Talk:Special unitary group

Odd

This page is odd.

The principal case that should be discussed is the complex special unitary group. Other fields need much more thought, to say what the group is.

'A common matrix' representation - well, this is the standard representation for SU(2). I'm not sure what 'generators' means here; in physics literature it usually implies a basis for the Lie algebra.

'Quantum relativity'?

Charles Matthews 13:30, 14 Jan 2004 (UTC)

Yup, I agree, odd. After the general definition this page should restrict to just the complex case. SU(2) is important enough in physics that it should probably get its own page instead of listing its specific properties here.

-- Fropuff 08:07, 2004 Feb 16 (UTC)

Whether its odd or not depends on your point of view. After 28 years of doing math, the idea that the complex version of SU(2) is the most importent case seems peculiar to me. The complexification of the lie algebra su_n is the complex Lie algebra sl_n. So the complexification of SU(2) is SL_2(C). Two by two matrices with complex entries of determinant 1.

The article would be better if it maintained an elemenatary tone throughout.

Real ?

Can I clarify that, in the first line, the matrix entries are real? Robinh 07:15, 29 September 2005 (UTC)

Unitary matrix doesn't imply real entries. In fact the 1×1 case is the unit circle in the complex plane. Charles Matthews 08:54, 29 September 2005 (UTC)

Thanks for this Charles. If this is the case, I get 6 degrees of freedom for SU(2) (that is, eight for the entries but subtract two for the condition of a unit determinant). How does this square with the later statement that SU(2) is isomorphic to the quaternions of absolute value 1 (which I figure to have three degrees of freedom)?

best wishes Robinh 09:17, 29 September 2005 (UTC)

Three is right. No time - I have to go. Count skew-hermitian matrices (Lie algebra) is easier. Charles Matthews 10:17, 29 September 2005 (UTC)

Back again. Look at things this way: unitay matrices are exp(iH) with H a hermitian matrix. For 2×2, the hermitian condition is real diagonal and complex conjugate off-diagonal entries, ie dimension 4. For the special unitaty group H should also have trace zero; so we get 3. Charles Matthews 13:05, 29 September 2005 (UTC)

Right, got it. Thanks! Is the isomorphism easy to write down?

best wishes Robinh 14:35, 29 September 2005 (UTC)

Take a look at the article on the 3-sphere under the section called group structure, or the article on the quaternions. -- Fropuff 15:31, 29 September 2005 (UTC)

Complexify???

What does this word Complexify mean please. Is it a real word? The subject is complex enough with out using difficult words to further obfuscate the matter.--Light current 00:59, 4 October 2005 (UTC)

To complexify a vector space means to pass to the complexification of that space. -- Fropuff 02:26, 4 October 2005 (UTC)

if v is an element of a real vector space, then the collection of linear combinations like (x+iy)v make up the complexification of that real vector space. like (1,2,6) is an element of R³, then (2i,4+7i,3) is an element of its complexification. See complexification for more. In the structure theory of Lie algebras, you often complexify the real Lie algebras because you want to find the eigenvalues of some stuffs, and eigenvalues are only guaranteed to exist when you work in an algebraically closed field. You lose some of information by doing this. For example, su(2) and sl(2,R) are different real Lie algebras, but their complexifications are both sl(2,C). Lethe | Talk 02:29, 4 October 2005 (UTC)

OK Thanks. If its a real word, and its linked to, and a simpler word won't do, then I suppose it will have to remain! But I really wish simpler words could be found.--Light current 02:39, 4 October 2005 (UTC)

what simpler word could there be that means "turn a real vector space into an associated complex vector space"? It seems to fit just right to me. -Lethe | Talk 06:43, 4 October 2005 (UTC)

Actually the chatty note in the Lie algebra discussion is out of place, and what is said is a bit confusing. Charles Matthews 08:29, 4 October 2005 (UTC)

Generalize to an arbitrary field

I think it would be helpful if the definition could be generalized to arbitrary fields as groups of lie type (IE the field is finite) are important in group theory. I've posted a similar request on the unitary group page. Unfortunately I don't know enough about these groups to generalize the definition so help would be appreciated. TooMuchMath 03:36, 14 April 2006 (UTC)

special?

Is it not true that the determinant should be 1 and not just any unit? Isn't that the difference between U(1) and SU(1) and the whole reason that S is used at all? Regardless what field its over, SL(n,p) should be invertible matrices with determinant one, right? i am really off if this is wrong, but i would love to hear what i am missing, and perhaps it could be added to the article.

