Editing Special unitary group

{{Short description|Group of unitary matrices with determinant of 1}}
{{redirect|SU(5)|the specific grand unification theory|Georgi–Glashow model}}
{{Group theory sidebar |Topological}}
{{Lie groups |Classical}}

In mathematics, the '''special unitary group''' of degree {{math|''n''}}, denoted {{math|SU(''n'')}}, is the [[Lie group]] of {{math|''n'' × ''n''}} [[Unitary matrix|unitary matrices]] with [[determinant]] 1.

The [[Matrix (mathematics)|matrices]] of the more general [[unitary group]] may have [[complex number|complex]] determinants with absolute value 1, rather than real 1 in the special case.

The group operation is [[matrix multiplication]]. The special unitary group is a [[normal 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#Some examples|standard inner product]] on <math>\mathbb{C}^n</math>.{{efn|For a characterization of {{math|U(''n'')}} and hence {{math|SU(''n'')}} in terms of preservation of the standard inner product on <math>\mathbb{C}^n</math>, see ''[[Classical group]]''.}} It is itself a subgroup of the [[general linear group]], <math>\operatorname{SU}(n) \subset \operatorname{U}(n) \subset \operatorname{GL}(n, \mathbb{C} ).</math>

The {{math|SU(''n'')}} groups find wide application in the [[Standard Model]] of [[particle physics]], especially {{math|[[#The_group_SU(2)|SU(2)]]}} in the [[electroweak interaction]] and {{math|[[#SU(3)|SU(3)]]}} in [[quantum chromodynamics]].<ref>{{cite book |author1=Halzen, Francis |authorlink1 = Francis Halzen |author2=Martin, Alan | authorlink2 = Alan Martin (physicist)| title=Quarks & Leptons: An Introductory Course in Modern Particle Physics |url=https://archive.org/details/quarksleptonsint0000halz |url-access=registration | publisher=John Wiley & Sons | year=1984 | isbn=0-471-88741-2}}</ref>

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 reciprocal|norm]] 1, and is thus [[diffeomorphic]] to the [[3-sphere]]. Since [[unit quaternion]]s can be used to represent rotations in 3-dimensional space (uniquely 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''}<!--this comment forces preceding brace to render as math-->}}.{{efn|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)]].}} Since the quaternions can be identified as the even subalgebra of the Clifford Algebra {{math|Cl(3)}}, {{math|SU(2)}} is in fact identical to one of the symmetry groups of [[spinor]]s, [[spin group|Spin]](3), that enables a spinor presentation of rotations.

== Properties ==
The special unitary group {{math|SU(''n'')}} is a strictly real [[Lie group]] (vs. a more general [[complex Lie group]]). Its dimension as a [[manifold|real manifold]] is {{math|''n''<sup>2</sup> − 1}}. Topologically, it is [[compact space|compact]] and [[simply connected]].<ref>{{harvnb|Hall|2015}}, Proposition 13.11</ref> Algebraically, it is a [[simple Lie group]] (meaning its [[Lie algebra]] is simple; see below).<ref>{{cite book |author-link=Brian Garner Wybourne |author=Wybourne, B.G. |year=1974 |title=Classical Groups for Physicists |publisher=Wiley-Interscience |isbn=0471965057}}</ref>

The [[center of a group|center]] of {{math|SU(''n'')}} is isomorphic to the [[cyclic group]] <math>\mathbb{Z}/n\mathbb{Z}</math>, and is composed of the diagonal matrices {{math|''ζ'' ''I''}} for {{math|''ζ''}} an {{math|''n''}}th root of unity and {{math|''I''}} the {{math|''n'' × ''n''}} identity matrix.

Its [[outer automorphism group]] for {{math|''n'' ≥ 3}} is <math>\mathbb{Z}/2\mathbb{Z},</math> while the outer automorphism group of {{math|SU(2)}} is the [[trivial group]].

A [[maximal torus]] of [[Algebraic torus#Split rank of a semisimple group|rank]] {{math|''n'' − 1}} is given by the set of diagonal matrices with determinant {{math|1}}. The [[Weyl group#The Weyl group of a connected compact Lie group|Weyl group]] of {{math|SU(''n'')}} 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 that the determinant is {{math|1}}).

The [[Lie algebra]] of {{math|SU(''n'')}}, denoted by <math>\mathfrak{su}(n)</math>, can be identified with the set of [[traceless]] [[antiHermitian|anti‑Hermitian]] {{math|''n'' × ''n''}} complex matrices, with the regular [[commutator]] as a Lie bracket. [[Particle physics|Particle physicists]] often use a different, equivalent representation: The set of traceless [[Hermitian]] {{math|''n'' × ''n''}} complex matrices with Lie bracket given by {{math|−''i''}} times the commutator.

