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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)
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)
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)
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)
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)
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)
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)
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.
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)
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.
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.
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)
Can someone explain this in a little more detail, please? Phoenix1304 ( talk) 16:41, 6 April 2008 (UTC)
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 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)
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)
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)
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).
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)
In the section for n=2, I think that the equation should actually read
Only the latter definition agrees with both my understanding and with the element given later,
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)
It looks like that the expression for anticommutator corresponds to . A correct form should read — Preceding unsigned comment added by 131.169.87.177 ( talk) 16:15, 7 May 2012 (UTC)
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)
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)
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)
Are the identities written in this section valid for that representation only? Paranoidhuman
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)
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)
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?
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)
The section "Isomorphism with unit quaternions" contains this passage:
"The complex matrix:"
"can be mapped to the quaternion:"
"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)
![]() | This article is rated B-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | ||||||||||
|
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)
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)
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)
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)
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)
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)
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)
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.
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)
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.
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.
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)
Can someone explain this in a little more detail, please? Phoenix1304 ( talk) 16:41, 6 April 2008 (UTC)
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 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)
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)
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)
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).
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)
In the section for n=2, I think that the equation should actually read
Only the latter definition agrees with both my understanding and with the element given later,
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)
It looks like that the expression for anticommutator corresponds to . A correct form should read — Preceding unsigned comment added by 131.169.87.177 ( talk) 16:15, 7 May 2012 (UTC)
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)
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)
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)
Are the identities written in this section valid for that representation only? Paranoidhuman
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)
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)
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?
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)
The section "Isomorphism with unit quaternions" contains this passage:
"The complex matrix:"
"can be mapped to the quaternion:"
"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)