View source for SU(2)

:''The following is modified from [[w:SU(2)]]''.

In [[mathematics]], the '''special unitary group''' of degree ''n'', denoted SU(''n''), is the [[group (mathematics)|group]] of ''n''&times;''n'' [[unitary matrices]] with [[determinant]] 1. The group operation is that of [[matrix multiplication]]. The special unitary group is a [[subgroup]] of the [[unitary group]] U(''n''), consisting of all ''n''&times;''n'' unitary matrices, which is itself a subgroup of the [[general linear group]] GL(''n'', '''C''').

The SU(n) groups find wide application in the [[Standard Model]] of [[particle physics]], especially SU(2) in the [[electroweak interaction]] and SU(3) in [[Quantum chromodynamics|QCD]].

The simplest case, SU(1), is the [[trivial group]], having only a single element. The group SU(2) is [[isomorphic]] to the group of [[quaternion]]s of [[Quaternion#Conjugation.2C_the_norm.2C_and_division|absolute value]] 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), we have a [[surjective]] [[homomorphism]] from SU(2) to the [[rotation group]] SO(3) whose [[Kernel (mathematics)|kernel]] is <math>\{+I, -I\}</math>.

==Properties==
The special unitary group SU(''n'') is a real [[matrix Lie group]] of dimension ''n''<sup>2</sup>&minus; 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 SU(''n'') is isomorphic to the [[cyclic group]] '''Z'''<sub>''n''</sub>. Its [[outer automorphism group]], for ''n'' ≥ 3, is '''Z'''<sub>2</sub>, while the outer automorphism group of SU(2) is the [[trivial group]].

The SU(n) algebra is generated by ''n''<sup>2</sup>  operators, which satisfy the commutator relationship (for i,j,k,l = 1, 2, ..., n)

:<math>\left [ \hat{O}_{ij} , \hat{O}_{kl} \right ] = \delta_{jk} \hat{O}_{il} - \delta_{il} \hat{O}_{kj}.</math>

Additionally, the operator

:<math>\hat{N} = \sum_{i=1}^n \hat{O}_{ii}</math>

satisfies

:<math>\left [ \hat{N}, \hat{O}_{ij} \right ] = 0</math>

which implies that the number of ''independent'' generators of SU(n) is n<sup>2</sup>-1.<ref>R.R. Puri, ''Mathematical Methods of Quantum Optics'', Springer, 2001.</ref>

==Generators==
In general the [[infinitesimal generator]]s of SU(n), ''T'', are [[Group representation|represented]] as [[Trace (linear algebra)|traceless]] [[Skew-Hermitian matrix| hermitian matrices]].  I.e:

:<math>\operatorname{tr}(T_a) = 0 \,</math>
and
:<math> T_a = T_a^\dagger. \,</math>

===Fundamental representation===
In the defining or fundamental representation the generators are represented by ''n''&times;''n'' matrices where:
:<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>
:where the ''f'' are the ''structure constants'' and are antisymmetric in all indices, whilst the ''d'' are symmetric in all indices.
As a consequence:
:<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>
:<math>\left[T_a, T_b \right]_- = i \sum_{c=1}^{n^2 -1}{f_{abc} T_c}. \,</math>

We also have
:<math>\sum_{c,e=1}^{n^2 -1}d_{ace}d_{bce}= \frac{n^2-4}{n}\delta_{ab} \,</math>

as a normalization convention.

===Adjoint representation===
In the [[adjoint representation]] the generators are represented by <math>(n^2-1)</math> <math>n</math>&times;<math>n</math> matrices whose elements are defined by the structure constants:
:<math> (T_a)_{jk} = -if_{ajk}. \,</math>

==SU(2)==
:<math>\operatorname{SU}_2(\mathbb{C})</math> and <math>\mathfrak{su}_2(\mathbb{C})</math>

A general matrix element of <math>\operatorname{SU}_2(\mathbb{C})</math> takes the form

:<math>U = 
\begin{pmatrix}
\alpha&-\overline{\beta}\\
\beta&\overline{\alpha}
\end{pmatrix}</math>

where <math>\alpha,\beta\in\mathbb{C}</math> such that <math>|\alpha|^2 + |\beta|^2 = 1</math>. We can consider the following map <math>\varphi : \mathbb{C}^2 \to \operatorname{M}(2,\mathbb{C})</math>, (where <math>\operatorname{M}(2,\mathbb{C})</math> denotes the set of 2 by 2 complex matrices), defined in the obvious way by