"Unit" actually means "one". Greets, David 14:17, 21 November 2006 (UTC)

"Unity" means "one". "Unit" is ambiguous. —The preceding unsigned comment was added by 164.67.229.214 (talk) 07:06, 28 February 2007 (UTC).

Merging

I think there should be information about representation theory of SU(N) in this article, and a good way to start is to merge it with Representation theory of SU(2). --Itinerant1 21:22, 16 February 2007 (UTC)

• Do not merge. SU(2) is a simple special case that is accessible to younger students, and is filled with interesting connections and spacial cases that don't hold true for SU(N). In particular, SU(2) has important applications in undergrad quantum mechanics, while the theory of the full SU(N) may be intimidating to such students (who will typically have onlya poor background in math). 19:24, 15 March 2007 (UTC)

• Do not merge Representation of SU(2) is a large enough topic (and important enough of a case) to have its own page. Jason Quinn 04:31, 30 March 2007 (UTC)

What are you talking about. At this stage I am only studying representation theory for SU(2). That's enough for now. Keep the pages seperate.

do not merge - worst idea ever - now there is a consensus i am removing the suggestion.

Sudden specialization to SU(2)

In the middle of the section on Lie algebras, whoever wrote it makes an example using SU(2), and then jumps back with "Back to general SU(n)." This is not very clear, especially with multiple paragraphs in the example discussion. (The generators of SU(3) do not anticommute, for example.) Someone please fix this.

This statement is incorrect.... wanna confirm?

The Lie algebra corresponding to SU(n) is denoted by \mathfrak{su}(n). It consists of the traceless antihermitian n \times n complex matrices, with the regular commutator as Lie bracket. Note that this is a real and not a complex Lie algebra, in the convention used by mathematicians. A factor i is often inserted by particle physicists who find the different, complex Lie algebra convenient.

The nature of the Lie algebra is NOT affected by the inclusion or lack of the factor i in the definition. A real Lie algebra, in a vector space sense, may have complex entries. From the history it appears that it mistake was made by rewording a sentence during a text-rearrangement. Jason Quinn 20:21, 6 April 2007 (UTC)

If you can understand what this page is on about then I think you must already have a pretty deep understanding of what SU groups are and therefore don't really need it. can anyone simplify the word-y bit? I'm a physics undergrad and I can't understand it at all. -could do with a basic oerview type explanation that doesn't just descibe it in terms of more jargon.Tashafairbairn (talk) 23:52, 18 November 2007 (UTC)

SU(2) diffeomorphic to 3-sphere?

Can someone explain this in a little more detail, please? Phoenix1304 (talk) 16:41, 6 April 2008 (UTC)

Confusion between O(n), SO(n) and Spin(n)

I fixed some incorrect isomorphisms (called identities) in the section Important Subgroups.

SU(n) is always both connected and simply connected. O(n) is neither. The 'determinant 1'-subgroup of O(n), SO(n) is connected but not simply connected. The universal cover (which per definition is simply connected) of SO(n) is Spin(n) when n > 2, where the spin group Spin(n) is the (unique) double cover of SO(n).

For the inclusions (called subgroups) I'm not sure, but I think O(n) and not just SO(n) can be included in SU(n). I don't think Spin(n) (or moreso Pin(n)) can be included in SU(n) in general though, so I'll just leave it as it stands. In the other direction SU(n) can be included in SO(2n) and not just in O(2n) because SU(n) interpreted in a straightforward way (underlying real space of vector representation) as a subgroup of $2n\times 2n$ invertable real matrices has determinant 1 for all its elements just as the SO(2n) in O(2n) interpreted in the standard way (vector representation) .85.224.18.125 (talk) 09:33, 4 August 2008 (UTC)

"General" representation of SU(n) generators?

I do not believe that the statement - "In general the generators of SU(n), T, are represented as traceless Hermitian matrices." is in fact true in general. This is certainly a case, but as far as I can see, you can, and people often do define them other ways, noteably to be anti-hermitian. —Preceding unsigned comment added by 128.230.195.31 (talk) 17:08, 11 August 2008 (UTC)

'Important Subgroups' suggestion

Can't we generalize the first statement from "p>1 and n-p>1" to "p>0 and n-p>0" since SU(1) is trivial, and SU(n) > SU(n-1) x U(1) ? —Preceding unsigned comment added by 67.175.94.215 (talk) 07:29, 25 May 2009 (UTC)

Lie Algebra generators

I don't believe identity can be included with the set of su(2) generators. When you do, you get u(2), not su(2).