== Lie algebra ==
{{main|Classical group#U(p, q) and U(n) – the unitary groups}}

The Lie algebra <math>\mathfrak{su}(n)</math> of <math>\operatorname{SU}(n)</math> consists of {{math|''n'' × ''n''}} [[skew Hermitian matrix|skew-Hermitian]] matrices with trace zero.<ref>{{harvnb|Hall|2015}} Proposition 3.24</ref> This (real) Lie algebra has dimension {{math|''n''<sup>2</sup> − 1}}. More information about the structure of this Lie algebra can be found below in ''{{slink|#Lie algebra structure}}''.

=== Fundamental representation ===
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 <math>i</math> from the mathematicians'. With this convention, one can then choose generators {{math|''T''<sub>''a''</sub>}} that are [[trace (linear algebra)|traceless]] [[Hermitian matrix|Hermitian]] complex {{math|''n'' × ''n''}} matrices, where:

<math display="block"> T_a \, T_b = \tfrac{1}{\, 2n \,}\,\delta_{ab}\,I_n + \tfrac{1}{2}\,\sum_{c=1}^{n^2 -1}\left(if_{abc} + d_{abc} \right) \, T_c </math>

where the {{math|''f''}} are the [[structure constants]] and are antisymmetric in all indices, while the {{math|''d''}}-coefficients are symmetric in all indices.

As a consequence, the commutator is:

<math display="block"> ~ \left[T_a, \, T_b\right] ~ = ~ i \sum_{c=1}^{n^2 -1} \, f_{abc} \, T_c \;,</math>

and the corresponding anticommutator is:

<math display="block"> \left\{T_a, \, T_b\right\} ~ = ~ \tfrac{1}{n} \, \delta_{ab} \, I_n + \sum_{c=1}^{n^2 -1}{d_{abc} \, T_c} ~.</math>

The factor of {{math|''i''}} in the commutation relation arises from the physics convention and is not present when using the mathematicians' convention.

The conventional normalization condition is


<math display="block">\sum_{c,e=1}^{n^2 - 1} d_{ace}\,d_{bce} = \frac{\, n^2 - 4 \,}{n} \, \delta_{ab}~ .</math>

The generators satisfy the Jacobi identity:<ref>{{Cite book |last=Georgi |first=Howard |url=https://www.taylorfrancis.com/books/9780429499210 |title=Lie Algebras in Particle Physics: From Isospin to Unified Theories |date=2018-05-04 |publisher=CRC Press |isbn=978-0-429-49921-0 |edition=1 |location=Boca Raton |language=en |doi=10.1201/9780429499210|bibcode=2018laip.book.....G }}</ref>

<math display="block">[T_a,[T_b,T_c]]+[T_b,[T_c,T_a]]+[T_c,[T_a,T_b]]=0.</math>

By convention, in the physics literature the generators <math>T_a</math> are defined as the traceless Hermitian complex matrices with a <math>1/2</math> prefactor: for the <math>SU(2)</math> group, the generators are chosen as <math>\frac{1}{2} \sigma_1, \frac{1}{2} \sigma_2, \frac{1}{2} \sigma_3</math> where <math>\sigma_a</math> are the [[Pauli matrices]], while for the case of <math>SU(3)</math> one defines <math>T_a = \frac{1}{2} \lambda_a</math> where <math>\lambda_a</math> are the [[Gell-Mann matrices]].<ref>{{Cite book |last=Georgi |first=Howard |url=https://www.taylorfrancis.com/books/9780429499210 |title=Lie Algebras in Particle Physics: From Isospin to Unified Theories |date=2018-05-04 |publisher=CRC Press |isbn=978-0-429-49921-0 |edition=1 |location=Boca Raton |language=en |doi=10.1201/9780429499210|bibcode=2018laip.book.....G }}</ref> With these definitions, the generators satisfy the following normalization condition:

<math display="block">Tr(T_a T_b) = \frac{1}{2} \delta_{ab}.</math>

=== Adjoint representation ===
In the {{math|(''n''<sup>2</sup> − 1)}}-dimensional [[adjoint representation of a Lie group|adjoint representation]], the generators are represented by {{math|(''n''<sup>2</sup> − 1) × (''n''<sup>2</sup> − 1)}} matrices, whose elements are defined by the structure constants themselves:

<math display="block">\left(T_a\right)_{jk} = -if_{ajk}.</math>

== The group SU(2) ==
{{see also|Versor|Pauli matrices|3D rotation group#A note on Lie algebras|Representation theory of SU(2)}}

Using [[matrix multiplication]] for the binary operation, {{math|SU(2)}} forms a group,<ref>{{harvnb|Hall|2015}} Exercise 1.5</ref>

<math display="block">\operatorname{SU}(2) = \left\{ \begin{pmatrix} \alpha & -\overline{\beta} \\ \beta & \overline{\alpha} \end{pmatrix}: \ \ \alpha,\beta \in \mathbb{C}, |\alpha|^2 + |\beta|^2 = 1 \right\}~,</math>

where the overline denotes [[complex conjugate|complex conjugation]].

