Talk:Representation theory of SL2(R)

Since the complexified Lie algebras agree, ${\mathfrak {su}}_{2}^{\mathbb {C} }={\mathfrak {sl}}_{2}^{\mathbb {C} }$ , the representation theory of the two is the same. Echsecutor (talk) 14:22, 8 April 2011 (UTC)[reply]

Since one is a compact Lie group, and the other isn't, the representation theory of the two groups is qualitatively quite different. The reasons that looking at the Lie algebras is not sufficient could be explained. But in any case the proposal to merge seems misconceived: the theories are not the same at all. Charles Matthews (talk) 19:18, 2 May 2011 (UTC)[reply]

Well, sorry for the bold statement. I should have phrased my suggestion more precisely:

As you correctly mentioned

{\mathfrak {su}}^{\mathbb {C} }

is the compact real form of the Lie algebra

{\mathfrak {sl}}_{2}^{\mathbb {C} }

and of course the representation theory of the corresponding groups is different.

I suggested the merge, because the article on

SU(2)

really only covers the representation theory of the Lie algebra

{\mathfrak {su}}^{\mathbb {C} }

and this part is redundant. Hence my suggestion is to take the discussion of the Lie algebra representation

{\mathfrak {su}}_{2}^{\mathbb {C} }

and either make it into a new article which should be referenced in the

SL(2)

article instead of repeating the story. Or, as indicated, merge the representation theory of

{\mathfrak {su}}_{2}^{\mathbb {C} }

(which I still think is the same as the one of

{\mathfrak {sl}}_{2}^{\mathbb {C} }

) into the

SL(2)

article completely. Anyway some content about the actual group

SU(2)

representations should be put in that article. Echsecutor (talk) 10:32, 13 May 2011 (UTC)[reply]

If these articles were about representation of Lie algebras, they certainly should be merged. Between these two cases, finite dimensional representations follow the same pattern and are equivalent in some sense, but of different Lie groups, indeed. And in the infinite dimensional case they are not equivalent at all. Do not merge. Incnis Mrsi (talk) 16:13, 22 August 2011 (UTC)[reply]