:<math>
\varphi(\alpha,\beta) =
\begin{pmatrix}
\alpha&-\overline{\beta}\\
\beta&\overline{\alpha}
\end{pmatrix}.
</math>

By considering <math>\mathbb{C}^2</math> [[diffeomorphism|diffeomorphic]] to <math>\mathbb{R}^4</math> and <math>\operatorname{M}(2,\mathbb{C})</math> diffeomorphic to <math>\mathbb{R}^8</math> we can see that <math>\varphi</math> is an injective real linear map and hence an embedding. Now considering the restriction of <math>\varphi</math> to the [[3-sphere]], denoted <math>S^3</math>, we can see that this is an embedding of the 3-sphere onto a compact submanifold of <math>\operatorname{M}(2,\mathbb{C})</math>. However it is also clear that <math>\varphi(S^3) = \operatorname{SU}_2(\mathbb{C})</math>. Therefore as a manifold we have <math>S^3</math> is diffeomorphic to <math>\operatorname{SU}_2(\mathbb{C})</math> and so <math>\operatorname{SU}_2(\mathbb{C})</math> is a compact, connected [[Lie group]]. 

Now considering the [[Lie algebra]] <math>\mathfrak{su}_2(\mathbb{C})</math>, a general element takes the form

:<math>
U' = 
\begin{pmatrix}
ix & -\overline{\beta}\\
\beta & -ix
\end{pmatrix}
</math>

where <math>x \in \mathbb{R}</math> and <math>\beta \in \mathbb{C}</math>. 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

:<math>
u_1 = \begin{pmatrix}
0 & i\\
i & 0
\end{pmatrix}
\qquad
u_2 = \begin{pmatrix}
0 & -1\\
1 & 0
\end{pmatrix}
\qquad
u_3 = \begin{pmatrix}
i & 0\\
0 & -i
\end{pmatrix}
</math>

which are easily seen to have the form of the general element specified above. These satisfy the relations <math>u_3u_2 = -u_2u_3 = -u_1</math> and <math>u_2u_1 = -u_1u_2 = -u_3</math>. The commutator bracket is therefore specified by

:<math>
[u_3,u_1]=2u_2, \qquad [u_1,u_2] = 2u_3, \qquad [u_2,u_3] = 2u_1.
</math>

The above generators are related to the [[Pauli matrices]] by <math>u_1 = i\sigma_1</math>, <math>u_2 = -i\sigma_2</math> and <math>u_3 = i\sigma_3</math>.

==SU(3)==
The generators of <math>\mathfrak{su}</math>(3), ''T'', in the defining representation, are:
:<math>T_a = \frac{\lambda_a }{2}.\,</math>
where <math>\lambda \,</math>, the [[Gell-Mann matrices]], are the SU(3) analog of the [[Pauli matrices]] for SU(2):

:{| border="0" cellpadding="5" cellspacing="0"
|<math>\lambda_1 = \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}</math>
|<math>\lambda_2 = \begin{pmatrix} 0 & -i & 0 \\ i & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}</math>
|<math>\lambda_3 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 0 \end{pmatrix}</math>
|-
|<math>\lambda_4 = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{pmatrix}</math>
|<math>\lambda_5 = \begin{pmatrix} 0 & 0 & -i \\ 0 & 0 & 0 \\ i & 0 & 0 \end{pmatrix}</math>
|<math>\lambda_6 = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix}</math>
|-
|<math>\lambda_7 = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & -i \\ 0 & i & 0 \end{pmatrix}</math>
|<math>\lambda_8 = \frac{1}{\sqrt{3}} \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -2 \end{pmatrix}.</math>
|
|}

Note that they are all [[Trace (linear algebra)|traceless]] [[Hermitian matrix|Hermitian matrices]] as required.

These obey the relations
:<math>\left[T_a, T_b \right] = i \sum_{c=1}^8{f_{abc} T_c} \,</math>
where the ''f'' are the ''structure constants'', as previously defined, and have values given by
:<math>f_{123} = 1 \,</math>
:<math>f_{147} = -f_{156} = f_{246} = f_{257} = f_{345} = -f_{367} = \frac{1}{2} \,</math>
:<math>f_{458} = f_{678} = \frac{\sqrt{3}}{2}, \,</math>
and all other <math>f_{abc}</math> not related to these by permutation are zero.

