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| {{redirect|SU(5)|the specific grand unification theory|Georgi–Glashow model}}
| | This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users. |
| {{Technical|date=December 2013}}
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| {{Group theory sidebar |Topological}}
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| {{Lie groups |Classical}}
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| The '''special unitary group''' of degree {{math|''n''}}, denoted {{math|SU(''n'')}}, is the [[Group (mathematics)|group]] of {{math|''n''×''n''}} [[Unitary matrix|unitary]] [[Matrix (mathematics)|matrices]] with [[determinant]] 1. The group operation is that of [[matrix multiplication]]. The special unitary group is a [[subgroup]] of the [[unitary group]] {{math|U(''n'')}}, consisting of all {{math|''n''×''n''}} unitary matrices. As a [[classical group|compact classical group]], {{math|U(''n'')}} is the group that preserves the [[Inner product space#Examples|standard inner product]] on {{math|'''C'''<sup>''n''</sup>}}.<ref group=nb>For a characterization of {{math|U(''n'')}} and hence {{math|SU(''n'')}} in terms of preservation of the standard inner product on {{math|ℂ<sup>''n''</sup>}}, see [[Classical group]].</ref> It is itself a subgroup of the [[general linear group]], {{math|SU(''n'') ⊂ U(''n'') ⊂ GL(''n'', '''C''')}}.
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| The {{math|SU(''n'')}} groups find wide application in the [[Standard Model]] of [[particle physics]], especially [[#n = 2|{{math|SU(2)}}]] in the [[electroweak interaction]] and [[#n = 3|{{math|SU(3)}}]] in [[Quantum chromodynamics|QCD]].<ref>{{cite book | author=Halzen, Francis; Martin, Alan | title=Quarks & Leptons: An Introductory Course in Modern Particle Physics | publisher=John Wiley & Sons | year=1984 | isbn=0-471-88741-2}}</ref>
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| The simplest case, {{math|SU(1)}}, is the [[trivial group]], having only a single element. The group {{math|SU(2)}} is [[isomorphic]] to the group of [[quaternion]]s of [[Quaternion#Conjugation, the norm, and division|norm]] 1, and is thus [[diffeomorphic]] to the [[3-sphere]]. Since [[unit quaternion]]s can be used to represent rotations in 3-dimensional space (up to sign), there is a [[surjective]] [[homomorphism]] from {{math|SU(2)}} to the [[rotation group SO(3)|rotation group {{math|SO(3)}}]] whose [[kernel (algebra)|kernel]] is {{math|{+''I'', −''I''}}}.<ref group=nb>For an explicit description of the homomorphism {{math|SU(2) → SO(3)}}, see [[Rotation group SO(3)#Connection between SO(3) and SU(2)|Connection between SO(3) and SU(2)]].</ref> {{math|SU(2)}} is also identical to one of symmetry groups of [[spinor]]s, [[Spin group|Spin]](3), that enables a spinor presentation of rotations.
| | '''MathML''' |
| | :<math forcemathmode="mathml">E=mc^2</math> |
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| ==Properties==
| | <!--'''PNG''' (currently default in production) |
| The special unitary group {{math|SU(''n'')}} is a real [[Lie group]] (though not a complex Lie group). Its dimension as a [[manifold|real manifold]] is {{nowrap|{{math|''n''<sup>2</sup> − 1}}}}. Topologically, it is [[Compact space|compact]] and [[simply connected]]. Algebraically, it is a [[simple Lie group]] (meaning its [[Lie algebra]] is simple; see below). The [[center of a group|center]] of {{math|SU(''n'')}} is isomorphic to the [[cyclic group]] {{math|Z<sub>''n''</sub>}}, and is composed of the diagonal matrices ''ζI'' for ''ζ'' an ''n''<sup>th</sup> root of unity and ''I'' the ''n''×''n'' identity matrix. Its [[outer automorphism group]], for {{nowrap|{{math|''n'' ≥ 3}}}}, is {{math|Z<sub>2</sub>}}, while the outer automorphism group of {{math|SU(2)}} is the [[trivial group]].
| | :<math forcemathmode="png">E=mc^2</math> |
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| A maximal torus, of rank {{math|''n'' − 1}}, is given by the set of diagonal matrices with determinant 1. The [[Weyl group]]
| | '''source''' |
| is the [[symmetric group]] {{math|''S<sub>n</sub>''}}, which is represented by [[Generalized permutation matrix#Signed permutation group|signed permutation matrices]] (the signs being necessary to ensure the
| | :<math forcemathmode="source">E=mc^2</math> --> |
| determinant is 1).