Pure jargon

I agree with Tashafairbairn above - while I don't want to seem ungrateful to the author(s), this topic, despite the obvious care and effort that's gone into it, is utterly impenetrable to me. It's an excellent example of a problem shared by some of Wikipedia's mathematical pages: they are incomprehensible on their own, and attempting to understand them gradually by following links leads to a frustrating dead end of circular definitions. To see what I mean, compare this page with SO(3) (especially the "Topology" section), which contains context-free analogies and clear explanation: I grasped that topic very well in a single read, but this one was a complete mystery from start to finish. 91.135.1.212 (talk) 00:24, 10 March 2012 (UTC)

I agree. Completely impenetrable Jargon. Like many technical articles on Wikipedia, apparently written by experts trying to show off to other experts with no attempt to make it accessible to non-experts. All the detail of what a Special Unitary Group actually is is obfuscated by links to yet more equally impenetrable articles. See WP:Jargon. A lead which means something to a reasonably intelligent non-expert is needed. 130.246.132.177 (talk) 15:09, 21 February 2013 (UTC)

Mistake in a definition?

In the section for n=2, I think that the equation $\varphi (\alpha ,\beta )={\begin{pmatrix}\alpha &-{\overline {\beta }}\\\beta &{\overline {\alpha }}\end{pmatrix}}$ should actually read $\varphi (\alpha ,\beta )={\begin{pmatrix}\alpha &-{\overline {\beta }}\\\beta &-\alpha \end{pmatrix}}$

Only the latter definition agrees with both my understanding and with the element given later, $u_{3}={\begin{pmatrix}i&0\\0&-i\end{pmatrix}}$

As I'm not an expert in the Special unitary group, could someone please verify that I'm correct? Vrmlguy (talk) 15:30, 21 April 2012 (UTC)

A misprint in anticommutator for SU(3)

It looks like that the expression for anticommutator $\{T^{a},T^{b}\}$ corresponds to $\{\lambda ^{a}\lambda ^{b}\}$. A correct form should read $\{T^{a},T^{b}\}={\frac {1}{3}}\delta ^{ab}+d^{abc}T^{c}$ — Preceding unsigned comment added by 131.169.87.177 (talk) 16:15, 7 May 2012 (UTC)

"Disambiguation needed" - Adjoint representation

I removed the tag - I could not see what the ambiguity was. The following equation defined and clarified the adjoint representation that was meant, for all intents and purposes, and can be seen extensively in the literature. See, e.g., Weinberg vol. 2. If the person who put the tag originally would like to clarify what was meant by the "disambiguation needed" perhaps this could lead to an improvement in readability. — Preceding unsigned comment added by 96.35.171.223 (talk) 17:30, 31 May 2012 (UTC)

'disambiguation needed' means the link does not go to an article but to a disambiguation page. Such pages should not be linked in articles, as they do not help the user following them: at best they waste users' time, at worst they don't help the user find the explanation they were looking for. I've fixed it by changing the link to one to adjoint representation of a Lie group.--JohnBlackburne^words_deeds 17:58, 31 May 2012 (UTC)

Okay, thanks for the clarification! Cheers. 96.35.171.223 (talk) 14:09, 20 June 2012 (UTC)

Physics vs mathematics convention

Hi!

I think it's important to define exactly the two conventions used for the Lie algebra, the exponential mapping, and the structure constants (more?) in mathematics and physics respectively. They differ by a factor of i. This can be done in two or three sentences + equations (sourced), and it should appear early, possibly directly following the lead.

However, I suspect that this particular problem persists in many articles. It may be better to create a separate article that could be linked. YohanN7 (talk) 18:42, 22 September 2012 (UTC)

I am not sure that all physicists have a single convention how to deal with ${\mathfrak {su}}$. YohanN7, turn on a spell checker for English, please. Three spelling mistakes in this post, not so long one, is awful. Incnis Mrsi (talk) 20:10, 22 September 2012 (UTC)

I can see only two spelling errors. Which ones do you see? I can see a grammatical error too.

