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Coset space of Lie groups

+ 5 like - 0 dislike
2279 views

I have recently asked this question on math SE, but no one has reacted so I thought maybe I can ask it here. (I've now deleted the question on math SE.)

I've been wondering for a while how one calculates the coset space of two Lie groups. For instance:
\begin{equation}
\mathrm{SO(3)} / \mathrm{SO(2)} \cong S^2 \tag{1}
\end{equation}
In most of the physics-oriented books I have read, they give a very intuitive reason why the above isomorphism is true. See for instance Einstein Gravity in a Nutshell by A. Zee:

As an example, the coset manifold $SO(3)/SO(2)$ is the familiar sphere $S^2$. Why? Let us go slow here. Every point $P$ on the sphere $S^2$ is uniquely associated with a unit vector $\hat{u}$ pointing from the center of the sphere to $P$. Denote by $\hat{z}$ the unit vector pointing to the north pole, that is, the unit vector pointing in the $z$ direction. Our first thought might be to associate the point $P$ with the rotation $g$ (that is, an element of $G = SO(3)$) that rotates $\hat{z}$ into $\hat{u}$. The problem is that the rotation $g$ is not uniquely determined. Denote by $H = SO(2)$ the subgroup of $G$ consisting of all rotations about the $z$-axis. Then two rotations $g_1$ and $g_2$ related by $g_1 = g_2h$, with $h$ an arbitrary element of $H$, would both rotate $\hat{z}$ into $\hat{u}$. In other words, $\hat{u} = g_1\hat{z} = g_2 h\hat{z} = g_2\hat{z}$. Thus, the point $P$ is not to be associated with the element $g_1$, but with the entire equivalence class $g_1$ belongs to. In other words, $P$ does not specify uniquely the rotation that would take $\hat{z}$ into the direction vector $\hat{u}$ associated with $P$.

This all makes perfect sense to me, but I was wondering if there is a more algebraic way to calculate the coset space? For instance, the book I'm reading from somehow (magically?) obtains equation $(1)$ from the fact that $\mathrm{SO(3)}$ acts on $S^2$ transitively. But I don't understand why one can conclude equation $(1)$ from the way $\mathrm{SO(3)}$ acts on $S^2$?

I still see the coset space as:

\begin{equation}
\mathrm{SO(3)} / \mathrm{SO(2)} = \{ g \cdot \mathrm{SO(2)} \mid g \in \mathrm{SO(3)}\}
\end{equation}

even though I know that $\mathrm{SO(2)}$ is not a normal subgroup of $\mathrm{SO(3)}$. Is the above correct?

asked Apr 10, 2014 in Mathematics by Hunter (510 points) [ revision history ]
edited Apr 11, 2014 by Hunter

You know the dimension of $SO(3)/SO(2)$, so you really just need to compute its Euler characteristic.

1 Answer

+ 5 like - 0 dislike

It is a well known theorem: when a Lie group $G$ acts smoothly and transitively on a connected manifold $M$, for a fixed $x\in M$, there is a Lie subgroup  $H \subset G$ defined by the group of elements $g\in G$ such that $gx=x$. It is possible to prove that changing $x\in M$, $H$ does not change. The theorem states that $G/H$ is diffeomorphic to $M$.

The fact that $H$ is (or is not) a normal subgroup is irrelevant here, since we are interested in the differentiable manifold structure of $G/H$. If we also want that this manifold receives a Lie group structure from $G$, we have to require that $H$ is a normal (Lie) subgroup of $G$.

Notice that a given manifold can be realized as different quotients $G/H$ if  there are different groups acting smoothly and transitively on it.

In the considered case $SO(3)$ acts smoothly and transitively on $S^2$ and $H= SO(2)$.

So, it is not so simple to compute the quotient, because the theorem works along  the other direction: One starts from a manifold equipped with a group of transformations and realizes the manifold as a quotient.

However the quotients of physically relevant Lie groups  are computed. See for instance

R. Glimore, Lie Groups, Lie Algebras, and some of Their Applications

A.O. Barut R. Raczka, Theory of group representations and applications.

answered Apr 11, 2014 by Valter Moretti (2,025 points) [ revision history ]
edited Apr 11, 2014 by Valter Moretti

Many thanks V. Moretti (again)! Of the two books you recommend, which one has a more emphasis on differential geometry? I'm interested to understand the grand picture of differential geometry and Lie theory (and topology).

The former is a "practical" book explicitly written for physicists. The latter is more mathematically minded, in some parts at least, however it was written for physicists, too.

If you are interested in books on these subjects stressing differential geometry aspects, I suggests  the classic:

F.W. Warner. Foundations of Differentiable Manifolds and Lie Groups

some chapters of the more advanced:

G.E. Bredon, Introduction to compact transformation groups

and also some chapters of

O' Neill, Semi Riemannian Geometry

Nice answer :-). BTW, if there are any SE posts of yours or others you would like to take have here, you can request them to be imported here.

@Dilation, I can safely say (in case V. Moretti is too humble) that most/all answers that V. Moretti has given on physics SE and maths SE are worthy of being imported to this forum.

@Hunter I agree, and I have "Import all of V. Moretti's posts" on my Physics Overflow To-Do list. 

@Hunter Thanks Hunter! I asked for importing some answers of mine indeed. At least those referring to graduate level questions. However it is not so simple to decide which ones truly deserve to be imported, I joined physics SE community just 4 months ago...

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