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| The [[Lie algebra]] of {{math|SU(''n'')}}, denoted by {{math|'''su'''(''n'')}} is generated by {{math|''n''<sup>2</sup> − 1}} operators, which satisfy the [[commutator relationship]] for {{nowrap|{{math|''i'', ''j'', ''k'', ''ℓ''}}}} = {{nowrap|{{math|1, 2, ..., ''n''}}}}
| | <span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples]. |
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| :<math>\left [ \hat{O}_{ij} , \hat{O}_{k \ell} \right ] = \delta_{jk} \hat{O}_{i \ell} - \delta_{i \ell} \hat{O}_{kj}.</math>
| | ==Demos== |
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| Additionally, the operator
| | Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]: |
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| :<math>\hat{N} = \sum_{i=1}^n \hat{O}_{ii}</math>
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| | ** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]]. |
| | ** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode. |
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| :<math>\left [ \hat{N}, \hat{O}_{ij} \right ] = 0,</math>
| | ==Test pages == |
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| which implies that the number of ''independent'' generators is {{nowrap|{{math|''n''<sup>2</sup> − 1}} }}.<ref>R.R. Puri, ''Mathematical Methods of Quantum Optics'', Springer, 2001.</ref>
| | To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages: |
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| ==Generators==
| | *[[Inputtypes|Inputtypes (private Wikis only)]] |
| In general the infinitesimal generators (elements of the Lie algebra) of {{math|SU(''n'')}}, {{math|''T''}}, are [[Group representation|represented]] as [[Trace (linear algebra)|traceless]] [[Hermitian matrix|hermitian matrices]]. I.e:
| | *[[Url2Image|Url2Image (private Wikis only)]] |
| | | ==Bug reporting== |
| :<math>\operatorname{tr}(T_a) = 0 \,</math>
| | If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de . |
| and
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| :<math> T_a = T_a^\dagger ~.</math>
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| ===Fundamental representation===
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| In the defining, or fundamental, representation the generators are represented by {{math|''n''×''n''}} matrices, where:
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| :<math>T_a T_b = \frac{1}{2n}\delta_{ab}I_n + \frac{1}{2}\sum_{c=1}^{n^2 -1}{(if_{abc} + d_{abc}) T_c} \,</math>
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| where the {{math|''f''}} are the '''[[structure constants]]''' and are antisymmetric in all indices, whilst the {{math|''d''}}-coefficients are symmetric in all indices.
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| As a consequence:
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| :<math>\left[T_a, T_b\right]_+ = \frac{1}{n}\delta_{ab} I_n+ \sum_{c=1}^{n^2 -1}{d_{abc} T_c} \,</math>
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| :<math>\left[T_a, T_b \right]_- = i \sum_{c=1}^{n^2 -1}{f_{abc} T_c} \, .</math>
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| We also take
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| :<math>\sum_{c,e=1}^{n^2 -1}d_{ace}d_{bce}= \frac{n^2-4}{n}\delta_{ab} \,</math>
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| as a normalization convention.
