Special unitary group#The group SU(2)

The special unitary group {{math|SU(n)}} is a strictly real Lie group (vs. a more general complex Lie group). Its dimension as a real manifold is {{math|n² − 1}}. Topologically, it is compact and simply connected.{{harvnb|Hall|2015}}, Proposition 13.11 Algebraically, it is a simple Lie group (meaning its Lie algebra is simple; see below).{{cite book |author-link=Brian Garner Wybourne |author=Wybourne, B.G. |year=1974 |title=Classical Groups for Physicists |publisher=Wiley-Interscience |isbn=0471965057}}

The center of {{math|SU(n)}} is isomorphic to the cyclic group $\backslash mathbb\{Z\}/n\backslash mathbb\{Z\}$, 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 $\backslash mathbb\{Z\}/2\backslash mathbb\{Z\},$ while the outer automorphism group of {{math|SU(2)}} is the trivial group.

A maximal torus of rank {{math|n − 1}} is given by the set of diagonal matrices with determinant {{math|1}}. The Weyl group of {{math|SU(n)}} is the symmetric group {{math|S_n}}, which is represented by signed permutation matrices (the signs being necessary to ensure that the determinant is {{math|1}}).

The Lie algebra of {{math|SU(n)}}, denoted by $\backslash mathfrak\{su\}(n)$, can be identified with the set of traceless anti‑Hermitian {{math|n × n}} complex matrices, with the regular commutator as a Lie bracket. 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.

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

The Lie algebra $\backslash mathfrak\{su\}(n)$ of $\backslash operatorname\{SU\}(n)$ consists of {{math|n × n}} skew-Hermitian matrices with trace zero.{{harvnb|Hall|2015}} Proposition 3.24 This (real) Lie algebra has dimension {{math|n² − 1}}. More information about the structure of this Lie algebra can be found below in {{slink|#Lie algebra structure}}.

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'. With this convention, one can then choose generators {{math|T_a}} that are traceless Hermitian complex {{math|n × n}} matrices, where:

$$T\_a\; \backslash ,\; T\_b\; =\; \backslash tfrac\{1\}\{\backslash ,\; 2n\; \backslash ,\}\backslash ,\backslash delta\_\{ab\}\backslash ,I\_n\; +\; \backslash tfrac\{1\}\{2\}\backslash ,\backslash sum\_\{c=1\}^\{n^2\; -1\}\backslash left(if\_\{abc\}\; +\; d\_\{abc\}\; \backslash right)\; \backslash ,\; T\_c$$

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:

$$~\; \backslash left[T\_a,\; \backslash ,\; T\_b\backslash right]\; ~\; =\; ~\; i\; \backslash sum\_\{c=1\}^\{n^2\; -1\}\; \backslash ,\; f\_\{abc\}\; \backslash ,\; T\_c\; \backslash ;,$$

and the corresponding anticommutator is:

$$\backslash left\backslash \{T\_a,\; \backslash ,\; T\_b\backslash right\backslash \}\; ~\; =\; ~\; \backslash tfrac\{1\}\{n\}\; \backslash ,\; \backslash delta\_\{ab\}\; \backslash ,\; I\_n\; +\; \backslash sum\_\{c=1\}^\{n^2\; -1\}\{d\_\{abc\}\; \backslash ,\; T\_c\}\; ~.$$

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

$$\backslash sum\_\{c,e=1\}^\{n^2\; -\; 1\}\; d\_\{ace\}\backslash ,d\_\{bce\}\; =\; \backslash frac\{\backslash ,\; n^2\; -\; 4\; \backslash ,\}\{n\}\; \backslash ,\; \backslash delta\_\{ab\}~\; .$$