The ''d'' take the values:
:<math>d_{118} = d_{228} = d_{338} = -d_{888} = \frac{1}{\sqrt{3}} \,</math>
:<math>d_{448} = d_{558} = d_{668} = d_{778} = -\frac{1}{2\sqrt{3}} \,</math>
:<math>d_{146} = d_{157} = -d_{247} = d_{256} = d_{344} = d_{355} = -d_{366} = -d_{377} = \frac{1}{2}. \,</math>

==Lie algebra==
The [[Lie algebra]] corresponding to SU(n) is denoted by <math>\mathfrak{su}(n)</math>. Its standard mathematical representation consists of the [[traceless]] [[antihermitian]] <math>n \times n</math> complex matrices, with the regular [[commutator]] as [[Lie bracket]]. A factor <math>i</math> 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>\mathfrak{su}(n)</math> is a Lie algebra over <math>\mathbb{R}</math>.

For example, the following antihermitian matrices used in [[quantum mechanics]] form a [[basis (linear algebra)|basis]] for <math>\mathfrak{su}(2)</math> over <math>\mathbb{R}</math>:
:<math>i\sigma_x = \begin{bmatrix} 0 & i \\ i & 0 \end{bmatrix}</math>
:<math>i\sigma_y = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}</math>
:<math>i\sigma_z = \begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix}</math>
(where <math>i</math> is the [[imaginary unit]].)

This representation is often used in [[quantum mechanics]] (see ''[[Pauli matrices]]'' and ''[[Gell-Mann matrices]]''), 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 [[quantum relativity]].

Note that the product of any two different generators is another generator, and that the generators [[anticommute]]. Together with the [[identity matrix]] (times <math>i</math>),
:<math> i I_2 = \begin{bmatrix} i & 0 \\ 0 & i \end{bmatrix}</math>
these are also generators of the Lie algebra <math>\mathfrak{u}(2)</math>.

Here it depends of course on the problem whether one works finally, as in non-relativistic quantum
mechanics, with 2-[[spinors]]; or, as in the relativistic [[Dirac equation|Dirac theory]], one needs an extension to 4-spinors; or in mathematics  even to [[Clifford algebra]]s.

''Note: make clearer the fact that under matrix multiplication (which is anticommutative in this case), we generate the Clifford algebra <math>\mathrm{Cl}_3</math>, whereas you generate the Lie algebra <math>\mathfrak{su}(2)</math> with commutator brackets instead.''

Back to general <math>\mathrm{SU}(n)</math>:

If we choose an (arbitrary) particular basis, then the [[linear subspace|subspace]] of traceless [[diagonal matrix|diagonal]] <math>n \times n</math> matrices with imaginary entries forms an <math>n - 1</math> dimensional [[Cartan subalgebra]].

[[Complexify]] the Lie algebra, so that any traceless <math>n \times n</math> 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>\mathrm{h}</math> is only <math>n - 1</math> 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</math>th basis vector is the matrix with <math>1</math> on the <math>i</math>th diagonal entry and zero elsewhere. Weights would then be given by <math>n</math> coordinates and the sum over all <math>n</math> coordinates has to be zero (because the unit matrix is only auxiliary).

So, <math>\mathrm{SU}(n)</math> has a [[Rank (linear algebra)|rank]] of <math>n - 1</math> and its [[Dynkin diagram]] is given by <math>A_{n - 1}</math>, a chain of <math>n - 1</math> vertices.

Its [[root system]] consists of <math>n(n - 1)</math> roots spanning a <math>n - 1</math> [[Euclidean space]]. Here, we use <math>n</math> redundant coordinates instead of <math>n - 1</math> to emphasize the symmetries of the root system (the <math>n</math> coordinates have to add up to zero). In other words, we are embedding this <math>n - 1</math> dimensional vector space in an <math>n</math>-dimensional one. Then, the roots consists of all the <math>n(n - 1)</math> permutations of <math>(1, -1, 0, \dots, 0)</math>. The construction given two paragraphs ago
explains why. A choice of [[simple root]]s is
:<math>(1, -1, 0, \dots, 0),\ </math>
:<math>(0, 1, -1, \dots, 0),\ </math>
:&hellip;,
:<math>(0, 0, 0, \dots, 1, -1).\ </math>

Its [[Cartan matrix]] is
:<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>

Its [[Weyl group]] or [[Coxeter group]] is the [[symmetric group]] <math>S_n</math>, the [[symmetry group]] of the <math>(n - 1)</math>-[[simplex]].

