Talk:Clifford algebra

message: The nice thing about this page ... is the way of doing the commutative diagram, which is cool.
Otherwise, this is really not up to 2004 standards of WP exposition.
Charles Matthews 16:40, 20 Sep 2004 (UTC)
thread_title: Early thread

message: Well okay, someone has gone through and switched the sign convention back to v2 = +Q(v). See my comment at Wikipedia talk:WikiProject Mathematics/Conventions -- Fropuff 15:18, 2005 Apr 26 (UTC)
In physics and algebra the v2 = +Q(v) sign convention is almost the only one used. In differential geometry the + and - conventions are about equally common, with the - convention favored 
mainly by those working in index theory. 
By the way, the book by Lawson and Michelsohn should not be used as a reference for spinors: it contains some rather bizarre definitions that are different from everyone elses, and there are several errors in the theorems they state. R.e.b. 16:43, 26 Apr 2005 (UTC)
thread_title: Sign convention

message: Hello, after proposing a rewrite of the hypercomplex number a couple of weeks ago (see Talk:Hypercomplex number) and a little bit of asking around, I've just replaced the article with a rewrite. There was a brief mention about Clifford algebras there before ("The Clifford algebras are another family of hypercomplex numbers." - nothing else), which I expanded a little bit. However, I am not versed with Clifford algebra, and would ask if anyone could have a look at the new hypercomplex number article version and see whether it can be improved. I appreciate any comment or help. Thanks, Jens Koeplinger 22:22, 31 July 2006 (UTC)
replies: text: Update: A nice and short high-level overview or introduction into Clifford algebras was just given here: http://en.wikipedia.org/w/index.php?title=Hypercomplex_number&oldid=119138089 - Is there any chance to incorporate a similar less-technical overview into the current article, and bring the more rigorous mathematical treatment later? This would follow an approach where articles begin on a high-level and in general terms, and become increasingly more technical the further one reads on. Thanks, Koeplinger 19:58, 31 March 2007 (UTC)
thread_title:  [[Hypercomplex number]] rewrite, mention of [[Clifford algebra]]s 

message: The section "Spinors" says "suppose that p+q=2n is even" (discussing spaces with signature (p,q)).  I suggest that this definition of n is uncommon: it is much more common to define n=p+q, this being the dimension of the space.  Thus, in this same article, in the section "Basis and dimension" we have "If the dimension of V is n...", and just below, in the section "Examples": "...where n=p+q is the dimension of the vector space...".  I suggest changing the section "Spinors" to bring this in line with the above, any possible objections? 131.111.8.96 22:18, 19 September 2006 (UTC)
thread_title:  small notation inconsistency 

message: Total noob here.  In the intro it is stipulated that
v^2 = Q(v).
However, Q is a quadratic form -- i.e. it maps VxV to the base field K -- and the algebra's multiplication operation maps VxV to V.  In other words, the left side v^2 of the above expression is an element of the algebra, while the right hand side is an element of the base field.  Some clarification would be nice.  (The same can be said about the subsequent expression uv + vu = 2B(u,v).)
:The algebra operation maps VxV not to V but to Cl(V) which is an algebra containing both V and K. It would be better to write v2 = Q(v)1 where 1 is the identity element of the algebra (1 is not an element of V nor is Q(v)1). Actually the base field K can be embedded in the algebra via k\mapsto k\cdot 1 so this is unambiguous. -- Fropuff 05:23, 4 May 2007 (UTC)
thread_title:  Confusing Intro 