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| ===Adjoint representation===
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| In the [[Adjoint representation of a Lie group|adjoint representation]], the generators are represented by {{nowrap|{{math|(''n''<sup>2</sup> − 1)}} }}× {{nowrap|{{math|(''n''<sup>2</sup> − 1)}} }} matrices, {{nowrap|{{math|''n''<sup>2</sup> − 1}} }} of them, whose elements are defined by the structure constants themselves:
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| :<math> (T_a)_{jk} = -if_{ajk}.</math>
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| == ''n'' = 2 ==
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| {{see also|Versor}}
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| {{math|SU(2)}} is the following group:
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| :<math> \mathrm{SU}(2) = \left \{ \begin{pmatrix} \alpha&-\overline{\beta}\\ \beta & \overline{\alpha} \end{pmatrix}: \ \ \alpha,\beta\in\mathbf{C}, |\alpha|^2 + |\beta|^2 = 1\right \} ~,</math>
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| where the overline denotes [[Complex conjugate|complex conjugation]]. Now consider the following map:
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| :<math> \begin{align} \varphi : \mathbf{C}^2 &\to \operatorname{M}(2,\mathbf{C}) \\ \varphi(\alpha,\beta) &= \begin{pmatrix} \alpha&-\overline{\beta}\\ \beta & \overline{\alpha}\end{pmatrix},\end{align} </math>
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| where {{math|M(2, '''C''')}} denotes the set of 2 by 2 complex matrices. By considering {{math|'''C'''<sup>2</sup>}} [[diffeomorphism|diffeomorphic]] to {{math|'''R'''<sup>4</sup>}} and {{math|M(2, '''C''')}} diffeomorphic to {{math|'''R'''<sup>8</sup>}} we can see that {{math|''φ''}} is an injective real linear map and hence an embedding. Now, considering the [[Restriction (mathematics)|restriction]] of {{math|''φ''}} to the [[3-sphere]] (since modulus is 1), denoted {{math|''S''<sup>3</sup>}}, we can see that this is an embedding of the 3-sphere onto a compact submanifold of {{math|M(2, '''C''')}}. However it is also clear that {{math|1=''φ''(''S''<sup>3</sup>) = SU(2)}}. Therefore as a manifold {{math|''S''<sup>3</sup>}} is diffeomorphic to {{math|SU(2)}} and so {{math|SU(2)}} is a compact, connected [[Lie group]].
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| The [[Lie algebra]] of {{math|SU(2)}} is:
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| :<math>\mathfrak{su} (2) = \left \{ \begin{pmatrix} ix & -\overline{\beta}\\ \beta & -ix \end{pmatrix}: \ x \in \mathbf{R}, \beta \in \mathbf{C} \right \}</math>
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| It is easily verified that matrices of this form have [[Trace (linear algebra)|trace]] zero and are [[Skew Hermitian matrix|antihermitian]]. The Lie algebra is then generated by the following matrices
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| :<math> u_1 = \begin{pmatrix}
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| 0 & i\\
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| i & 0
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| \end{pmatrix}
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| \qquad
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| u_2 = \begin{pmatrix}
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| 0 & -1\\
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| 1 & 0
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| \end{pmatrix}
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| \qquad
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| u_3 = \begin{pmatrix}
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| i & 0\\
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| 0 & -i
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| \end{pmatrix} ~,</math>
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| which are easily seen to have the form of the general element specified above. These satisfy {{math|1=''u''<sub>3</sub>''u''<sub>2</sub> = −''u''<sub>2</sub>''u''<sub>3</sub> = −''u''<sub>1</sub>}} and {{math|1=''u''<sub>2</sub>''u''<sub>1</sub> = −''u''<sub>1</sub>''u''<sub>2</sub> = −''u''<sub>3</sub>}}. The [[commutator bracket]] is therefore specified by
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| :<math>[u_3,u_1]=2u_2, \qquad [u_1,u_2] = 2u_3, \qquad [u_2,u_3] = 2u_1.</math>
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| The above generators are related to the [[Pauli matrices]] by {{math|1=''u''<sub>1</sub> = ''i σ''<sub>1</sub>,''u''<sub>2</sub> = −''i σ''<sub>2</sub>}} and {{math|1=''u''<sub>3</sub> = ''i σ''<sub>3</sub>}}. This representation is often used in [[quantum mechanics]] to represent the [[spin (physics)|spin]] of [[fundamental particle]]s such as [[electron]]s. They also serve as [[unit vector]]s for the description of our 3 spatial dimensions in [[loop quantum gravity]].
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| The Lie algebra is used to work out the [[Representation theory of SU(2)|representations of {{math|SU(2)}}]].