=== Diffeomorphism with the 3-sphere ''S''<sup>3</sup> ===
If we consider <math>\alpha,\beta</math> as a pair in <math>\mathbb{C}^2</math> where <math>\alpha=a+bi</math> and <math>\beta=c+di</math>, then the equation <math>|\alpha|^2 + |\beta|^2 = 1</math> becomes

<math display="block"> a^2 + b^2 + c^2 + d^2 = 1 </math>

This is the equation of the [[3-sphere|3-sphere S<sup>3</sup>]]. This can also be seen using an embedding: the map

<math display="block">\begin{align}
  \varphi \colon \mathbb{C}^2 \to{} &\operatorname{M}(2, \mathbb{C}) \\[5pt]
         \varphi(\alpha, \beta) ={} &\begin{pmatrix} \alpha & -\overline{\beta}\\ \beta & \overline{\alpha}\end{pmatrix},
\end{align}</math>

where <math>\operatorname{M}(2,\mathbb{C})</math> denotes the set of 2 by 2 complex matrices, is an injective real linear map (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>). Hence, the [[restriction (mathematics)|restriction]] of {{math|''φ''}} to the [[3-sphere]] (since modulus is 1), denoted {{math|''S''<sup>3</sup>}}, is an embedding of the 3-sphere onto a compact submanifold of <math>\operatorname{M}(2,\mathbb{C})</math>, namely {{math|1=''φ''(''S''<sup>3</sup>) = SU(2)}}.

Therefore, as a manifold, {{math|''S''<sup>3</sup>}} is diffeomorphic to {{math|SU(2)}}, which shows that  {{math|SU(2)}} is [[simply connected space|simply connected]] and that {{math|''S''<sup>3</sup>}} can be endowed with the structure of a compact, connected [[Lie group]].

=== Isomorphism with group of versors ===
[[Quaternion]]s of norm 1 are called [[versor]]s since they generate the [[Rotation group SO(3)#Using quaternions of unit norm|rotation group SO(3)]]:
The {{math|SU(2)}} matrix:

<math display="block"> \begin{pmatrix} 
    a + bi & c + di \\ 
    -c + di &  a - bi
  \end{pmatrix} \quad
  (a, b, c, d \in \mathbb{R})
</math>

can be mapped to the quaternion

<math display="block">a\,\hat{1} + b\,\hat{i} + c\,\hat{j} + d\,\hat{k}</math>

This map is in fact a [[group isomorphism]]. Additionally, the determinant of the matrix is the squared norm of the corresponding quaternion. Clearly any matrix in {{math|SU(2)}} is of this form and, since it has determinant&nbsp;{{math|1}}, the corresponding quaternion has norm {{math|1}}. Thus {{math|SU(2)}} is isomorphic to the group of versors.<ref>{{cite web |url=http://alistairsavage.ca/mat4144/notes/MAT4144-5158-LieGroups.pdf |series=MATH 4144 notes |title=LieGroups |last=Savage |first=Alistair}}</ref>

=== Relation to spatial rotations ===
{{Main|3D rotation group#Connection between SO(3) and SU(2)|Quaternions and spatial rotation}}
Every versor is naturally associated to a spatial rotation in 3 dimensions, and the product of versors is associated to the composition of the associated rotations. Furthermore, every rotation arises from exactly two versors in this fashion. In short: there is a 2:1 surjective homomorphism from {{math|SU(2)}} to [[3D rotation group|{{math|SO(3)}}]]; consequently {{math|SO(3)}} is isomorphic to the [[quotient group]] {{math|SU(2)/{{mset|±I}}}}, the manifold underlying {{math|SO(3)}} is obtained by identifying antipodal points of the 3-sphere {{math|''S''<sup>3</sup>}}, and {{math|SU(2)}} is the [[Covering group#Universal covering group|universal cover]] of {{math|SO(3)}}.

=== Lie algebra <span class="anchor" id="Lie algebra basis"></span> ===
The [[Lie algebra]] of {{math|SU(2)}} consists of {{math|2 × 2}} [[skew Hermitian matrix|skew-Hermitian]] matrices with trace zero.<ref>{{harvnb|Hall|2015}} Proposition 3.24</ref> Explicitly, this means

<math display="block">\mathfrak{su}(2) = \left\{ \begin{pmatrix} i\ a & -\overline{z} \\ z & -i\ a \end{pmatrix}:\ a \in \mathbb{R}, z \in \mathbb{C} \right\}~.</math>

The Lie algebra is then generated by the following matrices,

<math display="block">u_1 = \begin{pmatrix}
    0 & i \\
    i & 0
  \end{pmatrix}, \quad
  u_2 = \begin{pmatrix}
    0 & -1 \\
    1 &  0
  \end{pmatrix}, \quad
  u_3 = \begin{pmatrix}
    i &  0 \\
    0 & -i
  \end{pmatrix}~,
</math>

which have the form of the general element specified above.

This can also be written as <math>\mathfrak{s u}(2)=\operatorname{span}\left\{i \sigma_{1}, i \sigma_{2}, i \sigma_{3}\right\}</math> using the [[Pauli matrices#SU(2)|Pauli matrices]].