In the physics literature there is really only one convention commonly used. There is pretty much only two to choose from, and physicists almost always choose the one that give Hermitean matrices in the Lie algebra. Mathematicians almost always choose the other convention. YohanN7 (talk) 21:55, 22 September 2012 (UTC)

Generic element

The generic element does only display in MathJax. YohanN7 (talk) 15:17, 19 July 2015 (UTC)

Now fixed by ip hero. YohanN7 (talk) 21:12, 19 July 2015 (UTC)

Fundamental Representation

Are the identities written in this section valid for that representation only? Paranoidhuman

I moved your comment, misplaced at the top of the list, to the bottom, as per WP rules. The answer to your question is, of course, "yes", which you may instantly check by inspecting the dimensionality of the Identity, instead of asking your question. (Of course, the commutator expression is the definition of the Lie algebra and hence representation independent.) Still, analogous formulas for the anticommutator hold in all representations, mutatis mutandis. Naturally, the "effective" symmetric d coefficients in other reps are proportional to these ones, but, of course, the proportionality constant may be, and often is, zero; it is further proportional to the "anomaly coefficient". This, however, is not an article on the representation theory of SU(n)s. Cuzkatzimhut (talk) 20:45, 23 July 2015 (UTC)

Apparently without references. prokaryotes (talk) 21:09, 13 August 2015 (UTC)

Infinitesimal generators section

This section cannot be correct, if it is correct then su(n) would be isomorphic to gl(n) (or sl(n) if we ignore the nonsense about the identity operator being an element of su(n)). 67.242.104.88 (talk) 21:36, 2 August 2017 (UTC)

It is awkwardly worded, but actually close to correct. I'll make a tweak. YohanN7 (talk) 10:53, 7 August 2017 (UTC)

Isomorphism to SO(2,1) and SL(2,R)

The source [17], which is claiming that SO(2,1) and SL(2,R) are isormorphic, apparently uses the term "isomorphic" instead of "locally isomorphic". In fact, SO(2,1) and SL(2,R) are not isomorphic, as the first is not connected and the second is connected. So obviously, SU(1,1) cannot be isomorphic as Lie groups to both. The source [17] only shows something about the Lie algebras, which will, in general, not induce an isomorphism on Lie group level. See also this discussion on stackexchange, showing that SO(2,1) and SL(2,R) are not even isomorphic as abstract groups. IttalracS (talk) 09:17, 19 November 2019 (UTC)

citation Hall 2015

What is this document and how to find it? This "work" is cited repeatedly in the article, but the citation is very inadequate. How are we supposed to find this source with only what I presume to be the last name of the author and a year?

Is it https://www.springer.com/gp/book/9783319134666?

Surely, surely you checked the References section?? Cuzkatzimhut (talk) 00:46, 17 September 2020 (UTC)

Contradictory statements

In the section Fundamental representation, this passage appears:

"In the physics literature, it is common to identify the Lie algebra with the space of trace-zero Hermitian (rather than the skew-Hermitian) matrices. That is to say, the physicists' Lie algebra differs by a factor of i from the mathematicians'."

But at the end of the section Properties it is stated that the factor is -i.

Which is it, +i or -i ??? 2601:200:C000:1A0:C5A4:9C64:FAF7:1F39 (talk) 20:37, 24 May 2022 (UTC)

Isomorphic to what?

The section "Isomorphism with unit quaternions" contains this passage:

"The complex matrix:"

${\begin{pmatrix}a+bi&c+di\\-c+di&a-bi\end{pmatrix}}\quad (a,b,c,d\in \mathbb {R} )$

"can be mapped to the quaternion:"

$a\,{\hat {1}}+b\,{\hat {i}}+c\,{\hat {j}}+d\,{\hat {k}}$

"This map is in fact an isomorphism."

This last statement fails in a very crucial way: It does not say what is isomorphic to what.

In order for the statement to have any meaning, it is essential that it state both

a) what type of structures are isomorphic (Lie groups? Real algebras such as the quaternions? Or what?), and

b) which exact structures the isomorphism is between.

Without this information, the passage is meaningless. — Preceding unsigned comment added by 2601:200:c000:1a0:ac78:215c:9a5a:68c7 (talk • contribs)

Group isomorphism as SU(2) is a group. Clarification has been made, including use of versor in place of "unit quaternion". The article was marred by links in section titles which have now been removed. — Rgdboer (talk) 02:08, 12 January 2023 (UTC)