Talk:Projective unitary group

Notation for trivial group

Referring to "Therefore PU(1) = U(1)/U(1) = 0"

'0' is a weird name for the trivial group. Is that conventional? Sigfpe 20:47, 23 May 2006 (UTC)

Yes. The motivation I think comes from the fact that for an abelian group (like the trivial group) the group multiplication is often considered to be addition, and so the identity is called 0. The trivial group is abelian and it contains a single element, the identity, aka 0. So it's the group whose only element is 0, and thus 0 is a common name for it. JarahE 17:06, 29 May 2006 (UTC)

I'd have used 1 here simply because multiplicative notation is more common when talking about unitary groups. It's not all that important though... Sigfpe 21:13, 31 May 2006 (UTC)

Right versus left =

Is there any reason to say "right" multiplication in the article's 1st sentence? Why not just multiplication since U(1) is the center? —Preceding unsigned comment added by 24.1.245.141 (talk) 19:55, 11 December 2008 (UTC)

"Left" multiplication would work just as well. But a common source of confusion of U(1) actions on unitary groups is that the adjoint action is inequivalent, and it's easy to get confused about which one you want in a given situation. So I specified "right" so that there'd be no confusion with the adjoint action of U(1). In this case the U(1) is embedded in the center and so the adjoint action is trivial, but there are plenty of different embeddings leading to nontrivial adjoint actions that are often considered. JarahE (talk) 16:58, 13 December 2008 (UTC)

The same, or just homotopy equivalent?

The section The topology of PU(H)) contains this sentence:

"The space CP^∞ is also the same as the infinite-dimensional projective unitary group; see that article for additional properties and discussion."

The article on the projective unitary group explains why CP^∞ and PU(H) are homotopy equivalent: They are both Eilenberg-Mac Lane K(ℤ,2) spaces.

But that is different from saying they are "the same".

In any case, the phrase "the same" is meaningless unless it is specified in which category they are equivalent in.

I do not know the answer, but I hope someone knowledgeable about this subject will correct or clarify the article.50.205.142.50 (talk) 22:29, 26 April 2020 (UTC)