The generators satisfy the Jacobi identity:{{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 }}

$$[T\_a,[T\_b,T\_c]]+[T\_b,[T\_c,T\_a]]+[T\_c,[T\_a,T\_b]]=0.$$

By convention, in the physics literature the generators $T\_a$ are defined as the traceless Hermitian complex matrices with a $1/2$ prefactor: for the $SU(2)$ group, the generators are chosen as $\backslash frac\{1\}\{2\}\; \backslash sigma\_1,\; \backslash frac\{1\}\{2\}\; \backslash sigma\_2,\; \backslash frac\{1\}\{2\}\; \backslash sigma\_3$ where $\backslash sigma\_a$ are the Pauli matrices, while for the case of $SU(3)$ one defines $T\_a\; =\; \backslash frac\{1\}\{2\}\; \backslash lambda\_a$ where $\backslash lambda\_a$ are the Gell-Mann matrices.{{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 }} With these definitions, the generators satisfy the following normalization condition:

$$Tr(T\_a\; T\_b)\; =\; \backslash frac\{1\}\{2\}\; \backslash delta\_\{ab\}.$$

In the {{math|(n² − 1)}}-dimensional adjoint representation, the generators are represented by {{math|(n² − 1) × (n² − 1)}} matrices, whose elements are defined by the structure constants themselves:

$$\backslash left(T\_a\backslash right)\_\{jk\}\; =\; -if\_\{ajk\}.$$

{{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,{{harvnb|Hall|2015}} Exercise 1.5

where the overline denotes complex conjugation.

If we consider $\backslash alpha,\backslash beta$ as a pair in $\backslash mathbb\{C\}^2$ where $\backslash alpha=a+bi$ and $\backslash beta=c+di$, then the equation $|\backslash alpha|^2\; +\; |\backslash beta|^2\; =\; 1$ becomes

$$a^2\; +\; b^2\; +\; c^2\; +\; d^2\; =\; 1$$

This is the equation of the 3-sphere S³. This can also be seen using an embedding: the map

$$\backslash begin\{align\}$$

\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}

where $\backslash operatorname\{M\}(2,\backslash mathbb\{C\})$ denotes the set of 2 by 2 complex matrices, is an injective real linear map (by considering $\backslash mathbb\{C\}^2$ diffeomorphic to $\backslash mathbb\{R\}^4$ and $\backslash operatorname\{M\}(2,\backslash mathbb\{C\})$ diffeomorphic to $\backslash mathbb\{R\}^8$). Hence, the restriction of {{math|φ}} to the 3-sphere (since modulus is 1), denoted {{math|S³}}, is an embedding of the 3-sphere onto a compact submanifold of $\backslash operatorname\{M\}(2,\backslash mathbb\{C\})$, namely {{math|1=φ(S³) = SU(2)}}.

Therefore, as a manifold, {{math|S³}} is diffeomorphic to {{math|SU(2)}}, which shows that {{math|SU(2)}} is simply connected and that {{math|S³}} can be endowed with the structure of a compact, connected Lie group.

Quaternions of norm 1 are called versors since they generate the rotation group SO(3):

The {{math|SU(2)}} matrix:

$$\backslash begin\{pmatrix\}$$

a + bi & c + di \\

-c + di & a - bi

\end{pmatrix} \quad

(a, b, c, d \in \mathbb{R})

can be mapped to the quaternion

$$a\backslash ,\backslash hat\{1\}\; +\; b\backslash ,\backslash hat\{i\}\; +\; c\backslash ,\backslash hat\{j\}\; +\; d\backslash ,\backslash hat\{k\}$$

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 {{math|1}}, the corresponding quaternion has norm {{math|1}}. Thus {{math|SU(2)}} is isomorphic to the group of versors.{{cite web |url=http://alistairsavage.ca/mat4144/notes/MAT4144-5158-LieGroups.pdf |series=MATH 4144 notes |title=LieGroups |last=Savage |first=Alistair}}

{{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; 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|S³}}, and {{math|SU(2)}} is the universal cover of {{math|SO(3)}}.

The Lie algebra of {{math|SU(2)}} consists of {{math|2 × 2}} skew-Hermitian matrices with trace zero.{{harvnb|Hall|2015}} Proposition 3.24 Explicitly, this means

The Lie algebra is then generated by the following matrices,

$$u\_1\; =\; \backslash begin\{pmatrix\}$$

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}~,

which have the form of the general element specified above.

This can also be written as $\backslash mathfrak\{s\; u\}(2)=\backslash operatorname\{span\}\backslash left\backslash \{i\; \backslash sigma\_\{1\},\; i\; \backslash sigma\_\{2\},\; i\; \backslash sigma\_\{3\}\backslash right\backslash \}$ using the Pauli matrices.