These satisfy the [[quaternion]] relationships  <math>u_2\ u_3 = -u_3\ u_2 = u_1~,</math>  <math>u_3\ u_1 = -u_1\ u_3 = u_2~,</math>  and  <math>u_1 u_2 = -u_2\ u_1 = u_3~.</math> The [[commutator bracket]] is therefore specified by

<math display="block">\left[u_3, u_1\right] = 2\ u_2, \quad \left[u_1, u_2\right] = 2\ u_3, \quad \left[u_2, u_3\right] = 2\ u_1~.</math>

The above generators are related to the [[Pauli matrices]] by <math>u_1 = i\ \sigma_1~, \, u_2 = -i\ \sigma_2</math>  and  <math>u_3 = +i\ \sigma_3~.</math> This representation is routinely 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&nbsp;spatial dimensions in [[loop quantum gravity]]. They also correspond to the [[quantum logic gate#Pauli gates (X,Y,Z)|Pauli X, Y, and Z gates]], which are standard generators for the single qubit gates, corresponding to 3d rotations about the axes of the [[Bloch sphere]].

The Lie algebra serves to work out the [[representation theory of SU(2)|representations of {{math|SU(2)}}]].

== SU(3) ==
{{see also|Clebsch–Gordan coefficients for SU(3)}}

The group {{math|SU(3)}} is an 8-dimensional [[simple Lie group]] consisting of all {{math|3&nbsp;×&nbsp;3}} [[Unitary matrix|unitary]] [[Matrix (mathematics)|matrices]] with [[determinant]] 1.

=== Topology ===
The group {{math|SU(3)}} is a simply-connected, compact Lie group.<ref>{{harvnb|Hall|2015}} Proposition 13.11</ref> Its topological structure can be understood by noting that {{math|SU(3)}} acts [[transitive action|transitively]] on the unit sphere <math>S^5</math> in <math>\mathbb{C}^3 \cong \mathbb{R}^6</math>. The [[Group action (mathematics)#Fixed points and stabilizer subgroups|stabilizer]] of an arbitrary point in the sphere is isomorphic to {{math|SU(2)}}, which topologically is a 3-sphere. It then follows that {{math|SU(3)}} is a [[fiber bundle]] over the base {{math|''S''<sup>5</sup>}} with fiber {{math|''S''<sup>3</sup>}}. Since the fibers and the base are simply connected, the simple connectedness of {{math|SU(3)}} then follows by means of a standard topological result (the [[Homotopy group#Long exact sequence of a fibration|long exact sequence of homotopy groups]] for fiber bundles).<ref>{{harvnb|Hall|2015}} Section 13.2</ref>

The {{math|SU(2)}}-bundles over {{math|''S''<sup>5</sup>}} are classified by <math>\pi_4\mathord\left(S^3\right) = \mathbb{Z}_2</math> since any such bundle can be constructed by looking at trivial bundles on the two hemispheres <math>S^5_\text{N}, S^5_\text{S}</math> and looking at the transition function on their intersection, which is a copy of {{math|''S''<sup>4</sup>}}, so

<math display="block">S^5_\text{N} \cap S^5_\text{S} \simeq S^4</math>

Then, all such transition functions are classified by homotopy classes of maps

<math display="block">\left[S^4, \mathrm{SU}(2)\right] \cong \left[S^4, S^3\right] = \pi_4\mathord\left(S^3\right) \cong \mathbb{Z}/2</math>

and as <math>\pi_4(\mathrm{SU}(3)) = \{0\}</math> rather than <math>\mathbb{Z}/2</math>, {{math|SU(3)}} cannot be the trivial bundle {{math|SU(2) × ''S''<sup>5</sup> ≅ ''S''<sup>3</sup> × ''S''<sup>5</sup>}}, and therefore must be the unique nontrivial (twisted) bundle. This can be shown by looking at the induced long exact sequence on homotopy groups.

=== Representation theory ===
The representation theory of {{math|SU(3)}} is well-understood.<ref>{{harvnb|Hall|2015}} Chapter 6</ref> Descriptions of these representations, from the point of view of its complexified Lie algebra <math>\mathfrak{sl}(3; \mathbb{C})</math>, may be found in the articles on [[Lie algebra representation#Classifying finite-dimensional representations of Lie algebras|Lie algebra representations]] or [[Clebsch–Gordan coefficients for SU(3)#Representations of the SU(3) group|the Clebsch–Gordan coefficients for {{math|SU(3)}}]].

