Special unitary group

The group operation is matrix multiplication. The special unitary group is a subgroup of the unitary group $U(n)$ , consisting of all $n \times n$ unitary matrices. As a compact classical group, $U(n)$ is the group that preserves the standard inner product on $C n$ .^{[nb 1]} It is itself a subgroup of the general linear group, $SU(n) \subset U(n) \subset GL(n, C)$ .

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.^[1]

The simplest case, $SU(1)$ , is the trivial group, having only a single element. The group $SU(2)$ is isomorphic to the group of quaternions of norm 1, and is thus diffeomorphic to the 3-sphere. Since unit quaternions can be used to represent rotations in 3-dimensional space (up to sign), there is a surjective homomorphism from $SU(2)$ to the rotation group $SO(3)$ whose kernel is ${+ I, - I$ }.^{[nb 2]} $SU(2)$ is also identical to one of the symmetry groups of spinors, Spin(3), that enables a spinor presentation of rotations.

Properties

The special unitary group $SU(n)$ is a real Lie group (though not a complex Lie group). Its dimension as a real manifold is $n 2 - 1$ . Topologically, it is compact and simply connected.^[2] Algebraically, it is a simple Lie group (meaning its Lie algebra is simple; see below).^[3]

The center of $SU(n)$ is isomorphic to the cyclic group $Z n$ , and is composed of the diagonal matrices $ζ I$ for $ζ$ an $n$ ^th root of unity and $I$ the n×n identity matrix.

Its outer automorphism group, for $n \geq 3$ , is $Z 2$ , while the outer automorphism group of $SU(2)$ is the trivial group.

A maximal torus, of rank $n - 1$ , is given by the set of diagonal matrices with determinant 1. The Weyl group is the symmetric group $S n$ , which is represented by signed permutation matrices (the signs being necessary to ensure the determinant is 1).

The Lie algebra of $SU(n)$ , denoted by ${\mathfrak {su}}(n)$ , can be identified with the set of traceless antihermitian $n \times n$ complex matrices, with the regular commutator as Lie bracket. Particle physicists often use a different, equivalent representation: the set of traceless hermitian $n \times n$ complex matrices with Lie bracket given by $- i$ times the commutator.

Infinitesimal generators

The Lie algebra $u (n)$ can be generated by $n 2$ operators ${\hat {O}}_{ij}$ , $i, j = 1, 2, ..., n$ , which satisfy the commutator relationships

\left[{\hat {O}}_{ij},{\hat {O}}_{k\ell }\right]=\delta _{jk}{\hat {O}}_{i\ell }-\delta _{i\ell }{\hat {O}}_{kj}

for $i, j, k, ℓ$ = $1, 2, ..., n$ , where $δ jk$ denotes the Kronecker delta. One matrix implementation is given by

({\hat {O}}_{ij})_{k\ell }=\delta _{ik}\delta _{j\ell },

which has $1$ in the $ij$ -entry and zeros elsewhere. Since the identity operator commutes with every ${\hat {O}}_{ij},$ it may be subtracted from the ${\hat {O}}_{ii}$ to form traceless operators without disturbing the commutation relations,

{\hat {O}}_{ii}\rightarrow {\hat {O}}_{ii}-{\frac {1}{n}}{\hat {I}}_{n}.

The number of independent generators of $su (n)$ is then $n 2 - 1$ .^[4]

Fundamental representation

In the defining, or fundamental, representation of $su (n)$ the generators $T a$ are represented by traceless hermitian complex $n \times n$ matrices, where:

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}}\,

where the $f$ are the structure constants and are antisymmetric in all indices, while the $d$ -coefficients are symmetric in all indices. As a consequence, the anticommutator and commutator are:

\left\{T_{a},T_{b}\right\}={\frac {1}{n}}\delta _{ab}I_{n}+\sum _{c=1}^{n^{2}-1}{d_{abc}T_{c}}\,

\left[T_{a},T_{b}\right]=i\sum _{c=1}^{n^{2}-1}{f_{abc}T_{c}}\,.

We also take

\sum _{c,e=1}^{n^{2}-1}d_{ace}d_{bce}={\frac {n^{2}-4}{n}}\delta _{ab}\,

as a normalization convention.