message: I would like to add a section after the Clifford algebra#Examples:  real and complex Clifford algebras that constructs planar quaternions, quaternions and dual quaternions used to represent planar displacements, spatial rotations and spatial displacements of rigid bodies as Clifford algebras.  I will use vectors in R3 with the usual Euclidean metric and the negative sign for the Clifford multiplication, and select the even sub algebra to obtain quaternions.  A similar construction, but using a degenerate metric yields quaternions that represent planar displacements, and finally the same construction on R4 with a degenerate metric yields dual quaternions.  I hope this is viewed as a useful contribution to this article.Prof McCarthy (talk) 03:29, 16 August 2011 (UTC)
replies: text: I just added the section on quaternions and propose to added similar sections for dual quaternions and planar quaternions under the same heading.Prof McCarthy (talk) 05:57, 16 August 2011 (UTC)
replies: text: The material on rigid body kinematics might better appear at Geometric algebra, or a subarticle of that, which deals with some of the geometric interpretations and uses to which Clifford algebras can be put.  The convention there is to use a positive sign for a Euclidean metric, a negative one for additional hyperbolic directions (hence my edit yesterday, for which I apologise, not having seen you were proposing to use the opposite convention).  Jheald (talk) 11:15, 17 August 2011 (UTC)

text: Not a problem at all, I appreciate the help.  The goal here was to show how Euclidean geometry and the construction of a Clifford algebra yield quaternions and dual quaternions.  The information on rigid body kinematics appears in the articles on screw theory and dual quaternions.  I understand the issue that the current version of this article does not allow for a degenerate quadratic form and in fact implies that they are not used.  I am not enthusiastic about moving this material to the article on Geometric algebra, but if that is preferred I will do it. Prof McCarthy (talk) 17:12, 17 August 2011 (UTC)

text: One more thing, part of the reason for adding this explicit Clifford algebra based construction was a controversy that can be found on the discussion page of the article on dual quaternions, in which someone's web-page proposed that the dual unit may not commute with the quaternion units.  This made it into the Wikipedia article and confused several editors, and I am sure many readers as well. Prof McCarthy (talk) 17:18, 17 August 2011 (UTC)
replies: text: Fair enough.  I'd be interested in your assessment of the Conformal Geometric Algebra proposed by Hestenes and various others.  This essentially adds a second non-spatial direction to the dual quaternions, in a way that appears to generally neaten up the system, allowing translations to be represented using the same "sandwich" form as rotations (and rotations about a point), also happening to integrate in the whole of inversive geometry and the "screws by reflection" approach currently mentioned in passing in the screw theory article; and as a side bonus giving very tidy product-of-blades algebraic forms for spheres and circles and their intersections.  It's currently on my slate of articles to aim to work up (eventually), though I'm a bit heavily committed in real life just at the moment.  But if you're working on the screw theory article, it probably be worth examining there, particularly in the context of comparing and contrasting with the dual quaternion approach.  Jheald (talk) 10:18, 18 August 2011 (UTC)
replies: text: Thank you for your patience with this contribution on quaternions and dual quaternions.  I have met David Hestenes and several of his colleagues over the years and understand their enthusiasm for Geometric Algebra.  They have taken quite a broad view that extends far beyond my range.  Regarding Conformal Geometric Algebra, I can say that in the particular case of embedding spatial displacements in four dimensional space such that translations in a hyperplane are approximated by rotations, effectively about axes at infinity, actually works quite well.  Furthermore, the Clifford algebra constructed for this system yields Clifford's original biquaternions.  However, I am not able to judge the extensions of this idea that Geometric Algebra allows.  As for screws by reflection, this is an important tool in the geometry of relative spatial displacements, but I have been slow to update that particular section of the article.  I will pay more attention to it. Prof McCarthy (talk) 13:37, 18 August 2011 (UTC)
thread_title:  Section on Clifford algebras and rigid body kinematics 