=== Lie algebra ===
The generators, {{mvar|T}}, of the Lie algebra <math>\mathfrak{su}(3)</math> of {{math|SU(3)}} in the defining (particle physics, Hermitian) representation, are

<math display="block">T_a = \frac{\lambda_a}{2}~,  </math>

where {{math|''λ''<sub>a</sub>}}, the [[Gell-Mann matrices]], are the {{math|SU(3)}} analog of the [[Pauli matrices]] for {{math|SU(2)}}:

<math display="block">\begin{align}
  \lambda_1 ={} &\begin{pmatrix} 0 &  1 &  0 \\ 1 &  0 &  0 \\ 0 & 0 &  0 \end{pmatrix}, & 
  \lambda_2 ={} &\begin{pmatrix} 0 & -i &  0 \\ i &  0 &  0 \\ 0 & 0 &  0 \end{pmatrix}, & 
  \lambda_3 ={} &\begin{pmatrix} 1 &  0 &  0 \\ 0 & -1 &  0 \\ 0 & 0 &  0 \end{pmatrix}, \\[6pt]
  \lambda_4 ={} &\begin{pmatrix} 0 &  0 &  1 \\ 0 &  0 &  0 \\ 1 & 0 &  0 \end{pmatrix}, & 
  \lambda_5 ={} &\begin{pmatrix} 0 &  0 & -i \\ 0 &  0 &  0 \\ i & 0 &  0 \end{pmatrix}, \\[6pt]
  \lambda_6 ={} &\begin{pmatrix} 0 &  0 &  0 \\ 0 &  0 &  1 \\ 0 & 1 &  0 \end{pmatrix}, & 
  \lambda_7 ={} &\begin{pmatrix} 0 &  0 &  0 \\ 0 &  0 & -i \\ 0 & i &  0 \end{pmatrix}, & 
  \lambda_8 = \frac{1}{\sqrt{3}}
                &\begin{pmatrix} 1 &  0 &  0 \\ 0 &  1 &  0 \\ 0 & 0 & -2 \end{pmatrix}.
\end{align}</math>

These {{math|''λ''<sub>a</sub>}} span all [[trace (linear algebra)|traceless]] [[Hermitian matrix|Hermitian matrices]] {{mvar|H}} of the [[Lie algebra]], as required. Note that {{math|''λ''<sub>2</sub>, ''λ''<sub>5</sub>, ''λ''<sub>7</sub>}} are antisymmetric.

They obey the relations

<math display="block">\begin{align}
    \left[T_a, T_b\right] &= i \sum_{c=1}^8 f_{abc} T_c, \\
  \left\{T_a, T_b\right\} &= \frac{1}{3} \delta_{ab} I_3 + \sum_{c=1}^8 d_{abc} T_c,
\end{align}</math>

or, equivalently,

<math display="block">\begin{align}
            \left[\lambda_a, \lambda_b\right] &= 2i \sum_{c=1}^8 f_{abc} \lambda_c, \\
             \{\lambda_a, \lambda_b\} &= \frac{4}{3}\delta_{ab} I_3 + 2\sum_{c=1}^8{d_{abc} \lambda_c}.
\end{align}</math>

The {{mvar|f}} are the [[structure constants]] of the Lie algebra, given by

<math display="block">\begin{align}
                                                      f_{123} &= 1, \\
  f_{147} = -f_{156} = f_{246} = f_{257} = f_{345} = -f_{367} &= \frac{1}{2}, \\
                                           f_{458} =  f_{678} &= \frac{\sqrt{3}}{2},
\end{align}</math>

while all other {{math|''f<sub>abc</sub>''}} not related to these by permutation are zero. In general, they vanish unless they contain an odd number of indices from the set {{math|{{mset|2, 5, 7}}}}.{{efn|So fewer than {{frac|1|6}} of all {{math|''f<sub>abc</sub>''}}s are non-vanishing.}}

The symmetric coefficients {{math|''d''}} take the values

<math display="block">\begin{align}
                                             d_{118} = d_{228} = d_{338} = -d_{888} &=  \frac{1}{\sqrt{3}} \\
                                             d_{448} = d_{558} = d_{668} =  d_{778} &= -\frac{1}{2\sqrt{3}} \\
  d_{344} = d_{355} = -d_{366} = -d_{377} = -d_{247} = d_{146} = d_{157} =  d_{256} &=  \frac{1}{2} ~.
\end{align}</math>

They vanish if the number of indices from the set {{math|{{mset|2, 5, 7}}}} is odd.

A generic {{math|SU(3)}} group element generated by a traceless 3×3 Hermitian matrix {{mvar|H}}, normalized as {{math|tr(''H''<sup>2</sup>) {{=}} 2}}, can be expressed as a ''second order'' matrix polynomial in {{mvar|H}}:<ref>{{cite journal|last1=Rosen|first1=S P|title=Finite Transformations in Various Representations of SU(3)|journal=Journal of Mathematical Physics|volume=12|issue=4|year=1971|pages=673–681 |doi=10.1063/1.1665634|bibcode=1971JMP....12..673R}};  {{cite journal|doi=10.1016/S0034-4877(15)30040-9|title= Elementary results for the fundamental representation of SU(3)|author1= Curtright, T L|author2=Zachos, C K|year=2015|journal=Reports on Mathematical Physics|volume=76|issue=3|pages=401–404|bibcode=2015RpMP...76..401C|arxiv=1508.00868|s2cid= 119679825}}</ref>