Adjoint representation

In the $(n 2 - 1)$ -dimensional adjoint representation, the generators are represented by $(n 2 - 1)$ × $(n 2 - 1)$ matrices, whose elements are defined by the structure constants themselves:

(T_{a})_{jk}=-if_{ajk}.

n = 2

$SU(2)$ is the following group,^[5]

\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\}~,

where the overline denotes complex conjugation.

Isomorphism with S³

If we consider $\alpha ,\beta$ as a pair in $\mathbb{C}^2$ , then rewriting $|\alpha |^{2}+|\beta |^{2}=1$ with the underlying real coordinates becomes the equation

x_{1}^{2}+y_{1}^{2}+x_{2}^{2}+y_{2}^{2}=1

hence we have the equation of the sphere. This can also be seen using an embedding: the map

{\begin{aligned}\varphi \colon \mathbf {C} ^{2}&\to \operatorname {M} (2,\mathbf {C} )\\\varphi (\alpha ,\beta )&={\begin{pmatrix}\alpha &-{\overline {\beta }}\\\beta &{\overline {\alpha }}\end{pmatrix}},\end{aligned}}

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

Therefore, as a manifold, $S 3$ is diffeomorphic to $SU(2)$ and so $S 3$ is a compact, connected Lie group.

Lie Algebra

The Lie algebra of $SU(2)$ is

{\mathfrak {su}}(2)=\left\{{\begin{pmatrix}ia&-{\overline {z}}\\z&-ia\end{pmatrix}}:\ a\in \mathbf {R} ,z\in \mathbf {C} \right\}~.

Matrices of this form have trace zero and are antihermitian. The Lie algebra is then generated by the following matrices,

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}}~,

which have the form of the general element specified above.

These satisfy $u 3 u 2 = - u 2 u 3 = - u 1$ and $u 2 u 1 = - u 1 u 2 = - u 3$ . The commutator bracket is therefore specified by

[u_{3},u_{1}]=2u_{2},\qquad [u_{1},u_{2}]=2u_{3},\qquad [u_{2},u_{3}]=2u_{1}~.

The above generators are related to the Pauli matrices by $u 1 = i σ 1, u 2 = - i σ 2$ and $u 3 = i σ 3$ . 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.

The Lie algebra serves to work out the representations of $SU(2)$ .

n = 3

The generators of $SU (3)$ , $T$ , in the defining representation, are:

T_{a}={\frac {\lambda _{a}}{2}}.\,

where $λ$ , the Gell-Mann matrices, are the $SU(3)$ analog of the Pauli matrices for $SU(2)$ :

$\lambda _{1}={\begin{pmatrix}0&1&0\\1&0&0\\0&0&0\end{pmatrix}}$	$\lambda _{2}={\begin{pmatrix}0&-i&0\\i&0&0\\0&0&0\end{pmatrix}}$	$\lambda _{3}={\begin{pmatrix}1&0&0\\0&-1&0\\0&0&0\end{pmatrix}}$
$\lambda _{4}={\begin{pmatrix}0&0&1\\0&0&0\\1&0&0\end{pmatrix}}$	$\lambda _{5}={\begin{pmatrix}0&0&-i\\0&0&0\\i&0&0\end{pmatrix}}$
$\lambda _{6}={\begin{pmatrix}0&0&0\\0&0&1\\0&1&0\end{pmatrix}}$	$\lambda _{7}={\begin{pmatrix}0&0&0\\0&0&-i\\0&i&0\end{pmatrix}}$	$\lambda _{8}={\frac {1}{\sqrt {3}}}{\begin{pmatrix}1&0&0\\0&1&0\\0&0&-2\end{pmatrix}}.$

These $λ a$ span all traceless Hermitian matrices $H$ of the Lie algebra, as required. Note that $λ 2, λ 5, λ 7$ are antisymmetric.

