Definition of regular Lie group action

[mnb96]

Hello,

in group theory a regular action on a G-set S is such that for every x,y∈S, there exists exactly one g such that g⋅x = y.
I noticed however that in the theory of Lie groups the definition of regular action is quite different (see Definition 1.4.8 at this link).

Is there a connection between the two definition? I would like to understand why a regular Lie group action is defined in that way.

Thanks.

 

[fresh_42]

Your link does not work for me. Can you type it out?

 

[mnb96]

Yes. According to the authors of the book in the link I posted, a Lie group action G×M→M is regular when:

(i) all orbits have the same dimension,
(ii) for each z∈M, there are arbitrarily small neighborhoods $\mathcal{U}(z)$ of $z$ such that for all $z'\in \mathcal{U}(z)$, the set $\mathcal{U}(z) \cap \mathcal{O}(z')$ is connected.

The notation $\mathcal{O}(z')$ denotes the orbit of $z'$.

 

[fresh_42]

Those are basically two different types of an operation. The properties of "acting on" are the same, thou.
The first is a conception to study groups, any groups, e.g. finite groups. In this case we have a priori no further information than the properties which define an arbitrary group.
The second one is especially conceived for Lie groups, i.e. for a well-behaved (regular) operation on manifolds M which won't make sense in the general case for any group. From special interest here are the Lie group action on itself and there on the connected component of 1. These two conditions are mainly for compatibility and comparison of different orbits. They are made to respect the given analytic structures which we don't have in general.
Condition (ii) can be used to define a path between two points of M by connecting one small neighbourhood after the other which should result in the condition of regularity for arbitrary groups. Condition (i) guarantees that orbits behave non-degenerate for in a next step the tangent spaces of orbits probably will be examined. There you want to have isomorphisms of vector spaces such that you can restrict examinations to the tangent space in 1.

 

[mnb96]

Condition (ii) can be used to define a path between two points of M by connecting one small neighbourhood after the other which should result in the condition of regularity for arbitrary groups

Could you elaborate a bit more on this?
Can't we just "define a path between two points of M by connecting one small neighborhood after the other" even without invoking any group action?

Also, if condition (ii) should result in the condition of regularity for arbitrary groups, then why can't we just use the definition of regularity for arbitrary groups (which is much easier to understand)?

 

[fresh_42]

Also, if condition (ii) should result in the condition of regularity for arbitrary groups, then why can't we just use the definition of regularity for arbitrary groups (which is much easier to understand)?

Because it would not respect the fact that we are very much interested in local behaviour and analytic properties such as differentiability, continuity or functions being analytic. Manifolds are by their construction topological spaces which behave locally like Euclidean spaces. There is no such thing as local in the general case. We have additional properties and to get specific results we have to find a way to respect these properties. The general regularity simply isn't suitable, resp. not fulfilled in the situation we want to examine. (see below.)

Can't we just "define a path between two points of M by connecting one small neighborhood after the other" even without invoking any group action?

Path is meant as part of the entire orbit ($\mathcal{U}(z) \cap \mathcal{O}(z')$). Without group action how will you define "for every x,y∈S, there exists exactly one g such that g⋅x = y" on M (=S)? You can do it by means of a metric on M. But neither have we mentioned a metric nor do we bother since we want to deduce ordinary regularity of group action. The metric properties will fall in naturally.

Could you elaborate a bit more on this?

I'll try. Since a manifold $M$ is a second-countable space there is a countable set $\{U_n)\}$of overlapping open neighbourhouds connecting two points $s, t \in M$. Let the $s_i \in U_i ∩ U_{i+1}$. We know that the orbits (of the group $G$ action) in all neighbourhoods are connected so we can find a $g_i$ with $g_i . s_i = s_{i+1}$. At the starting point we choose a $h_1 \in G$ to bring us from $s$ to $s_1$ and similar at the end a $h_2 \in G$ for the last step to $t$. In total we have $g.s = t$ with $g = h_2 ∏_{n∈ℕ} g_i h_1$.

I admit I cannot see how to get rid of the infinity. But before we start to think about transfinite induction or convergence let's turn this fact into a discovery:
The basic structures of Lie groups and manifolds are by their definitions local structures. Therefore we need to define group actions locally. Gluing open neighbourhoods brings us from one point to another. However, it doesn't make it a global property.

 

[mnb96]

Without group action how will you define "for every x,y∈S, there exists exactly one g such that g⋅x = y" on M (=S)?

Ok. I understand your point.
Is there a straightforward example of a Lie group action that is regular according to the simpler definition for arbitrary groups but that is not regular according to the definition for Lie groups? Such an example would be incredibly helpful.