<math display="block">\begin{align}
  \exp(i\theta H) ={}
          &\left[-\frac{1}{3} I\sin\left(\varphi + \frac{2\pi}{3}\right) \sin\left(\varphi - \frac{2\pi}{3}\right) - \frac{1}{2\sqrt{3}}~H\sin(\varphi) - \frac{1}{4}~H^2\right]
           \frac{\exp\left(\frac{2}{\sqrt{3}}~i\theta\sin(\varphi)\right)}
                {\cos\left(\varphi + \frac{2\pi}{3}\right) \cos\left(\varphi - \frac{2\pi}{3}\right)} \\[6pt]
    & {} + \left[-\frac{1}{3}~I\sin(\varphi) \sin\left(\varphi - \frac{2\pi}{3}\right) - \frac{1}{2\sqrt{3}}~H\sin\left(\varphi + \frac{2\pi}{3}\right) - \frac{1}{4}~H^{2}\right]
           \frac{\exp\left(\frac{2}{\sqrt{3}}~i\theta \sin\left(\varphi + \frac{2\pi}{3}\right)\right)}
                {\cos(\varphi) \cos\left(\varphi - \frac{2\pi}{3}\right)} \\[6pt]
    & {} + \left[-\frac{1}{3}~I\sin(\varphi) \sin\left(\varphi + \frac{2\pi}{3}\right) - \frac{1}{2\sqrt{3}}~H \sin\left(\varphi - \frac{2\pi}{3}\right) - \frac{1}{4}~H^2\right]
           \frac{\exp\left(\frac{2}{\sqrt{3}}~i\theta \sin\left(\varphi - \frac{2\pi}{3}\right)\right)}
                {\cos(\varphi)\cos\left(\varphi + \frac{2\pi}{3}\right)}
\end{align}</math>
LP
where

<math display="block">\varphi \equiv \frac{1}{3}\left[\arccos\left(\frac{3\sqrt{3}}{2}\det H\right) - \frac{\pi}{2}\right].</math>

== Lie algebra structure ==
As noted above, the Lie algebra <math>\mathfrak{su}(n)</math> of {{math|SU(''n'')}} consists of {{math|''n'' × ''n''}} [[skew Hermitian matrix|skew-Hermitian]] matrices with trace zero.<ref>{{harvnb|Hall|2015}} Proposition 3.24</ref>

The [[complexification]] of the Lie algebra <math>\mathfrak{su}(n)</math> is <math>\mathfrak{sl}(n; \mathbb{C})</math>, the space of all {{math|''n'' × ''n''}} complex matrices with trace zero.<ref>{{harvnb|Hall|2015}} Section 3.6</ref> A [[Cartan subalgebra]] then consists of the diagonal matrices with trace zero,<ref>{{harvnb|Hall|2015}} Section 7.7.1</ref> which we identify with vectors in <math>\mathbb C^n</math> whose entries sum to zero. The [[root system|roots]] then consist of all the {{math|''n''(''n'' − 1)}} permutations of {{math|(1, −1, 0, ..., 0)}}.

A choice of [[simple root (root system)|simple root]]s is

<math display="block">\begin{align}
  (&1, -1,  0, \dots, 0,  0), \\
  (&0,  1, -1, \dots, 0,  0), \\
   &\vdots                    \\
  (&0,  0,  0, \dots, 1, -1).
\end{align}</math>

So, {{math|SU(''n'')}} is of [[Rank (linear algebra)|rank]] {{math|''n'' − 1}} and its [[Dynkin diagram]] is given by {{math|A<sub>''n''−1</sub>}}, a chain of {{math|''n'' − 1}} nodes: {{Dynkin|node|3|node|3|node|3}}...{{Dynkin|3|node}}.<ref>{{harvnb|Hall|2015}} Section 8.10.1</ref> Its [[Cartan matrix]] is

<math display="block">\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<sub>''n''</sub>}}, the [[symmetry group]] of the {{math|(''n'' − 1)}}-[[simplex]].

== Generalized special unitary group ==
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 {{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]].

Specifically, fix a [[Hermitian matrix]] {{math|''A''}} of signature {{math|''p'' ''q''}} in <math>\operatorname{GL}(n, \mathbb{R})</math>, then all

<math display="block">M \in \operatorname{SU}(p, q, \mathbb{R})</math>

satisfy

<math display="block">\begin{align}
  M^{*} A M &= A \\
     \det M &= 1.
\end{align}</math>

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>\mathbb C</math> and this gives one of the classical [[Lie group]]s. The standard choice for {{math|''A''}} when <math>\operatorname{F} = \mathbb{C}</math> is

<math display="block">A = \begin{bmatrix}
     0 & 0       & i \\
     0 & I_{n-2} & 0 \\
    -i & 0       & 0
  \end{bmatrix}.
</math>

However, there may be better choices for {{math|''A''}} for certain dimensions which exhibit more behaviour under restriction to subrings of <math>\mathbb C</math>.

