Talk:Lie group decomposition

Cartan decomp redirect

Perhaps Cartan decomposition shouldn't redirect here. for semisimple Lie algebras, Cartan decomposition can be explained via Cartan pairs or Cartan involutions in fairly accessible terms, with some nice e.g's. IMHO, it deserves its own page. Mct mht 10:20, 27 June 2006 (UTC)[reply]

It now has it's own page. Rich Farmbrough, 17:22, 1 November 2011 (UTC).[reply]

Bruhat decompositions

It would be nice if Bruhat decompositions had their own page. Owen Jones 16:27, 5 January 2007 (UTC)[reply]

Looks like they do too. Rich Farmbrough, 17:30, 1 November 2011 (UTC).[reply]

polar decomposition??

The section "polar decomposition KAK" seems to need sorting out:

1) the linked to page does not discuss the Lie group case

2) what the linked to page describes is (what it [correctly] calls) the "matrix polar decomposition" KP (=the matrix version of the Cartan decomposition) not the KAK decomposition (note of course that the positive definite Hermitian matrices P are different to the unitary matrices K)

Shouldn't KAK be called the (Lie group analogue of the) singular value decomposition? (Kostant [Ann. Sci. ENS 1973] (1.2.1) just writes "one knows that G=KAK" without giving it a name). 129.199.2.17 (talk) 21:08, 4 January 2013 (UTC)[reply]

I think the formula G = KAK should not be called a decomposition for most Lie groups, since the map

(k_{1},a,k_{2})\mapsto k_{1}ak_{2}

is usually nowhere close to being injective - for example, if G is itself compact. (By contrast, if

G=GL_{n}(\mathbb {R} )

then the map is injective on an open dense set.) So I am removing it from this page, which is about decompositions which apply to all Lie groups... David9550 (talk) 00:29, 2 April 2013 (UTC)[reply]