I'll try. Since a manifold $M$ is a second-countable space there is a countable set $\{U_n)\}$of overlapping open neighbourhouds connecting two points $s, t \in M$. Let the $s_i \in U_i ∩ U_{i+1}$. We know that the orbits (of the group $G$ action) in all neighbourhoods are connected so we can find a $g_i$ with $g_i . s_i = s_{i+1}$. At the starting point we choose a $h_1 \in G$ to bring us from $s$ to $s_1$ and similar at the end a $h_2 \in G$ for the last step to $t$. In total we have $g.s = t$ with $g = h_2 ∏_{n∈ℕ} g_i h_1$.

Did I misunderstand something, or your example does not work for any pair of points s,t in M? It seems to me it can only work for pair of points in the same orbit. Am I right?

 

[fresh_42]

To begin with: please correct me, if you find mistakes. It's been quite some time since I've been into Lie group actions.

Is there a straightforward example of a Lie group action that is regular according to the simpler definition for arbitrary groups but that is not regular according to the definition for Lie groups? Such an example would be incredibly helpful.

Let us take a direct sum of a one-dimensional and a two-dimensional Lie group acting regular in the sense of ordinary groups on the direct sum of a straight and a plane, resp. Then we have orbits of unequal dimensions. To violate the second condition will be tough, because we would have to find not connected orbits in arbitrary small neighborhoods, i.e. a group-regular operation that may not be continuous or at least has strange local orbits. Maybe one can construct something exotic like that, probably highly discrete, but I have no idea how.

Did I misunderstand something, or your example does not work for any pair of points s,t in M? It seems to me it can only work for pair of points in the same orbit. Am I right?

You are right, I need some kind of connectivity, i.e. countable many overlapping open sets between s and t plus some transitivity in those neighbourhoods to find some $g \in G$ to get from $z$ to $z'$ in the intersection with the next neighbourhood. With that I could construct the path.

To construct counterexamples is not an easy task for we needed Lie groups, i.e. analytic groups, a strange operation (most are at least continuous) with orbits that are disjoint in arbitrary neighbourhoods to prevent us getting to the next one.

I have an example of a Lie regular operation which is not group-regular:
Consider all real orthogonal matrices with determinate 1 acting on all real orthogonal matrices which have determinate ±1. There is no group element to get from an s with determinate 1 to a t with determinate -1.

By the way, my textbook on Lie groups by V.S.Varadarajan avoids the term "regular operation" for Lie groups and speaks of $G$-spaces and transitivity instead.

 

[mnb96]

Hello again,

thanks for your help, and for bringing those interesting examples.
I have been "meditating" for some time about this issue and after reading more carefully the book I was referring to, I think I made some progress.
I now think I have a clearer idea of why the authors define a regular Lie group action in that way. In order to understand it I constructed the following action:

$$*: \mathbb{R} \times \mathbb{C}^\times \rightarrow \mathbb{C}^\times$$

where the operation of the group $\mathbb{R}$ is the addition of reals and $\mathbb{C}^\times$ is the punctured complex plane. If we represent a complex number z in polar form as: $$z=(r,\theta)$$ we can define the action as:

$$g*z = (r^{\exp(g)},\; \theta + g)$$

Such an action has the following properties:

i) points on the unit circle orbit along the unit circle itself (see attached figure, green line)
ii) points inside the (punctured) unit disk orbit along "spirals" that get arbitrarily close to the unit circle, but never reach it (red lines)
iii) points outside the unit disk orbit along spirals that get arbitrarily close to the unit circle, but never reach it (blue lines).

From this example it is clear that if we consider a neighborhood of a point z on the unit circle, the neighborhood will contain orbits of points outside the unit circle, but such orbits are essentially spirals that get "infinitely squeezed" towards the unit circle, thus we can find an arbitrarily small neighborhood that contains disconnected pieces of spirals. This is why the authors impose that regularity requires the orbits to be connected.

At this point, one may ask why is it important to avoid such cases? From what I understood from that book, the authors deliberately want "regular Lie group action" to be exactly those action that foliate the space on which they act. I haven't familiarized yet with the formal definition of foliation, but it seems that a group action that acts regularly (according to the definition for arbitrary groups) on the orbits will surely partition the space, but it may not foliate it (as in the above example).

 

[mathwonk]

I never heard of the definition you gave of regular group action. in my terminology, your definition says the action is transitive (just one orbit) and has no fixed points (the only group element fixing anything is the identity). i forget the common term for having no fixed points. the author of your book probably wanted a shorter term than "transitive with no non trivial fixed points". the problem here is just that there are not very many adjectives in common use in mathematics and so the same words get defined differently by different people. common terms used a lot are good, or nice, or regular, or something positive, with more imaginative people using terms like innocuous, or some such. there is a story of a mathematician calling some special type of object "nice" and it getting transcribed incorrectly as "mice", and they say that to this day these objects are called mice in that subject. you probably know a regular topological space is defined in a way having nothing to do with group theory, and in analysis a regular function is one thing (holomorphic) and in algebraic geometry a regular map is something different again (locally polynomial), but in both cases meaning essentially "no poles".