These satisfy the quaternion relationships $u\_2\backslash \; u\_3\; =\; -u\_3\backslash \; u\_2\; =\; u\_1~,$ $u\_3\backslash \; u\_1\; =\; -u\_1\backslash \; u\_3\; =\; u\_2~,$ and $u\_1\; u\_2\; =\; -u\_2\backslash \; u\_1\; =\; u\_3~.$ The commutator bracket is therefore specified by

$$\backslash left[u\_3,\; u\_1\backslash right]\; =\; 2\backslash \; u\_2,\; \backslash quad\; \backslash left[u\_1,\; u\_2\backslash right]\; =\; 2\backslash \; u\_3,\; \backslash quad\; \backslash left[u\_2,\; u\_3\backslash right]\; =\; 2\backslash \; u\_1~.$$

The above generators are related to the Pauli matrices by $u\_1\; =\; i\backslash \; \backslash sigma\_1~,\; \backslash ,\; u\_2\; =\; -i\backslash \; \backslash sigma\_2$ and $u\_3\; =\; +i\backslash \; \backslash sigma\_3~.$ This representation is routinely used in quantum mechanics to represent the spin of fundamental particles such as electrons. They also serve as unit vectors for the description of our 3 spatial dimensions in loop quantum gravity. They also correspond to the 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).

{{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 × 3}} unitary matrices with determinant 1.

The group {{math|SU(3)}} is a simply-connected, compact Lie group.{{harvnb|Hall|2015}} Proposition 13.11 Its topological structure can be understood by noting that {{math|SU(3)}} acts transitively on the unit sphere $S^5$ in $\backslash mathbb\{C\}^3\; \backslash cong\; \backslash mathbb\{R\}^6$. The 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|S⁵}} with fiber {{math|S³}}. 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 long exact sequence of homotopy groups for fiber bundles).{{harvnb|Hall|2015}} Section 13.2

The {{math|SU(2)}}-bundles over {{math|S⁵}} are classified by $\backslash pi\_4\backslash mathord\backslash left(S^3\backslash right)\; =\; \backslash mathbb\{Z\}\_2$ since any such bundle can be constructed by looking at trivial bundles on the two hemispheres $S^5\_\backslash text\{N\},\; S^5\_\backslash text\{S\}$ and looking at the transition function on their intersection, which is a copy of {{math|S⁴}}, so

$$S^5\_\backslash text\{N\}\; \backslash cap\; S^5\_\backslash text\{S\}\; \backslash simeq\; S^4$$

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

$$\backslash left[S^4,\; \backslash mathrm\{SU\}(2)\backslash right]\; \backslash cong\; \backslash left[S^4,\; S^3\backslash right]\; =\; \backslash pi\_4\backslash mathord\backslash left(S^3\backslash right)\; \backslash cong\; \backslash mathbb\{Z\}/2$$

and as $\backslash pi\_4(\backslash mathrm\{SU\}(3))\; =\; \backslash \{0\backslash \}$ rather than $\backslash mathbb\{Z\}/2$, {{math|SU(3)}} cannot be the trivial bundle {{math|SU(2) × S⁵ ≅ S³ × S⁵}}, and therefore must be the unique nontrivial (twisted) bundle. This can be shown by looking at the induced long exact sequence on homotopy groups.

The representation theory of {{math|SU(3)}} is well-understood.{{harvnb|Hall|2015}} Chapter 6 Descriptions of these representations, from the point of view of its complexified Lie algebra $\backslash mathfrak\{sl\}(3;\; \backslash mathbb\{C\})$, may be found in the articles on Lie algebra representations or Clebsch–Gordan coefficients for SU(3)#Representations of the SU(3) group.

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

$$T\_a\; =\; \backslash frac\{\backslash lambda\_a\}\{2\}~,$$

where {{math|λ_a}}, the Gell-Mann matrices, are the {{math|SU(3)}} analog of the Pauli matrices for {{math|SU(2)}}:

$$\backslash begin\{align\}$$

\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}

These {{math|λ_a}} span all traceless Hermitian matrices {{mvar|H}} of the Lie algebra, as required. Note that {{math|λ₂, λ₅, λ₇}} are antisymmetric.