=== Example ===
An important example of this type of group is the [[Picard modular group]] <math>\operatorname{SU}(2, 1; \mathbb{Z}[i])</math> which acts (projectively) on complex hyperbolic space of dimension two, in the same way that <math>\operatorname{SL}(2,9;\mathbb{Z})</math> 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 |eprint=math/0509708 |date=September 2005 |title=An explicit fundamental domain for the Picard modular group in two complex dimensions |last1=Francsics |first1=Gabor |last2=Lax |first2=Peter D.}}</ref>

A further example is <math>\operatorname{SU}(1, 1; \mathbb{C})</math>, which is isomorphic to <math>\operatorname{SL}(2, \mathbb{R})</math>.

== Important subgroups ==
In physics the special unitary group is used to represent [[fermionic]] 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 {{math|''p'' > 1, ''n'' − ''p'' > 1}},

<math display="block">\operatorname{SU}(n) \supset \operatorname{SU}(p) \times \operatorname{SU}(n - p) \times \operatorname{U}(1),</math>

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]]&nbsp;1.

For completeness, there are also the [[orthogonal group|orthogonal]] and [[compact symplectic group|symplectic]] subgroups,

<math display="block">\begin{align}
   \operatorname{SU}(n) &\supset \operatorname{SO}(n), \\
  \operatorname{SU}(2n) &\supset \operatorname{Sp}(n).
\end{align}</math>

Since the [[rank of a Lie group|rank]] of {{math|SU(''n'')}} is {{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,

<math display="block">\begin{align}
   \operatorname{SO}(2n) &\supset \operatorname{SU}(n) \\
    \operatorname{Sp}(n) &\supset \operatorname{SU}(n) \\
  \operatorname{Spin}(4) &= \operatorname{SU}(2) \times \operatorname{SU}(2) \\
      \operatorname{E}_6 &\supset \operatorname{SU}(6) \\
      \operatorname{E}_7 &\supset \operatorname{SU}(8) \\
      \operatorname{G}_2 &\supset \operatorname{SU}(3)
\end{align}</math>
See ''[[Spin group]]'' and ''[[Simple Lie group]]'' for {{math|E<sub>6</sub>}}, {{math|E<sub>7</sub>}}, and {{math|G<sub>2</sub>}}.

There are also the [[spin group#Exceptional isomorphisms|accidental isomorphisms]]: {{math|1 = SU(4) = Spin(6)}}, {{math|1 = SU(2) = Spin(3) = Sp(1)}},{{efn|{{math|Sp(''n'')}} is the [[compact real form]] of <math>\operatorname{Sp}(2n, \mathbb{C})</math>. It is sometimes denoted {{math|USp(''2n'')}}. The dimension of the {{math|Sp(''n'')}}-matrices is {{math|2''n'' × 2''n''}}.}} and {{math|1 = U(1) = Spin(2) = SO(2)}}.

One may 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]].

== SU(1, 1) ==
<math>\mathrm{SU}(1,1) = \left \{ \begin{pmatrix}u & v \\ v^* & u^* \end{pmatrix} \in M(2,\mathbb{C}): u u^* - v v^* = 1 \right \},</math> where <math>~u^*~</math> denotes the [[complex conjugate]] of the complex number {{mvar|u}}.

This group is isomorphic to {{math|SL(2,ℝ)}} and {{math|Spin(2,1)}}<ref>{{cite book |first=Robert |last=Gilmore |year=1974 |title=Lie Groups, Lie Algebras and some of their Applications |pages=52, 201−205 |publisher=[[John Wiley & Sons]] |mr=1275599}}</ref> where the numbers separated by a comma refer to the [[signature (quadratic form)|signature]] of the [[quadratic form]] preserved by the group. The expression <math>~u u^* - v v^*~</math> in the definition of {{math|SU(1,1)}} is an [[Hermitian form]] which becomes an [[isotropic quadratic form]] when {{mvar|u}} and {{math|''v''}} are expanded with their real components.

An early appearance of this group was as the "unit sphere" of [[coquaternion]]s, introduced by [[James Cockle]] in 1852. Let

<math display="block">
  j = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\,, \quad
  k = \begin{bmatrix} 1 & \;~0 \\ 0 & -1 \end{bmatrix}\,, \quad
  i = \begin{bmatrix} \;~0 & 1 \\ -1 & 0 \end{bmatrix}~.
</math>

Then <math>~j\,k = \begin{bmatrix} 0 & -1 \\ 1 & \;~0 \end{bmatrix} = -i ~,~</math> <math>~ i\,j\,k = I_2 \equiv \begin{bmatrix} 1 & 0 \\ 0 &  1 \end{bmatrix}~,~</math> the 2×2 identity matrix, <math>~k\,i = j ~,</math> and <math>\;i\,j = k \;,</math> and the elements {{mvar|i, j,}} and {{mvar|k}} all [[anticommutative property|anticommute]], as in [[quaternion]]s. Also <math>i</math> is still a square root of {{math|−''I''{{sub|2}}}} (negative of the identity matrix), whereas <math>~j^2 = k^2 = +I_2~</math> are not, unlike in quaternions. For both quaternions and [[coquaternion]]s, all scalar quantities are treated as implicit multiples of {{mvar|I}}{{sub|2}} and notated as {{math|'''1'''}}.