message: The passage: " It turns out that every one of the algebras Cℓp,q(R) and Cℓn(C) is isomorphic to a matrix algebra over R, C, or H or to a direct sum of two such algebras." Has a major problem and a minor problem. Firstly, it leaves open the possibility that a Clifford algebra may be the direct sum of a matrix ring over C and a matrix ring over H. Using A⊕A is meant to clarify this. Secondly, the use of "over F" is overloaded here: a standard interpretation of "a matrix algebra over F" means that you are talking about the ring of matrices with entries from F as an F algebra. In our case here, we never talk about an H algebra, and in a few cases the matrix ring over C is an R algebra rather than a C algebra. Saying "with entries from" is both descriptive of the object and neutral as to whether the field is R or C.
My revision doesn't fully succeed in bringing the sentence in line with the standard interpretation, but I felt any more qualifications would be overcomplicated. (Eg. ... or the matrix algebra M(C) over R or...) In any case I think it's less likely to mislead the reader about the base field. Rschwieb (talk) 17:00, 17 December 2011 (UTC)
replies: text: Sorry, I missed the first point - that one's worth having clear.  The second point does not seem to be making an intelligible distinction between overloads to me: you seem to be describing the same thing, so what you're saying is going over my head.  When we are dealing with a matrix ring with H entries, is it not also an H-algebra? When is a matrix ring over C not a C-algebra? A separate point: we may want to say "full matrix ring". — Quondumtc 18:41, 17 December 2011 (UTC)

text: That the definitions are different is obvious, but that they lead to anything different is less so. That H is not a field does complicate the use of the wording: one does have to assume a definition expanded to an algebras over rings. I guess my intuition was that any matrix algebra with elements of field K must be a K-algebra, and I suppose the generalization to non-commutative rings like H seems so natural (there are different definitions?) that I felt this would be implied, which of course I shouldn't do: it hasn't been defined.  As to classification, an algebra can be over many things, e.g. every C-algebra is an R-algebra, and we can call it by either name. It makes more sense in classification to use a standard field.  In my mind a shortcoming of Classification of Clifford algebras is: that for (and only for) even n, Cℓn(C) is a real Clifford algebra is not even mentioned but left for the curious reader to discover by inspection, is not outlining the classification properly. In conclusion, I fully concede your point. I'm sure initially I interpreted exactly "a matrix with entries" in the phrase "a matrix algebra over". You might want to cast your eye over Matrix ring for exactly this (ab)use of the phrase. — Quondumtc 04:33, 18 December 2011 (UTC)
thread_title:  Elaboration 

message: From a mathematical point of view this is a nice article, especially that description of universality. From the physical point of view there is missing something. Before mentioning the Dirac matrices, one should mention the Pauli-matrices as well, since they are a representation of the Clifford-algebra in an Euclidean R3 (the Duffin-Kemmer-Petiau matrices of physics are not). And creation and annihilation operators of quantum Fermi-Dirac statistics are another example.  — Preceding unsigned comment added by 130.133.155.68 (talk) 16:57, 24 September 2012 (UTC)
thread_title:  Use in Physics 

message: Hi!
I think it would help if the maps in the exact sequences were given names and/or descriptions. My guess is that the first two are basically trivial injections, and that the last one is trivial too (everything->identity operator on V)? The second to last one follows from the canonical map from V to Cl(V)? The field K is the same thing as k times the identity in Cl(V) for all k in K? Blue links to kernel and exact sequence would probably help too. Given that the subject is presented in such generality (the "characteristic 2 disclaimer is everywhere"), and with some degree of claim of completeness, unnecessarily compact notation is uncalled for. As is, parts of the text cares a lot more about "not saying anything provably wrong" and "not forgetting special cases" than "explicitly saying whats right". I am not asking for a readable exposition for an eighth-grader. But, I guess I am asking for painfully spelled out definitions and plenty of blue links. There are things that are wrong in a strict sense (i.e. there is no norm defined through a (even non-degenerate) non-positive definite quadratic form), but these errors pass through the otherwise meticulous "correctness" filter because the abuse of notation and terminology is so obvious to the experienced. Such errors are generally caught too by those knowing something already about the subject. If these implicit definitions and understandings are slightly beyond the present level of understanding (or need reminding of), then they are unlikely to be picked up correctly by the reader. 
I know that this is asking for a lot, but I feel the article and the subject is worth it. (I did have concrete suggestions too;)) YohanN7 (talk) 13:32, 4 January 2013 (UTC)
replies: text: While we're on this topic, I want to throw out there that it looks like "Clifford group" could probably be taken and transplanted into its own page. It is not a vital organ to explaining Clifford algebras, so I think we should consider splitting it out. Rschwieb (talk) 14:53, 4 January 2013 (UTC)
replies: text: I agree that it makes sense to move this to its own article. As a group, it stands on its own; it does not require the context of a Clifford algebra. —Quondum 22:10, 1 March 2014 (UTC)
thread_title:  [[Clifford algebra#Clifford group|Clifford group]] 