==Generalized special unitary group==
For a [[field (mathematics)|field]] ''F'', the '''generalized special unitary group over F''', SU(''p'',''q'';''F''), is the [[Group (mathematics)|group]] of all [[linear transformation]]s of [[determinant]] 1 of a [[vector space]] of rank ''n'' = ''p'' + ''q''   over ''F'' which leave invariant a [[nondegenerate form|nondegenerate]], [[Hermitian form]] of [[signature of a quadratic form|signature]] (''p'', ''q''). This group is often referred to as the '''special unitary group of signature p q over F'''. The field ''F'' can be replaced by a [[commutative ring]], in which case the vector space is replaced by a [[free module]].

Specifically, fix a [[Hermitian matrix]] ''A'' of signature ''p'' ''q'' in GL(''n'',''R''), then all

:<math>M \in SU(p,q,R)</math>

satisfy

:<math>M^{*} A M = A \,</math>

:<math>\det M = 1. \,</math>

Often one will see the notation <math>SU_{p,q}</math> without reference to a ring or field, in this case the ring or field being referred to is '''C''' and this gives one of the classical [[Lie groups]]. The standard choice for ''A'' when ''F'' = '''C''' is
: <math>
  A
  =
  \begin{bmatrix}
    0 & 0 & i \\
    0 & I_{n-2} & 0 \\
    -i & 0 & 0
  \end{bmatrix}.
</math>
However there may be better choices for ''A'' for certain dimensions which exhibit more behaviour under restriction to subrings of '''C'''.

===Example===
A very important example of this type of group is the [[Picard modular group]] SU(2,1;'''Z'''[''i'']) which acts (projectively) on [[complex hyperbolic space]] of degree two, in the same way that SL(2,'''Z''') acts (projectively) on real [[hyperbolic space]] of dimension two. In 2003 [[Gábor Francsics]] and [[Peter Lax]] computed a fundamental domain for the action of this group on <math>HC^2</math>, see [http://www.esi.ac.at/Preprint-shadows/esi1273.html].
Another example is SU(1,1;'''C''') which is isomorphic to SL(2,'''R''').

==Important subgroups==
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 SU(n) that are important in [[Grand unification theory|GUT physics]] are, for p>1, n-p>1:
:<math>
SU(n) \supset SU(p)\times SU(n-p) \times U(1).
</math>
For completeness there are also the [[orthogonal]] and [[symplectic group|symplectic]] subgroups:
:<math>
SU(n) \supset O(n)
</math>
:<math>
SU(2n) \supset USp(2n).
</math>
Since the [[rank of a Lie group|rank]] of SU(n) is n-1 and of 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. SU(n) is a subgroup of various other lie groups:
:<math>
SO(2n) \supset SU(n)
</math>
:<math>
USp(2n) \supset SU(n)
</math>
:<math>
Spin(4) = SU(2) \times SU(2)
</math> (see [[Spin group]])
:<math>
E_6 \supset SU(6) 
</math>
:<math>
E_7 \supset SU(8)
</math>
:<math>
G_2 \supset SU(3)
</math> (see [[Simple Lie groups]] for E<sub>6</sub>, E<sub>7</sub>, and G<sub>2</sub>).
There are also the identities '''SU(4)=Spin(6)''', '''SU(2)=Spin(3)=USp(2)''' and '''U(1)=Spin(2)=SO(2)''' .

One should finally mention that SU(2) is the [[double covering group]] of SO(3), a relation that plays an important role in the theory of rotations of 2-[[spinor]]s in non-relativistic [[quantum mechanics]].

==See also==
{{Portal|Mathematics}}
* [[Group theory]]
* [[Representation theory]]

==Notes==
{{More footnotes|date=November 2009}}
{{Reflist}}

==References==
* {{cite book | author=Halzen, Francis
|author2=Martin, Alan |
 title=Quarks & Leptons: An Introductory Course in Modern Particle Physics | publisher=John Wiley & Sons | year=1984 | isbn=0-471-88741-2}}
*[http://arxiv.org/abs/math/0605784v1 Maximal Subgroups of Compact Lie Groups ]

==External links==
*[http://courses.washington.edu/phys55x/Physics%20558_lec1_03.htm Physics 558 - Lecture 1, Winter 2003]

{{DEFAULTSORT:Special Unitary Group}}
[[Category:Lie groups]]