Talk:Special unitary group

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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.

== 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)

Contents 1 Complexify??? 2 Generalize to an arbitrary field 3 special? 4 Merging 5 Sudden specialization to SU(2) 6 This statement is incorrect.... wanna confirm?

[edit] 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)

[edit] 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)

[edit] 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).

[edit] 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.

[edit] 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.

[edit] 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)