message: In Élie Cartan (1966), The Theory of Spinors, I found the term reversal being defined as an improper rotation, whereas in this article it is defined as a particular antiautomorphism, something that is entirely different. The latter sense presumably alludes to the effect of reversing the ordering a product of 1-vectors.  The former (as an improper rotation) might be more interesting in the context of geometric algebra.  Should we clarify the multiple use of the term (or even remove it)?  Also, I find it strange that Élie Cartan is not cited in this article. —Quondum 19:54, 20 November 2014 (UTC)
thread_title:  Use of "reversal" 

message: In my opinion the sentence "Clifford algebras are a type of associative algebra" is not the most appropriate for signifying the intended meaning. I propose
           "Clifford algebras constitute/comprise a subset of associative algebras"
or the simple
           "a Clifford algebra is an associative algebra" 
SoSivr (talk) 14:19, 23 December 2014 (UTC)
replies: text: The "simple" version above has the disadvantage that it seems to assert that every associative algebra is a Clifford algebra.  I've updated the statement to say something more meaningful/definitive.  —Quondum 20:35, 20 December 2016 (UTC)
thread_title: First Line

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thread_title:  External links modified 

message: The tensor algebra has two coalgebras on it, one of which is a Hopf algebra. Is the universal construction of the Clifford algebra from the the tensor algebra compatible with the Hopf algebra? If so, can this be articulated? If not, why not, and can further deformations be made to make it compatible? 67.198.37.16 (talk) 20:59, 4 May 2019 (UTC)
thread_title:  Hopf algebra? 

message: Lars Ahlfors and W.K. Clifford assume basis elements are imaginary units with negative one for their squares. See Lars Ahlfors (1985) "Mobius transformations and Clifford numbers", (pages 65 to 73 in Differential Geometry and Complex Analysis, H.E. Rauch Memorial Volume, I. Chavel & H.M. Farkas editors, Springer books). Expansion of the category of Clifford algebras to include basis elements that square to plus one came with those who would include split-complex numbers and split-quaternions. These latter algebras are composition algebras and are an ill-fit for traditional Clifford algebra. — Rgdboer (talk) 02:47, 2 December 2019 (UTC)
thread_title:  Traditional definition 

message: An editor has asked for a discussion to address the redirect BRD algebra. Please participate in the redirect discussion if you wish to do so.  signed, Rosguill talk 19:58, 30 January 2020 (UTC)
thread_title:  "BRD algebra" listed at [[Wikipedia:Redirects for discussion|Redirects for discussion]] 

message: I am inclined to replace the notation "Cℓ" with "Cl" due to font issues, notwithstanding its use in some texts.  In the context, there is no ambiguity.  Any objections?  —Quondum 17:25, 29 March 2020 (UTC)
replies: text: —Quondum 03:21, 10 April 2020 (UTC)
thread_title:  Notation "Cℓ" 

message: In the section "Examples: real and complex Clifford algebras", it should also be mentioned that:
*  is isomorphic to the algebra of split-complex numbers;
*  is isomorphic to the algebra of split-quaternions;
*  is isomorphic to the algebra of biquaternions.
129.104.241.214 (talk) 01:39, 11 November 2023 (UTC)
replies: text: Yes, this seems correct. 67.198.37.16 (talk) 00:15, 22 May 2024 (UTC)
thread_title:  Some unmentioned examples 

message: It should be mentioned that every element of the Lipschitz group is homogeneous, and in fact, equal to the product of at most \mathrm{dim}(V) vectors with non-zero quadratic form. I'm just not adding it because I'm not sure on the specifics (does this work for nondegenerate forms? what about char 2?) – viiii (talk) 03:28, 14 December 2023 (UTC)
thread_title:  Properties of the Lipschitz group