They obey the relations

\left[T_{a},T_{b}\right]=i\sum _{c=1}^{8}{f_{abc}T_{c}}\,

\{T_{a},T_{b}\}={\frac {1}{3}}\delta _{ab}I_{3}+\sum _{c=1}^{8}{d_{abc}T_{c}}\,

(or, equivalently,

\{\lambda _{a},\lambda _{b}\}={\frac {4}{3}}\delta _{ab}I_{3}+2\sum _{c=1}^{8}{d_{abc}\lambda _{c}}

The $f$ are the structure constants of the Lie algebra, given by:

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}},\,

while all other $f abc$ not related to these by permutation are zero. In general, they vanish, unless they contain an odd number of indices from the set {2,5,7}.^{[nb 3]}

The symmetric coefficients $d$ take the values:

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_{146}=d_{157}=-d_{247}=d_{256}=d_{344}=d_{355}=-d_{366}=-d_{377}={\frac {1}{2}}~.

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

As a topological space, SU(3) is a non-trivial, locally trivial bundle of a 3-sphere over a 5-sphere.

A generic SU(3) group element generated by a traceless 3×3 hermitian matrix $H$ , normalized as $tr(H 2) = 2$ , is given by^[6]

{\begin{aligned}&\exp(i\theta H)\\={}&\left[-{\tfrac {1}{3}}I\sin \left(\varphi +2\pi /3\right)\sin \left(\varphi -2\pi /3\right)-{\tfrac {1}{2{\sqrt {3}}}}~H\sin(\varphi )-{\tfrac {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)}}\\[4pt]&{}+\left[-{\tfrac {1}{3}}~I\sin(\varphi )\sin \left(\varphi -{\frac {2\pi }{3}}\right)-{\tfrac {1}{2{\sqrt {3}}}}~H\sin \left(\varphi +{\frac {2\pi }{3}}\right)-{\tfrac {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)}}\\[4pt]&{}+\left[-{\tfrac {1}{3}}~I\sin(\varphi )\sin \left(\varphi +{\frac {2\pi }{3}}\right)-{\tfrac {1}{2{\sqrt {3}}}}~H\sin \left(\varphi -{\frac {2\pi }{3}}\right)-{\tfrac {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{aligned}}

where

\varphi \equiv {\tfrac {1}{3}}\left(\arccos \left({\tfrac {3}{2}}{\sqrt {3}}\det H\right)-{\tfrac {\pi }{2}}\right).

Lie algebra structure

The above representation bases generalize to $n > 3$ , using generalized Pauli matrices.

If we choose an (arbitrary) particular basis, then the subspace of traceless diagonal $n \times n$ matrices with imaginary entries forms an $(n - 1)$ -dimensional Cartan subalgebra.^[7]

Complexify the Lie algebra, so that any traceless $n \times n$ matrix is now allowed. The weight eigenvectors are the Cartan subalgebra itself, as well as the matrices with only one nonzero entry which is off diagonal. Even though the Cartan subalgebra $h$ is only $(n - 1)$ -dimensional, to simplify calculations, it is often convenient to introduce an auxiliary element, the unit matrix which commutes with everything else (which is not an element of the Lie algebra!) for the purpose of computing weights—and that only. So, we have a basis where the $i$ -th basis vector is the matrix with 1 on the $i$ -th diagonal entry and zero elsewhere. Weights would then be given by $n$ coordinates and the sum over all $n$ coordinates has to be zero (because the unit matrix is only auxiliary).

So, $SU(n)$ is of rank $n - 1$ and its Dynkin diagram is given by $A n -1$ , a chain of $n - 1$ vertices, o−o−o−o---o.^[8] Its root system consists of $n (n - 1)$ roots spanning a $n - 1$ Euclidean space. Here, we use $n$ redundant coordinates instead of $n - 1$ to emphasize the symmetries of the root system (the $n$ coordinates have to add up to zero).

In other words, we are embedding this $n - 1$ dimensional vector space in an $n$ -dimensional one. Thus, the roots consists of all the $n (n - 1)$ permutations of $(1, -1, 0, ..., 0)$ . The construction given above explains why. A choice of simple roots is

{\begin{aligned}(&1,-1,0,\dots ,0),\\(&0,1,-1,\dots ,0),\\&\qquad \vdots \\(&0,0,0,\dots ,1,-1).\end{aligned}}

Its Cartan matrix is

{\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 $S n$ , the symmetry group of the $(n - 1)$ -simplex.