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| == ''n'' = 3 ==
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| The generators of {{math|'''su'''(3)}}, {{math|''T''}}, in the defining representation, are:
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| :<math>T_a = \frac{\lambda_a }{2}.\,</math>
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| where {{math|λ}} the [[Gell-Mann matrices]], are the {{math|SU(3)}} analog of the Pauli matrices for {{math|SU(2)}}:
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| :{| border="0" cellpadding="8" cellspacing="0"
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| |<math>\lambda_1 = \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}</math>
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| |<math>\lambda_2 = \begin{pmatrix} 0 & -i & 0 \\ i & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}</math>
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| |<math>\lambda_3 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 0 \end{pmatrix}</math>
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| |-
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| |<math>\lambda_4 = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{pmatrix}</math>
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| |<math>\lambda_5 = \begin{pmatrix} 0 & 0 & -i \\ 0 & 0 & 0 \\ i & 0 & 0 \end{pmatrix}</math>
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| |-
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| |<math>\lambda_6 = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix}</math>
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| |<math>\lambda_7 = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & -i \\ 0 & i & 0 \end{pmatrix}</math>
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| |<math>\lambda_8 = \frac{1}{\sqrt{3}} \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -2 \end{pmatrix} .</math>
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| |}
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| Note that the <math>\lambda_a</math> span all [[Trace (linear algebra)|traceless]] [[Hermitian matrix|Hermitian matrices]] as required.
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| These obey the relations
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| :<math>\left[T_a, T_b \right] = i \sum_{c=1}^8{f_{abc} T_c} \,</math>
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| :<math> \{T_a, T_b\} = \frac{1}{3}\delta_{ab} + \sum_{c=1}^8{d_{abc} T_c} \,</math>,
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| :(or, equivalently, <math> \{\lambda_a, \lambda_b\} = \frac{4}{3}\delta_{ab} + 2\sum_{c=1}^8{d_{abc} \lambda_c} </math>).
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| The {{math|''f''}} are the structure constants, given by:
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| :<math>f_{123} = 1 \,</math>
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| :<math>f_{147} = -f_{156} = f_{246} = f_{257} = f_{345} = -f_{367} = \frac{1}{2} \,</math>
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| :<math>f_{458} = f_{678} = \frac{\sqrt{3}}{2}, \,</math>
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| while all other <math>f_{abc}</math> not related to these by permutation are zero.
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| The {{math|''d''}} take the values:
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| :<math>d_{118} = d_{228} = d_{338} = -d_{888} = \frac{1}{\sqrt{3}} \,</math>
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| :<math>d_{448} = d_{558} = d_{668} = d_{778} = -\frac{1}{2\sqrt{3}} \,</math>
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| :<math>d_{146} = d_{157} = -d_{247} = d_{256} = d_{344} = d_{355} = -d_{366} = -d_{377} = \frac{1}{2}. \,</math>
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| ==Lie algebra==
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| The above representation bases generalize to [[Generalizations of Pauli matrices|{{nowrap|{{math|''n'' > 3}}}}]].
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| The [[Lie algebra]] corresponding to {{math|SU(''n'')}} is denoted by {{math|'''su'''(''n'')}}. Its standard mathematical representation consists of the [[traceless]] [[antihermitian]] {{math|''n''×''n''}} complex matrices, with the regular [[commutator]] as Lie bracket. A factor {{math|''i''}} is often inserted by [[particle physics|particle]] [[physicist]]s, so that all matrices become Hermitian. This is simply a different, more convenient, representation of the same real Lie algebra. Note that {{math|'''su'''(''n'')}} is a Lie algebra over {{math|'''R'''}}.
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| If we choose an (arbitrary) particular basis, then the [[linear subspace|subspace]] of traceless [[diagonal matrix|diagonal]] {{math|''n''×''n''}} matrices with imaginary entries forms an {{nowrap|{{math|(''n'' − 1)}}}}-dimensional [[Cartan subalgebra]].
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| [[Complexify]] the Lie algebra, so that any traceless {{math|''n''×''n''}} matrix is now allowed. The [[weight (representation theory)|weight]] [[eigenvector]]s are the Cartan subalgebra itself and the matrices with only one nonzero entry which is off diagonal. Even though the Cartan subalgebra {{math|'''h'''}} is only {{nowrap|{{math|(''n'' − 1)}}-dimensional}}, to simplify calculations, it is often convenient to introduce an auxiliary element, the unit matrix which commutes with everything else (which should not be thought of as an element of the Lie algebra!) for the purpose of computing weights and that only. So, we have a basis where the {{math|''i''}}-th basis vector is the matrix with 1 on the {{math|''i''}}-th diagonal entry and zero elsewhere. Weights would then be given by {{math|''n''}} coordinates and the sum over all {{math|''n''}} coordinates has to be zero (because the unit matrix is only auxiliary).
