Talk:Indefinite orthogonal group

Name of this page[edit]

I've come to feel that Generalized orthogonal group is a bad name for this page. It gives the impression that the group O(p,q) is not actually a orthogonal group but some generalization thereof. What is meant, of course, is that it is a generalization of the classical orthogonal group O(n) preserving a positive-definite quadratic form on Rⁿ. Perhaps orthogonal group (indefinite signature) would be a better name. Comments or suggestions? -- Fropuff 17:35, 7 March 2006 (UTC)[reply]

What name is used for these groups in the literature? Pierreback 18:43, 8 March 2006 (UTC)[reply]

They are usually just referred to as "orthogonal groups", the same as O(n). To distinguish from O(n) they are sometimes called "noncompact orthogonal groups" (although O(n,C) is also noncompact) or "orthogonal groups with indefinite signature". -- Fropuff 19:18, 8 March 2006 (UTC)[reply]

Perhaps "indefinite orthogonal groups" is the best name for these groups? A search at www.ams.org/mathscinet gives three relevant hits with articles by e.g. A. Knapp and Peter Trapa. Pierreback 23:26, 24 April 2006 (UTC)[reply]

The name "indefinite orthogonal group" is also used in Wolf: Spaces of constant curvature p. 335. This book also have some interesting statements about these groups. Pierreback 23:32, 16 May 2006 (UTC)[reply]

This was taken care of long long ago. 67.198.37.16 (talk) 20:20, 3 December 2020 (UTC)[reply]

Title needs to be changed[edit]

In representation theory groups and generally throughout Lie theory, these groups are definitely called "indefinite orthogonal groups" (sorry for the pun!). "Noncompact orthogonal groups" is descriptive, but hardly, if ever, used. I would have changed the title here and now, but it appears that one needs administrative priviledges to do it. Is this correct? Arcfrk 12:29, 10 March 2007 (UTC)[reply]

This was taken care of long long ago. 67.198.37.16 (talk) 20:19, 3 December 2020 (UTC)[reply]

More references[edit]

It would be very nice with more references. Pierreback 13:19, 9 July 2007 (UTC)[reply]

Presumption[edit]

Suppose one has two "indefinite orthogonal transformations" (that preserve the quadratic form of a pseudo-Euclidean space), can it be shown that the composition of these transformations also satisfies this property ? In other words, given the quadratic form, is there a transformation group beyond the identity that respects the form. Note that the Hurwitz problem has restricted solutions. The structure of composition algebras is also limited. The presumption of the title of this article, though found in "reliable sources", might be addressed by a reminder of the 1890s scandal that arose with hyperbolic quaternions. Unwarranted assumptions serve no one, and this Talk provides a place to improve the article, the encyclopedia, and mathematical physics. — Rgdboer (talk) 21:44, 27 June 2016 (UTC)[reply]

What are you trying to say? The first two sentences seem to represent a question with an easy answer (yes, see classical group for preservation of bilinear forms), but then there follows seemingly unrelated sentences about "assumptions" and "presumptions" and scandals of the 1890s. I don't think I understand what you mean.

If you mean if a group can preserve two different bilinear forms on the same space, then there answer is the intersection of the groups preserving the individual forms. This may or may not be the trivial group. YohanN7 (talk) 13:08, 28 June 2016 (UTC)[reply]

Thank you for the referral to Classical groups. The construction of Aut(φ) given there is useful. Trying now to show that two elements multiply to a third.— Rgdboer (talk) 22:28, 28 June 2016 (UTC)[reply]

Okay, that is straightforward. Its a group. Thank you for your response. — Rgdboer (talk) 22:40, 1 July 2016 (UTC)[reply]

Why non-compact?[edit]

Why this group is not compact? It seems to be something really important in this contex but nobody gave a proof. — Preceding unsigned comment added by 128.178.14.131 (talk) 12:08, 20 February 2018 (UTC)[reply]

Basically that is the case because the sets of fixed norms are not compact anymore. --Jobu0101 (talk) 11:50, 6 January 2022 (UTC)[reply]

Complex indefinite groups[edit]

The article claims there is no indefinite unitary group:

The group O(p, q) is defined for vector spaces over the reals. For complex spaces, all groups O(p, q; C) are isomorphic to the usual orthogonal group O(p + q; C), since the transform $z_{j}\mapsto iz_{j}$ changes the signature of a form.

This seems wrong to me: if we have an bilinear form $x^{\dagger }diag(1,..,1,-1,..,-1)x$ , then sending $x_{i}\rightarrow ix_{i}$ for some component x_i does not change it, as (i*)i=-i^2=1. — Preceding unsigned comment added by 188.107.44.78 (talk) 16:03, 15 April 2018 (UTC)[reply]

The explanation given in the article is correct, and you appear to have produced a negative sign out of thin air. Removing the errant 'dubious' tag. Student298 (talk) 21:00, 11 December 2018 (UTC)[reply]

It's great my question has recieved some thought eight months after having been posted. My mistake was assuming the natural generalisation of “preserving a bilinear form on a real vector space” to be preserving a sesquilinear form, a mistake that would have been apparent to a careful reader by my use of the symbol

\dagger

or the word “unitary”. I hope this removes all doubts about the nature of the “thin air” i produced a minus sign from.

I still think that indefinite unitary groups are an encylopedically relevant topic that is related to indefinite orthogonal groups, and would deserve a mention somewhere on this page. --94.219.145.120 (talk) 17:23, 27 March 2019 (UTC)[reply]

I have now edited the lede to mention Unitary group#Indefinite forms and Sesquilinear form. Sorry about the slow responses. Wikipedia is not a place to get answers to questions; for that you should use Math exchange. The comments here are meant to be discussions about improving the article itself, and are ... ignored... until someone knowledgeable gets around to reading and responding to them. With additional prowling about, you would have discovered that Unitary group does cover the indefinite case (although briefly) and you could have added the sentence I just added, yourself. :-) 67.198.37.16 (talk) 20:17, 3 December 2020 (UTC)[reply]

Requested expansion: Clifford algebra[edit]

The article on Clifford algebra is mostly focused on the orthogonal group. It would be nice to have a section here about "what's different" for the indefinite orthogonal group, as well as a discussion of spinors. Also:

What are the automorphisms? (the discrete group automorphisms)? Are there any, beyond the ones that move between the connected components? i.e. automorphisms of the single component SO^+(p,q) ?
Which of these are inner automorphisms?

67.198.37.16 (talk) 20:40, 3 December 2020 (UTC)[reply]