They obey the relations

$$\backslash begin\{align\}$$

\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}

or, equivalently,

$$\backslash begin\{align\}$$

\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}

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

$$\backslash begin\{align\}$$

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}

while all other {{math|f_abc}} 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_abc}}s are non-vanishing.}}

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

$$\backslash begin\{align\}$$

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}

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(H²) {{=}} 2}}, can be expressed as a second order matrix polynomial in {{mvar|H}}:{{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}}

$$\backslash begin\{align\}$$

\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}

LP

where

$$\backslash varphi\; \backslash equiv\; \backslash frac\{1\}\{3\}\backslash left[\backslash arccos\backslash left(\backslash frac\{3\backslash sqrt\{3\}\}\{2\}\backslash det\; H\backslash right)\; -\; \backslash frac\{\backslash pi\}\{2\}\backslash right].$$

As noted above, the Lie algebra $\backslash mathfrak\{su\}(n)$ of {{math|SU(n)}} consists of {{math|n × n}} skew-Hermitian matrices with trace zero.{{harvnb|Hall|2015}} Proposition 3.24

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

A choice of simple roots is

$$\backslash begin\{align\}$$

(&1, -1, 0, \dots, 0, 0), \\

(&0, 1, -1, \dots, 0, 0), \\

&\vdots \\

(&0, 0, 0, \dots, 1, -1).

\end{align}

So, {{math|SU(n)}} is of rank {{math|n − 1}} and its Dynkin diagram is given by {{math|A_n−1}}, a chain of {{math|n − 1}} nodes: {{Dynkin|node|3|node|3|node|3}}...{{Dynkin|3|node}}.{{harvnb|Hall|2015}} Section 8.10.1 Its Cartan matrix is

$$\backslash begin\{pmatrix\}$$

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}.

Its Weyl group or Coxeter group is the symmetric group {{math|S_n}}, the symmetry group of the {{math|(n − 1)}}-simplex.

For a field {{math|F}}, the generalized special unitary group over F, {{math|SU(p, q; F)}}, is the group of all linear transformations of determinant 1 of a vector space of rank {{math|1=n = p + q}} over {{math|F}} which leave invariant a nondegenerate, Hermitian form of 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 $\backslash operatorname\{GL\}(n,\; \backslash mathbb\{R\})$, then all

$$M\; \backslash in\; \backslash operatorname\{SU\}(p,\; q,\; \backslash mathbb\{R\})$$

satisfy

$$\backslash begin\{align\}$$

M^{*} A M &= A \\

\det M &= 1.

\end{align}

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 $\backslash mathbb\; C$ and this gives one of the classical Lie groups. The standard choice for {{math|A}} when $\backslash operatorname\{F\}\; =\; \backslash mathbb\{C\}$ is

$$A\; =\; \backslash begin\{bmatrix\}$$

0 & 0 & i \\

0 & I_{n-2} & 0 \\

-i & 0 & 0

\end{bmatrix}.

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

An important example of this type of group is the Picard modular group $\backslash operatorname\{SU\}(2,\; 1;\; \backslash mathbb\{Z\}[i])$ which acts (projectively) on complex hyperbolic space of dimension two, in the same way that $\backslash operatorname\{SL\}(2,9;\backslash mathbb\{Z\})$ 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|HC²}}.{{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.}}

A further example is $\backslash operatorname\{SU\}(1,\; 1;\; \backslash mathbb\{C\})$, which is isomorphic to $\backslash operatorname\{SL\}(2,\; \backslash mathbb\{R\})$.

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 GUT physics are, for {{math|p > 1, n − p > 1}},

$$\backslash operatorname\{SU\}(n)\; \backslash supset\; \backslash operatorname\{SU\}(p)\; \backslash times\; \backslash operatorname\{SU\}(n\; -\; p)\; \backslash times\; \backslash operatorname\{U\}(1),$$

where × denotes the direct product and {{math|U(1)}}, known as the circle group, is the multiplicative group of all complex numbers with absolute value 1.

