Cosets for lie groups

Question

I am looking for a general way of determining cosets for $(G\times H)/H$, where $G$ and $H$ are Lie groups.

For example what are the cosets $(SU(3)\times SU(2))/SU(2)$. Is there a general method of determining it? (I am actually trying to use it to find the triviality of a fiber bundle whose base space is Grassmann and fiber is $O(n)$.)

This post imported from StackExchange Physics at 2014-06-01 18:58 (UCT), posted by SE-user user44895

Comments

@Danu: I agree it's off-topic, but the question is relevant for certain topics in quantum field theory. [At author]: Perhaps you could somehow link it to physics by motivating the question with a reference to some physics problem, c.f. spontaneous symmetry breaking?

This post imported from StackExchange Physics at 2014-06-01 18:58 (UCT), posted by SE-user JamalS

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You might be interested in this link: physicsoverflow.org/14447/coset-space-of-lie-groups

This post imported from StackExchange Physics at 2014-06-01 18:58 (UCT), posted by SE-user Hunter

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@HUNTER:It is different actually. I have asked something same before [1]:physics.stackexchange.com/q/110148

This post imported from StackExchange Physics at 2014-06-01 18:58 (UCT), posted by SE-user user44895

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@user44895 no, I don't think so. The question you refer to is about the intuition of a coset space, whereas the link I refer to is about how to calculate that in a more general way. It might be worthwile for you to read the answer given in that link.

This post imported from StackExchange Physics at 2014-06-01 18:58 (UCT), posted by SE-user Hunter

Answer 1

The way the question is phrased is a little ambiguous. How does $H$ sit inside $G\times H$ as a subgroup? If it sits inside it in the canonical way as $\{1\}\times H$, then the space of cosets is canonically isomorphic to $G$ and each coset is simply $G \times \{g\}$ for $g$ an element of $H$. I.e., for each element of H there is a different coset. There is nothing to do.

Now if $H$ sits inside a little differently, as it might in your example, since $H\subseteq G$ also, the concrete forms of the cosets will differ. But the picture will look the same, the coset space will be isomorphic to the above, it's just that the cosets will concretely be different.

The main issue is whether you have $H$ sitting inside $G\times H$ as a normal subgroup or not. In the first case above, it is normal, and the coset space happens to be a group itself. But if you have put SU(2) inside of SU(3) in any of the infinitely many diferent ways, then it is not a normal subgroup.

This post imported from StackExchange Physics at 2014-06-01 18:58 (UCT), posted by SE-user joseph f. johnson

Comments

A simple version would be in which case (cases) $(G \times H)/H$ will not be different from $G$.

This post imported from StackExchange Physics at 2014-06-01 18:58 (UCT), posted by SE-user user44895