This article is rated Start-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | |||||||||||
|
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)
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)
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)
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)
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)
It would be very nice with more references. Pierreback 13:19, 9 July 2007 (UTC)
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)
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)
Okay, that is straightforward. Its a group. Thank you for your response. — Rgdboer ( talk) 22:40, 1 July 2016 (UTC)
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)
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 changes the signature of a form.
This seems wrong to me: if we have an bilinear form , then sending 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)
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:
67.198.37.16 ( talk) 20:40, 3 December 2020 (UTC)
This article is rated Start-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | |||||||||||
|
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)
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)
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)
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)
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)
It would be very nice with more references. Pierreback 13:19, 9 July 2007 (UTC)
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)
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)
Okay, that is straightforward. Its a group. Thank you for your response. — Rgdboer ( talk) 22:40, 1 July 2016 (UTC)
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)
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 changes the signature of a form.
This seems wrong to me: if we have an bilinear form , then sending 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)
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:
67.198.37.16 ( talk) 20:40, 3 December 2020 (UTC)