For completeness, there are also the orthogonal and symplectic subgroups,

$$\backslash begin\{align\}$$

\operatorname{SU}(n) &\supset \operatorname{SO}(n), \\

\operatorname{SU}(2n) &\supset \operatorname{Sp}(n).

\end{align}

Since the 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,

$$\backslash begin\{align\}$$

\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}

See Spin group and Simple Lie group for {{math|E₆}}, {{math|E₇}}, and {{math|G₂}}.

There are also the 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 $\backslash operatorname\{Sp\}(2n,\; \backslash mathbb\{C\})$. It is sometimes denoted {{math|USp(2n)}}. The dimension of the {{math|Sp(n)}}-matrices is {{math|2n × 2n}}.}} 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-spinors in non-relativistic quantum mechanics.

where $~u^*~$ denotes the complex conjugate of the complex number {{mvar|u}}.

This group is isomorphic to {{math|SL(2,ℝ)}} and {{math|Spin(2,1)}}{{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}} where the numbers separated by a comma refer to the signature of the quadratic form preserved by the group. The expression $~u\; u^*\; -\; v\; v^*~$ 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 coquaternions, introduced by James Cockle in 1852. Let

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}~.

Then the 2×2 identity matrix, $~k\backslash ,i\; =\; j\; ~,$ and $\backslash ;i\backslash ,j\; =\; k\; \backslash ;,$ and the elements {{mvar|i, j,}} and {{mvar|k}} all anticommute, as in quaternions. Also $i$ is still a square root of {{math|−I{{sub|2}}}} (negative of the identity matrix), whereas $~j^2\; =\; k^2\; =\; +I\_2~$ are not, unlike in quaternions. For both quaternions and coquaternions, all scalar quantities are treated as implicit multiples of {{mvar|I}}{{sub|2}} and notated as {{math|1}}.

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

Note that the 2-sheet hyperboloid $\backslash left\backslash \{\; x\; i\; +\; y\; j\; +\; z\; k\; :\; x^2\; -\; y^2\; -\; z^2\; =\; 1\; \backslash right\backslash \}$ corresponds to the imaginary units 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, might view elliptical polarization as an exhibit of the elliptical shape of a wave with {{nowrap|pole $~p\; \backslash ne\; \backslash pm\; i~$.}} The Poincaré sphere model used since 1892 has been compared to a 2-sheet hyperboloid model,{{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 }} and the practice of 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 motions of the Poincaré disk model of hyperbolic plane geometry. Indeed, for a point {{math|[z, 1]}} in the complex projective line, the action of {{math|SU(1,1)}} is given by

since in projective coordinates $(\backslash ;u\backslash ,z\; +\; v,\; \backslash ;\; v^*\backslash ,z\; +u^*\backslash ;)\; \backslash thicksim\; \backslash left(\backslash ;\backslash frac\{\backslash ,u\backslash ,z\; +\; v\backslash ,\}\{v^*\backslash ,z\; +u^*\},\; \backslash ;\; 1\; \backslash ;\backslash right).$

Writing $\backslash ;suv\; +\; \backslash overline\{suv\}\; =\; 2\backslash ,\backslash Re\backslash mathord\backslash bigl(\backslash ,suv\backslash ,\backslash bigr)\backslash ;,$ complex number arithmetic shows

$$\backslash bigl|u\backslash ,z\; +\; v\backslash bigr|^2\; =\; S\; +\; z\backslash ,z^*\; \backslash quad\; \backslash text\{\; and\; \}\; \backslash quad\; \backslash bigl|v^*\backslash ,z\; +\; u^*\backslash bigr|^2\; =\; S\; +\; 1~,$$

where $S\; =\; v\backslash ,v^*\; \backslash left(z\backslash ,z^*\; +\; 1\backslash right)\; +\; 2\backslash ,\backslash Re\backslash mathord\backslash bigl(\backslash ,uvz\backslash ,\backslash bigr).$

Therefore, so that their ratio lies in the open disk.{{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}}

{{Portal|Mathematics}}

• Unitary group
• Projective special unitary group, {{math|PSU(n)}}
• Orthogonal group
• Generalizations of Pauli matrices
• Representation theory of SU(2)

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{{Reflist|25em}}

• {{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