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| So, {{math|SU(''n'')}} is of [[Rank (linear algebra)|rank]] {{nowrap|{{math|''n'' − 1}}}} and its [[Dynkin diagram]] is given by {{math|A<sub>''n''−1</sub>}}, a chain of {{nowrap|{{math|''n'' − 1}} }} vertices. Its [[root system]] consists of {{nowrap|{{math|''n''(''n'' − 1)}} }} roots spanning a {{nowrap|{{math|''n'' − 1}} }} [[Euclidean space]]. Here, we use {{math|''n''}} redundant coordinates instead of {{nowrap|{{math|''n'' − 1}}}} to emphasize the symmetries of the root system (the {{math|''n''}} coordinates have to add up to zero). In other words, we are embedding this {{nowrap|{{math|''n'' − 1}}}} dimensional vector space in an {{math|''n''}}-dimensional one. Then, the roots consists of all the {{nowrap|{{math|''n''(''n'' − 1)}}}} permutations of {{math|(1, −1, 0, ..., 0)}}. The construction given two paragraphs ago explains why. A choice of [[simple root (root system)|simple root]]s is
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| :<math>(1, -1, 0, \dots, 0),\ </math>
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| :<math>(0, 1, -1, \dots, 0),\ </math>
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| :…,
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| :<math>(0, 0, 0, \dots, 1, -1).\ </math>
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| Its [[Cartan matrix]] is
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| :<math> \begin{pmatrix} 2 & -1 & 0 & \dots & 0 \\-1 & 2 & -1 & \dots & 0 \\ 0 & -1 & 2 & \dots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \dots & 2 \end{pmatrix}.\ </math>
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| Its [[Weyl group]] or [[Coxeter group]] is the [[symmetric group]] {{math|S<sub>''n''</sub>}}, the [[symmetry group]] of the {{nowrap|{{math|(''n'' − 1)}}}}-[[simplex]].
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| ==Generalized special unitary group==
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| For a [[field (mathematics)|field]] {{math|''F''}}, the '''generalized special unitary group over ''F''''', {{math|SU(''p'', ''q''; ''F'')}}, is the [[Group (mathematics)|group]] of all [[linear transformation]]s of [[determinant]] 1 of a [[vector space]] of rank {{nowrap|{{math|1=''n'' = ''p'' + ''q''}}}} over {{math|''F''}} which leave invariant a [[nondegenerate form|nondegenerate]], [[Hermitian form]] of [[signature of a quadratic form|signature]] {{math|(''p'', ''q'')}}. This group is often referred to as the '''special unitary group of signature {{math|''p q''}} over {{math|''F''}}'''. The field {{math|''F''}} can be replaced by a [[commutative ring]], in which case the vector space is replaced by a [[free module]].
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| Specifically, fix a [[Hermitian matrix]] {{math|''A''}} of signature {{math|''p q''}} in {{math|GL(''n'', '''R''')}}, then all
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| :<math>M \in \mathrm{SU}(p, q, R)</math>
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| satisfy
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| :<math>M^{*} A M = A \,</math>
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| :<math>\det M = 1. \,</math>
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| Often one will see the notation {{math|SU(''p'', ''q'')}} without reference to a ring or field; in this case, the ring or field being referred to is {{math|'''C'''}} and this gives one of the classical [[Lie groups]]. The standard choice for {{math|''A''}} when {{nowrap|{{math|1=''F'' = '''C'''}}}} is
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| : <math> A = \begin{bmatrix}
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| 0 & 0 & i \\
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| 0 & I_{n-2} & 0 \\
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| -i & 0 & 0
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| \end{bmatrix}. </math>
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| However there may be better choices for {{math|''A''}} for certain dimensions which exhibit more behaviour under restriction to subrings of {{math|'''C'''}}.