Generalized special unitary group

For a field $F$ , the generalized special unitary group over F, $SU(p, q; F)$ , is the group of all linear transformations of determinant 1 of a vector space of rank $n = p + q$ over $F$ which leave invariant a nondegenerate, Hermitian form of 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

M\in \mathrm {SU} (p,q,R)

satisfy

{\begin{aligned}M^{*}AM&=A\\\det M&=1.\end{aligned}}

Often one will see the notation $SU(p, q)$ 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

A={\begin{bmatrix}0&0&i\\0&I_{n-2}&0\\-i&0&0\end{bmatrix}}.

However, there may be better choices for $A$ for certain dimensions which exhibit more behaviour under restriction to subrings of $C$ .

Example

An 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 $SL(2,9; Z)$ 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 $HC 2$ .^[9]

A further 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 GUT physics are, for $p > 1, n - p > 1$ ,

\mathrm {SU} (n)\supset \mathrm {SU} (p)\times \mathrm {SU} (n-p)\times \mathrm {U} (1),

where × denotes the direct product and $U(1)$ , 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,

{\begin{aligned}\mathrm {SU} (n)&\supset \mathrm {SO} (n),\\\mathrm {SU} (2n)&\supset \mathrm {Sp} (n).\end{aligned}}

Since the 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,

{\begin{aligned}\mathrm {SO} (2n)&\supset \mathrm {SU} (n)\\\mathrm {Sp} (n)&\supset \mathrm {SU} (n)\\\mathrm {Spin} (4)&=\mathrm {SU} (2)\times \mathrm {SU} (2)\\\mathrm {E} _{6}&\supset \mathrm {SU} (6)\\\mathrm {E} _{7}&\supset \mathrm {SU} (8)\\\mathrm {G} _{2}&\supset \mathrm {SU} (3)\end{aligned}}

See spin group, and simple Lie groups for E₆, E₇, and G₂.

There are also the accidental isomorphisms: $SU(4) = Spin(6)$ , $SU(2) = Spin(3) = Sp(1)$ ,^{[nb 4]} and $U(1) = Spin(2) = SO(2)$ .

One may 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-spinors in non-relativistic quantum mechanics.

Remarks

↑ For a characterization of $U(n)$ and hence $SU(n)$ in terms of preservation of the standard inner product on $ℂ n$ , see Classical group.
↑ For an explicit description of the homomorphism $SU(2) \to SO(3)$ , see Connection between SO(3) and SU(2).
↑ So fewer than 1/6 of all $f abc$ s are non-vanishing.
↑ $Sp(n)$ is the compact real form of $Sp(2 n, C)$ . It is sometimes denoted $USp(2n)$ . The dimension of the $Sp(n)$ -matrices is $2 n \times 2 n$ .

Notes

↑ Halzen, Francis; Martin, Alan (1984). Quarks & Leptons: An Introductory Course in Modern Particle Physics. John Wiley & Sons. ISBN 0-471-88741-2.
↑ Hall 2015 Proposition 13.11
↑ Wybourne, B G (1974). Classical Groups for Physicists, Wiley-Interscience. ISBN 0471965057 .
↑ R.R. Puri, Mathematical Methods of Quantum Optics, Springer, 2001.
↑ Hall 2015 Exercise 1.5
↑ Rosen, S P (1971). "Finite Transformations in Various Representations of SU(3)". Journal of Mathematical Physics. 12 (4): 673. doi:10.1063/1.1665634. ; Curtright, T L; Zachos, C K (2015). "Elementary results for the fundamental representation of SU(3)". Reports On Mathematical Physics. 76: 401–404. doi:10.1016/S0034-4877(15)30040-9.
↑ Hall 2015 Section 7.7.1
↑ Hall 2015 Section 8.10.1
↑ Francsics, Gabor; Lax, Peter D. "An Explicit Fundamental Domain For The Picard Modular Group In Two Complex Dimensions". arXiv:math/0509708v1 .

References

Hall, Brian C. (2015), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, 222 (2nd ed.), Springer
Iachello, Francesco (2006), Lie Algebras and Applications, Lecture Notes in Physics, 708, Springer, ISBN 3540362363

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