 

[fresh_42]

Wow, awesome attractors and a beautiful vector field!

This is why the authors impose that regularity requires the orbits to be connected.

Without having it considered in detail, however, aren't those orbits (topologically) connected?! Transitivity is lacking.

I still try to get the definition of a Lie regular operation above into context. My textbook examines analytic operations which makes a lot more sense. May it be the case that the authors call an operation regular if the orbits are regular submanifolds?

 

[zinq]

mnb96 wrote: "From what I understood from that book, the authors deliberately want "regular Lie group action" to be exactly those action that foliate the space on which they act."

A few comments on this:

1) The author uses a strictly nonstandard definition of foliation, in which she permits leaves to have varying dimensions (p. 33). But all topologists and differential geometers that I've encountered since I learned about foliations in 1970 use "foliation" strictly for the case that all leaves are of the same dimension. (And locally must fit together topologically like the disks

E^k x {pt}'s​

in the product

E^k x E^n-k​

(where here I am using E^r to denote an open disk of dimension r, and pt means any point).

2) A "regular" group action as defined in this book (p. 35) requires strictly more than that the action merely foliate the space (either in the standard sense or in the author's private nonstandard sense): It is the part ii) of the author's definition of regular that you cite above that is the extra condition:

"(ii) for each z∈M, there are arbitrarily small neighborhoods U(z) of z such that for all z′∈U(z), the set U(z)∩O(z′) is connected."​

Any smooth vector field without stationary points automatically foliates the space it's defined on via its integral curves. Such as your clever example of a vector field with a limit cycle on ℂ - {0}, which as you point out violates this extra condition (ii).

3) Where you write: "This is why the authors impose that regularity requires the orbits to be connected," (or to be exact, that the intersection of arbitrarily small neighborhoods of a point intersect an orbit in a connected piece of it), I don't see your giving any explanation of why the author imposes regularity condition.

The reason for this is so that a bit later in the book (p. 115) she can find a local cross-section to a free and regular group action that, when it intersects any orbit, the intersection occurs solely at a single point.

BUT: My impression is that this is not a good book. She spends a lot of time on the concept of a moving frame, but doesn't even know what that means, and defines it in a way that totally misses the point. Her definition of foliation is nonstandard. Etc., etc.

 

[mnb96]

Thanks a lot zinq for taking the trouble of going through the pages of that book.
After reading your post I feel that things are starting to make sense.
Let me try to summarize your points, and please, correct me if you find any incorrect statements.

1) What the author of that book is (probably) trying to achieve is to find cross-sections to Lie group actions such that the intersections with each orbit occur at a single point of the cross section. Intuitively, this allows one to establish a 1-to-1 correspondence between each point of the cross-section and each orbit, and thus to find a complete set of invariants for that group action.

2) The type of actions that allow the existence of cross-sections having the properties of "point 1)" are the ones that the authors define as "free and regular".

3) If an action foliates the space, it does not necessarily mean that such an action is regular.

4) The explicit example I gave in my previous post illustrates an action that does foliate the punctured complex plane, but that is not regular (because it violates condition ii))Regarding your personal opinion on the book, I partly agree with you, since after reading the first 30 pages of that book I did find several mistakes and incorrect statements. However, since I am willing to become more familiar with the concept/techniques of the moving frame, I would like to hear why you think the author "doesn't even know what that means".

 

[zinq]

mnb96, you have paraphrased me correctly.

First of all, a "frame" means a basis for the tangent space (of n-dimensional Euclidean space or of any n-dimensional manifold) at a point. This tangent space at any point is a copy of ℝⁿ.

A "moving frame" is a smoothly varying family of frames, one for each point of some open set. Such an open set U is taken to be small enough so that, above this open set, the tangent bundle is just the product U x ℝⁿ. Which is always possible to do.

Then, the choice of a frame at each point of U just amounts to a smooth mapping

F: U → GL(n,ℝ)

where GL(n,ℝ) is the group of invertible n x n matrices with entries in ℝ — this is the same as the space of bases for ℝⁿ (just think of all the rows of the matrix as the basis vectors).

Many local calculations concerning the manifold in question, such as its curvature, can be made from having a moving frame defined on an open set like U, so this can be a useful technique to use.

 

[mnb96]

Thanks zinq for the clarifications.

I suspect that the author of that book is just using a more modern and abstract definition of "moving frame" that seems to originate from the work of Olver et al. (see here). It is possible that under specific circumstances the definition that you gave is a special case of their general definition based on equivariant maps, but I am not sure.

 

[zinq]

Well, I strongly disapprove of mathematicians' redefining established terminology, because it just creates confusion. And make no mistake, "moving frames" is very established terminology, first used by Elie Cartan in 1937 (in French the term is "repère mobile").