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| ===Example===
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| A very important example of this type of group is the [[Picard modular group]] {{math|SU(2, 1; '''Z'''[''i''])}} which acts (projectively) on [[complex hyperbolic space]] of degree two, in the same way that {{math|SL(2,9;'''Z''')}} acts (projectively) on real [[hyperbolic space]] of dimension two. In 2005 [[Gábor Francsics]] and [[Peter Lax]] computed an explicit fundamental domain for the action of this group on {{math|HC<sup>2</sup>}}.<ref>{{cite arXiv |arxiv=math/0509708v1.pdf |title=An Explicit Fundamental Domain For The Picard Modular Group In Two Complex Dimensions |last1=Francsics |first1=Gabor |last2=Lax |first2=Peter D. }}</ref> Another example is {{math|SU(1, 1; '''C''')}} which is isomorphic to {{math|SL(2,'''R''')}}.
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| ==Important subgroups==
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| In physics the special unitary group is used to represent [[bosonic]] symmetries. In theories of [[symmetry breaking]] it is important to be able to find the subgroups of the special unitary group. Subgroups of {{math|SU(''n'')}} that are important in [[Grand unification theory|GUT physics]] are, for {{nowrap|{{math|''p'' > 1, ''n'' − ''p'' > 1}} }}:
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| :<math>\mathrm{SU}(n) \supset \mathrm{SU}(p)\times \mathrm{SU}(n-p) \times \mathrm{U}(1)</math>,
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| where × denotes the [[Direct product of groups|direct product]] and {{math|U(1)}}, known as the [[circle group]], is the multiplicative group of all [[complex number]]s with [[Absolute value#Complex numbers|absolute value]] 1.
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| For completeness there are also the [[orthogonal]] and [[compact symplectic group|symplectic]] subgroups:
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| :<math>\mathrm{SU}(n) \supset \mathrm{SO}(n),</math>
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| :<math>\mathrm{SU}(2n) \supset \mathrm{Sp}(n).</math>
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| Since the [[rank of a Lie group|rank]] of {{math|SU(n)}} is {{nowrap|{{math|''n'' − 1}} }} and of {{math|U(1)}} is 1, a useful check is that the sum of the ranks of the subgroups is less than or equal to the rank of the original group. {{math|SU(''n'')}} is a subgroup of various other Lie groups:
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| :<math>\mathrm{SO}(2n) \supset \mathrm{SU}(n)</math>
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| :<math>\mathrm{Sp}(n) \supset \mathrm{SU}(n)</math>
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| :<math>\mathrm{Spin}(4) = \mathrm{SU}(2) \times \mathrm{SU}(2)</math> (see [[Spin group]])
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| :<math>\mathrm{E}_6 \supset \mathrm{SU}(6)</math>
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| :<math>\mathrm{E}_7 \supset \mathrm{SU}(8)</math>
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| :<math>\mathrm{G}_2 \supset \mathrm{SU}(3)</math> (see [[Simple Lie groups]] for E<sub>6</sub>, E<sub>7</sub>, and G<sub>2</sub>).
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| There are also the identities {{nowrap|{{math|1 = SU(4) = Spin(6)}} }}, {{nowrap|{{math|1 = SU(2) = Spin(3) = Sp(1)}} }},<ref>{{math|Sp(''n'')}} is the [[compact real form]] of {{math|Sp(2''n'', '''C''')}}. It is sometimes denoted {{math|USp(''2n''}}. The dimension of the {{math|Sp(''n'')}}-matrices is {{math|2''n'' × 2''n''}}.</ref> and {{nowrap|{{math|1 = U(1) = Spin(2) = SO(2)}} }}.
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| One should finally mention that {{math|SU(2)}} is the [[double covering group]] of {{math|SO(3)}}, a relation that plays an important role in the theory of rotations of 2-[[spinor]]s in non-relativistic [[quantum mechanics]].
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| ==See also==
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| {{Portal|Mathematics}}
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| * [[Projective special unitary group]], {{math|PSU(''n'')}}
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| * [[Generalizations of Pauli matrices]]
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| ==Remarks==
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| {{Reflist|group=nb}}
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| ==Notes==
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| {{More footnotes|date=November 2009}}
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| {{Reflist}}
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| ==References==
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| *[http://arxiv.org/abs/math/0605784v1 Maximal Subgroups of Compact Lie Groups ]
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| {{DEFAULTSORT:Special Unitary Group}}
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| [[Category:Lie groups]]
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