , x2 , f2 ) we define u1 u2 =
(a1 + a2 + f1 (x2 ), x1 + x2 , f1 + f2 ) where, f1 (x2 = w(x2 , f1 ) then H(w) becomes a Hausdorff topological group.In the case of a normed space X and a
canonical bilinear function w : X × X ∗ → R we write H(X) instead of H(w).
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Relatively Minimal Subgroups
Definition 3.1. Let X be a subgroup of a Hausdorff topological group (G, τ ).
We say that X is relatively minimal in G if every coarser Hausdorff
¯ group
¯
topology σ ⊂ τ of G induces on X the original topology.That is, σ ¯X = τ ¯X .
Theorem 3.1. The subgroups X and X ∗ are reletively minimal in the generalized Heisenberg group H(X) = (R × X) h X ∗ for every normed space X.
We will need some lemmas to prove this theorem.
Lemma 3.2. Let G be a topological group and let e be the identity element
then: Given a neighbourhood U of e, there exist a neighbourhood V of e such
that V V ⊂ U .
Proof. Since the function f : (x, y) 7→ xy is continuous at (e, e) and f (e, e) =
e there exist a nbd O of (e, e) such that f (O) ⊂ U .There exist W1 , W2 nbd
of e such that W1 × W2 ⊂ O .Take V to be W1 ∩ W2 .
Lemma 3.3. A homomorphism f of a topological group G into a topological
group G0 is continuous iff it continuous at one point of G.
Proof. Suppose f is continuous at a point a ∈ G; then if V 0 is any nbd of
f (a), V = f −1 (V 0 ) is a nbd of a. Hence if x is any point of G, we have
f (xa−1 V ) = f (x)[f (a)]−1 V 0 , and therefore f is continuous at x.
We shall now quote without proving a theorem derived from Hahn-Banach
theorem.
Theorem 3.4. Let X be a normed space 0 6= x ∈ X, then there exist f ∈ X ∗
such that ||f || = 1 and f (x) = ||x||.
We can now prove 3.1.
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Proof. Let τ be the given topology of H(X) and suppose that σ ⊂ τ is a
coarser Hausdorff group topology on H(X). Denote by X × X ∗ the normed
space with respect to the norm ||(x, f )|| : = max {||x||, ||f ||}. We prove
in fact that the map q : (H(X), σ) → X × X ∗ , (r, x, f ) 7→ (x, f ) is continuous. This will imply that the natural retractions (H(X), σ) → X ∗ and
(H(X), σ) →
It guarantees
the ¯continuity of the identity
¯ X are continuous.
¯
¯
¯ and (X ∗ , σ ¯ ∗ ) → (X ∗ , τ ¯ ∗ ). By the inclusion we
maps¯ (X, σ ¯¯X ) → (X,
τ
X
X
¯ X ¯
get σ ¯X = τ ¯X and σ ¯X ∗ = τ ¯X ∗ . This will mean that X and X∗ are relatively
minimal subgroups in H(x). Assuming the contrary there exists a coarser
Hausdorff group topology σ on H(X) such that q : (H(X), σ) → X × X ∗ is
not continuous. Since (H(X), σ) is a Housdorff topological group,and from
the continuity of the mapping:(u, v) 7→ [u, v] = uvu−1 v −1 we can choose a σnbd V of the neutral element 0 := (0, 0X , 0X ∗ ) such that 1 := (1, 0X , 0X ∗ ) 6=
[u, v] for every u, v ∈ V .
From lemma 3.3 we know that q is not σ continuous at 0, hence there exista
a© positive δ such that q(U ) is not embedded
ª into the ball B((0X , 0X ∗ , δ) :=
∗
(x, f ) ∈ X × X : max{||x||, ||f ||} < δ for every σ-nbd U of 0. Then it
follows that q(U ) is norm-unbounded in X × X ∗ . Indeed ,by lemma 3.2 we
get that ,for every n ∈ N we can choose a σ-nbd W of 0 such that
W.W
| {z. . . W} ⊆ U.
n
As we already know ,q(W ) is not a subset of B((0X , 0X ∗ ), δ). Therefore for
every n ∈ N there exists a triple tn := (rn , xn , fn ) ∈ W such that the pair
(xn , fn ) = q(tn ) satisfies ||(xn , fn )|| := max{||xn ||, ||fn || ≥ δ in X ×X ∗ . Then
by the definition of the group operation in H(X) we get tnn = (sn , nxn , nfn )
(for some sn ∈ R). Thus, ||q(tnn )|| = ||nxn , nfn || ≥ nδ. Therefore, q(U ) is
norm unbounded . Then there exists a sequence S := {un := (an , yn , φn }n∈N
in U such that at least one of the sets A := {yn }n∈N and B := {φn }n∈N is
unbounded . Suppose
first that A is unbounded.¯ The ¯intersection VX ∗ :=
¯
∗
¯
V ∩ X is a σ X ∗ - nbd of 0X ∗ in X ∗ . Clearly,σ ¯X ⊆ τ ¯X . Therefore ,VX ∗
contains a ball B(0X ∗ , ²) of X ∗ for some ² > 0. Since A is norm-unbounded
and B(0X ∗ , ²) ⊆ VX ∗ we can derived from theorem 3.4 that the set
< A, B(0X ∗ , ²) >= {< yn , f >= f (yn ) : n ∈ N, f ∈ B(0X ∗ , ²)}
is unbounded in R, since there exits c ∈ [0, 1] such that for every n ∈ N,there
exits f ∈ B(0X ∗ , ²) such that f (yn ) = c||yn ||. In fact we have
< A, B(0X ∗ , ²) >= R
because <, >: X × X ∗ → R is bilinear and mB(0X ∗ , ²) ⊆ B(0X ∗ , ²) for every
m ∈ [−1, 1] . On the other hand, for every un = (an , yn , φn ) ∈ S ⊂ V , and
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f = (0, 0X , f ) ∈ VX ∗ ⊂ V ,the commutator [f, un ] is (f (yn ), 0X , 0X ∗ ). Hence
,[V, V ] = {[a, b] : a, b ∈ V } contains the subgroup R × {0X } × {0X ∗ }. But
then 1 ∈ [V, V ]. This contradicts our assumption. The case of unbounded
B = {φn }n∈N is similar. Indeed, observe that we have
< V, V >⊇< VX , B >⊇< B(0X , ²), B >= R
for every B(0X , ²) ⊆ VX := V ∩X. On the other hand ,[un , x] = (φn (x), 0X , 0X ∗ )
for every un ∈ S and every x := (0, X, 0X ∗ ) ∈ VX ∗ . As before this implies
that 1 ∈ [V, V ].
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References
[1] M. Megrelishvili, Generelized Heisenberg groups and Shtern’s question,
2004.
[2] N. Bourbaki, General Topology, part 2, 1966.
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