The coquaternion <math>~q = w + x\,i + y\,j + z\,k~</math> with scalar {{mvar|w}}, has conjugate <math>~q = w - x\,i - y\,j - z\,k~</math> similar to Hamilton's quaternions. The quadratic form is <math>~q\,q^* = w^2 + x^2 - y^2 - z^2.</math>

Note that the 2-sheet [[hyperboloid]] <math>\left\{ x i + y j + z k : x^2 - y^2 - z^2 = 1 \right\}</math> corresponds to the [[imaginary unit]]s in the algebra so that any point {{mvar|p}} on this hyperboloid can be used as a '''pole''' of a sinusoidal wave according to [[Euler's formula]].

The hyperboloid is stable under {{math|SU(1, 1)}}, illustrating the isomorphism with {{math|Spin(2, 1)}}. The variability of the pole of a wave, as noted in studies of [[polarization (waves)|polarization]], might view [[elliptical polarization]] as an exhibit of the elliptical shape of a wave with {{nowrap|pole <math>~p \ne \pm i~</math>.}} The [[Poincaré sphere (optics)|Poincaré sphere]] model used since 1892 has been compared to a 2-sheet hyperboloid model,<ref>{{cite journal |first1=R.D. |last1=Mota |first2=D. |last2=Ojeda-Guillén |first3=M. |last3=Salazar-Ramírez |first4=V.D. |last4=Granados |year=2016 |title=SU(1,1) approach to Stokes parameters and the theory of light polarization |journal=Journal of the Optical Society of America B |volume=33 |issue=8 |pages=1696–1701 |arxiv=1602.03223 |doi=10.1364/JOSAB.33.001696|bibcode=2016JOSAB..33.1696M |s2cid=119146980 }}</ref> and the practice of [[SU(1,1) interferometry|{{math|SU(1, 1)}} interferometry]] has been introduced.

When an element of {{math|SU(1, 1)}} is interpreted as a [[Möbius transformation]], it leaves the [[unit disk]] stable, so this group represents the [[motion (geometry)|motion]]s of the [[Poincaré disk model]] of hyperbolic plane geometry. Indeed, for a point {{math|<big>[</big>z, 1<big>]</big>}} in the [[complex projective line]], the action of {{math|SU(1,1)}} is given by

<math display="block">\begin{pmatrix}u & v \\ v^* & u^* \end{pmatrix}\,\bigl[\;z,\;1\;\bigr] = [\;u\,z + v, \, v^*\,z +u^*\;] \, = \, \left[\;\frac{uz + v}{v^*z +u^*}, \, 1 \;\right]</math>

since in [[projective coordinates]] <math>(\;u\,z + v, \; v^*\,z +u^*\;) \thicksim \left(\;\frac{\,u\,z + v\,}{v^*\,z +u^*}, \; 1 \;\right).</math>

Writing <math>\;suv + \overline{suv} = 2\,\Re\mathord\bigl(\,suv\,\bigr)\;,</math> complex number arithmetic shows

<math display="block">\bigl|u\,z + v\bigr|^2 = S + z\,z^* \quad \text{ and } \quad \bigl|v^*\,z + u^*\bigr|^2 = S + 1~,</math>
where <math>S = v\,v^* \left(z\,z^* + 1\right) + 2\,\Re\mathord\bigl(\,uvz\,\bigr).</math>

Therefore, <math>z\,z^* < 1 \implies \bigl|uz + v\bigr| < \bigl|\,v^*\,z + u^*\,\bigr|</math> so that their ratio lies in the open disk.<ref>{{cite book |author-link=C. L. Siegel |last=Siegel |first=C. L. |year=1971 |title=Topics in Complex Function Theory |volume=2 |pages=13–15 |translator1-last=Shenitzer |translator1-first=A. |translator2-last=Tretkoff |translator2-first=M. |publisher=Wiley-Interscience |isbn=0-471-79080 X}}</ref>

== See also ==
{{Portal|Mathematics}}
* [[Unitary group]]
* [[Projective special unitary group]], {{math|PSU(''n'')}}
* [[Orthogonal group]]
* [[Generalizations of Pauli matrices]]
* [[Representation theory of SU(2)]]

== Footnotes ==
{{notelist|1}}

== Citations ==
{{Reflist|25em}}

== References ==
* {{Citation| last=Hall|first=Brian C.|title=Lie Groups, Lie Algebras, and Representations: An Elementary Introduction|edition=2nd|series=Graduate Texts in Mathematics|volume=222|publisher=Springer|year=2015|isbn=978-3319134666}}
* {{Citation| last=Iachello|first=Francesco|title=Lie Algebras and Applications|series=Lecture Notes in Physics|volume=708|publisher=Springer|year=2006|isbn=3540362363}}

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