1. Introduction
Let
G be a compact connected Lie group and
be an automorphism on
G [
1]. The twisted conjugate action
[
2] of
G on itself associated to
is defined as
Two elements
and
of
G are called
-conjugate if they are in one twisted orbit of
G associated to
, i.e., there exists an element
such that
Based on this, given a compact connected Lie group
G with an additional
-module structure and
T a maximal compact torus of
, An J. [
3] defined the twisted Weyl group
of
G associated to
-module and reduced the calculation of
to the action of
on
T, where
is the first nonabelian cohomology of
with coefficients in
G [
2].
Motivated by the underlying work, in this paper we consider the case of
G with an nonabelian
-module structure, where
is the one-dimensional torus. Picking a topological generator
of
, which can be regarded as an automorphism of
G [
4], we first define the twisted Weyl group
of
G associated to
and then define the twisted Weyl group
associated to
-module, where
T is a maximal compact torus in
. Closed by the definition, we study the action of
on
T and show that two elements in
T are
-conjugate if and only if they are in one
-orbit. Furthermore, we prove that
is a finite group, which is the same as the case of classical Weyl groups.
Based on the underlying properties of
, we study the action of
on the first cohomology
of the compact Lie group
with coefficients in
T [
5], and prove that the natural map
induced by the natural embedding
is a bijection. Using the result, one can reduce the calculation for
to the calculation for the orbit space
. Indeed, using this formula, one can simplify some calculations in dynamical systems theory, especially in fractional dynamics and fractional-wavelet analysis of some positive definite distributions [
6,
7,
8].
In
Section 2, we exhibit the definition of the twisted Weyl group
of
G associated to
-module together with its some properties. In
Section 3, we construct a one-to-one correspondence between the orbit space of the action of
on
and
. In
Section 4, we discuss some new developments in the field as well as its relations with amenability of groups [
9,
10,
11,
12]. For basic knowledge on compact Lie groups and twisted conjugate actions, one can refer [
2,
13,
14]; for the nonabelian cohomology of Lie groups, one can refer [
5,
15,
16,
17].
2. Twisted Weyl Groups of Compact Lie Groups
Let G be a compact connected Lie group with an -module structure and be a topological generator of . From the definition of -module structure, can be regarded as an automorphism on G. Denote by the twisted conjugate action of G associated to .
First of all, we define the twisted Weyl group of
G associated to
.
Definition 1. Definewhere T is a maximal compact torus of . It is easy to know that
and
are both closed subgroups of
G and that
is a normal subgroup of
.
Definition 2. Define is called the twisted Weyl group of G associated to δ. As an abstract group, the group operations on
are defined as following. For all
,
where
,
,
represents the equivalence class of
in
; for all
,
where
.
Next, we exhibit the definition of the twisted Weyl group of
G associated to
-module.
Definition 3. Define,where T is a maximal compact torus of . It is clear that they are both closed subgroups of
G and that
is a normal subgroup of
.
Proposition 1. Let δ be a topological generator of . Then, Proof. It suffices to prove
. Let
and set
For the underlying
, it is obvious that
. Thus, for all
,
Since
is a topological generator of
,
is dense in
. Again
is a closed subgroup of
,
Then,
,
. So,
□
By Proposition 1,
is connected. For the underlying two subgroups
and
of
G, we claim that:
Proposition 2. For a given topological generator of and a maximal compact torus T of ,
- (i)
;
- (ii)
.
Proof. (i) It suffices to show that .
Suppose that
. Then
Define
Then,
is a dense subset of
. In fact, for all
,
Since
is a topological generator of
, the set which is generated by
is a dense subset of
. Again for all
,
is dense in
. It is obvious that
is also a closed subgroup of
. Hence,
which shows that
for all
. So,
, i.e.,
(ii) Similarly, it suffices to show that .
Suppose that
. Then we have
for all
.
For
, for any
, define
Analogously,
So,
□
From Proposition 2, one can get that the subgroups
and
of
G are independent with the choice of the topological generator
of
.
Definition 4. Let δ be a topological generator of and be defined as above. Define, is called the twisted Weyl group of G associated to -module. Following, we present some properties of
.
Proposition 3. is independent with the choice of T.
Proof. Let
is another maximal compact torus of
. Then by ([
3], Proposition 2.11), there exists an element
such that
. Thus,
for all
. Hence,
Similarly, we have
Therefore,
Analogously,
Then,
which shows that
is independent with the choice of
T. □
For the reason, we write
as
and write
as
in simplified forms, respectively.
Lemma 1. Denote by Lie functor. Then, Proof. Above all, we show the first equality. It suffices to show the reverse inclusion for the clear fact that
. For all
, we have
Then,
. Now, we show
for all
. In fact,
implies that
. Since
and
are both
semisimple (see in [
3]),
is
semisimple. Hence,
. Then, we have
for all
.
Next, we show the equality
. For all
,
. Take
e for the unit of
T. Then
, i.e.,
. Thus,
and hence
. So,
Moreover,
which shows
. Then,
Therefore, □
Proposition 4. As an abstract group, is a finite group.
Proof. By Lemma 1, it is clear. □
Remark 1. In the case of classical Weyl groups, let G be a compact connected Lie group, T be a maximal torus of G, and and be the Lie algebras of G and T respectively. Denote by and the complexifications of and , and denote be the set of roots of with respect to . In analytical level, Weyl group is defined as the quotient of normalizer by centralizer . In algebraical level, Weyl group is defined as the subgroup of the orthogonal group on generated by the root reflections for . When is considered as acting on , coincides with [14]. In the case of twisted Weyl groups, let G be a compact connected Lie group with an module structure, T be a maximal torus of , and , , and described as above. In [3], An J. defined the twisted Weyl groupwhere δ is a generator of . Motivated by the underlying equality between and , one can also consider the algebraic twisted Weyl group. Since δ can be regarded as an automorphism on G, its differential can be thought as an automorphism on . DefineIf , α is called a twisted root of with respect to . Denote by the set of twisted roots of with respect to . For , the twisted root reflection is defined asDefine as the group generated by the twisted root reflections for . Similar as the proof in ([14], Theorem 4.54), one can obtain thatwhen is considered as acting on . For analytical twisted Weyl group , we have an analogous algebraical counterpart described as above. The twisted conjugate action
of
G associated to
naturally induces the action of twisted Weyl group
on
where
is a topological generator of
,
. Now, we show the following property of the action of
on
T.
Proposition 5. Let G be a compact connected Lie group associated to an -module, δ be a topological generator of , T be a maximal compact torus of and be the twisted Weyl group of G associated to -module. Then two elements of T are δ-conjugate if and only if they are in one -orbit.
Proof. Denote by
the
-orbit over
, i.e.,
⇒) If
, then there exist
,
such that
,
. Hence,
. So,
which shows that
,
. So,
are
-conjugate.
⇐) If
are
-conjugate. i.e., there exists an element
such that
. It needs to show that
are in one
-orbit, i.e., to find an element
such that
or to find an element
such that
Define
Then
, which holds for the fact
T is connected and
for all
. Again,
where
represents the automorphism induced by
. Thus,
Then by ([
2], Theorem 2.1),
where
represents the dimension of the maximal compact torus of
. Hence,
T is maximal compact torus of
. In fact, for the underlying
,
is also a maximal compact torus of
for any
, we have
which shows that
. Thus,
T and
are both torus of
. Then there exists an element
such that
. Pick
, then
.
Now we show
and
. By
we get
; by
we get
. □
3. Twisted Weyl Groups of Compact Lie Groups and Nonabelian Cohomology
In this section, we discuss the relationship between twisted Weyl group and the first nonabelian cohomology of with coefficients in G.
Let
T be a maximal compact torus of
. As
T is abelian and
acts trivially on
T,
coincides with
and
is a group homomorphism
. Thus,
naturally acts on
, i.e.,
where
. Now, we show that the definition is well-defined. It suffices to show that
.
In fact, for any
, we have
which shows that the underlying definition is well-defined. Thus, we have
Hence,
is a group homomorphism.
The natural embedding
induces the natural map
and
can be reduced to
where
is the
-orbit over
in
. Thus, we construct a correspondence between
and
by
.
Theorem 1. Let G be a compact connected Lie group associated to an -module, δ be a topological generator of , T be a maximal compact torus of and be the twisted Weyl group of G associated to -module. Then the mapis a bijection. Proof. By ([
17], Theorem 2.5), the natural map
is a surjection and hence
is also a surjection.
Now we show
is an injection. Suppose that
have the same image under
. Then, there exists an element
such that
for all
. Picking a topological generator
of
. Thus, we have
which shows that
and
are
cojugate. By Proposition 5,
and
are in one
-orbit. In other words, there exists an element
such that
which also means that there exists
such that
and
Since
is a topological generator of
, one can obtain that
for all
. Thus,
which shows that
and
are in one
-orbit. So,
and thus,
which shows that
is an injection. □
Remark 2. Let A be a general compact group [4] and G be a compact connected Lie group with an A-module structure. Our motivation for this paper is to define the twisted Weyl group of G associated A-module. For this aim, we have to deal with the existence of the maximal compact torus in , where is the the identity connected component ofHowever, for a general compact group A, maximal compact torus in may not exist [1]. Even for the existence of invariant maximal compact torus in G, it is not certain too [17]. From the discussions for the existence of maximal compact torus in and , we find the following two open problems. Problem 1. Under what conditions there exists a nontrivial maximal compact torus in ?
Problem 2. If T is a nontrivial maximal compact torus of . Can the maximal compact torus T generalize to an invariant maximal compact torus of G?
4. Discussions
In [
9], Bartholdi studied the amenability of
-set, here
is a group, which was induced by John Von Neumann in 1929. Fundamentally, the notion exhibited the following property of a group acting on a
-set
X: The
-set
X right is called amenable if there exists a
-invariant mean
m on the power set
of
X, namely a function
satisfying
and
for all
and
.
In [
9], Bartholdi presented some criterions to show a
-set
X amenable. By ([
9], Proposition 2.3), one can get that the underlying twisted Weyl group
is indeed amenable. For the amenability of twisted Weyl groups, we will study it in a sole paper.