Special unitary group

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It has been suggested that Representation theory of SU(2) be merged into this article or section. (Discuss)

In mathematics, the special unitary group of degree n, denoted SU(n), is the group of n×n unitary matrices with unit determinant. 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×n unitary matrices, which is itself a subgroup of the general linear group GL(n, C).

The simplest case, SU(1), is a trivial group, having only a single element. The group SU(2) is isomorphic to the group of quaternions of absolute value 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), we have a surjective homomorphism from SU(2) to the rotation group SO(3) whose kernel is ${ + I, - I}$ .

1 Properties
2 Generators
- 2.1 SU(2)
- 2.2 SU(3)
3 Lie algebra
4 Generalized special unitary group
- 4.1 Example
5 See also
6 References
7 External links

[edit] Properties

The special unitary group SU(n) is a real matrix Lie group of dimension n² − 1. Topologically, it is compact and simply connected. Algebraically, it is a simple Lie group (meaning its Lie algebra is simple; see below). The center of SU(n) is isomorphic to the cyclic group Z_n. Its outer automorphism group, for n ≥ 3, is Z₂, while the outer automorphism group of SU(2) is the trivial group.

The SU(n) algebra is generated by n² operators, which satisfy the commutator relationship (for i,j,k,l = 1, 2, ..., n)

$\left [ \hat{O}_{ij} , \hat{O}_{kl} \right ] = \delta_{jk} \hat{O}_{il} - \delta_{il} \hat{O}_{kj}$

Additionally, for the operator

$\hat{N} = \sum_{i=1}^n \hat{O}_{ii}$

it satisfies

$\left [ \hat{N}, \hat{O}_{ij} \right ] = 0$

which implies that the number of independent generators of SU(n) is n²-1.^[1]

[edit] Generators

[edit] SU(2)

For SU(2), the generators are known as the Pauli matrices:

$\sigma_1 = \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix}$

$\sigma_2 = \begin{pmatrix} 0&-i\\ i&0 \end{pmatrix}$

$\sigma_3 = \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix}$

[edit] SU(3)

The analog of the Pauli matrices for SU(3) are Gell-Mann matrices:

$\lambda_1 = \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}$	$\lambda_2 = \begin{pmatrix} 0 & -i & 0 \\ i & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}$	$\lambda_3 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 0 \end{pmatrix}$
$\lambda_4 = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{pmatrix}$	$\lambda_5 = \begin{pmatrix} 0 & 0 & -i \\ 0 & 0 & 0 \\ i & 0 & 0 \end{pmatrix}$	$\lambda_6 = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix}$
$\lambda_7 = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & -i \\ 0 & i & 0 \end{pmatrix}$	$\lambda_8 = \frac{1}{\sqrt{3}} \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -2 \end{pmatrix}$

The generators of SU(3) are defined as T by the relation

$T_a = \frac{\lambda_a}{2} .\,$

These obey the relations

$\left[T_a, T_b \right] = i \sum_{c=1}^8{f_{abc} T_c} \,$

where f is a structure constant and has a value given by

$f^{123} = 1 \,$

$f^{147} = f^{165} = f^{246} = f^{257} = f^{345} = f^{376} = \frac{1}{2} \,$

$f^{458} = f^{678} = \frac{\sqrt{3}}{2} \,$

$\operatorname{tr}(T_a) = 0 \,$

[edit] Lie algebra

The Lie algebra corresponding to $SU(n)$ is denoted by $\mathfrak{su}(n)$ . It consists of the traceless antihermitian $n \times n$ complex matrices, with the regular commutator as Lie bracket. Note that this is a real and not a complex Lie algebra, in the convention used by mathematicians. A factor $i$ is often inserted by particle physicists who find the different, complex Lie algebra convenient.

For example, the following matrices used in quantum mechanics form a basis for $\mathfrak{su}(2)$ over $\mathbb{R}$ :

$i\sigma_x = \begin{bmatrix} 0 & i \\ i & 0 \end{bmatrix}$

$i\sigma_y = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}$

$i\sigma_z = \begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix}$

(where $i$ is the imaginary unit.)

This representation is often used in quantum mechanics (see Pauli matrices and Gell-Mann matrices), 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 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 $i$ ),

$i I_2 = \begin{bmatrix} i & 0 \\ 0 & i \end{bmatrix}$

these are also generators of the Lie algebra $\mathfrak{u}(2)$ .

Note: make clearer the fact that under matrix multiplication (which is anticommutative in this case), we generate the Clifford algebra $Cl 3$ , whereas you generate the Lie algebra $\mathfrak{u}(2)$ with commutator brackets instead.

Back to general $SU(n)$ :

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.

Complexify the Lie algebra, so that any traceless $n \times n$ matrix is now allowed. The weight eigenvectors are the Cartan subalgebra itself and 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 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 $i$ 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, $\mathfrak{su}(n)$ has a rank of $n - 1$ and its Dynkin diagram is given by $A n - 1$ , a chain of $n - 1$ vertices.

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. Then, the roots consists of all the $n (n - 1)$ permutations of $(1, -1, 0, \dots, 0)$ . The construction given two paragraphs ago explains why. A choice of simple roots is

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

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

…,

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

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.

[edit] Generalized special unitary group

The generalized special unitary group over R, SU(p,q;F), is the group of all linear transformations of determinant 1 of a free module of rank n = p + q over a commutative ring R 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 R.

Specifically, fix a hermitian matrix A of signature p q in GL(n,R), then all

$M \in SU(p,q,F)$

satisfy

M * A M = A

det M = 1

Often one will see the notation $S U p, q$ without reference to a ring, in this case the ring being referred to is C which is one of the classical Lie groups. The standard choice for A when R = 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.

[edit] 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 D. Lax computed a fundamental domain for the action of this group on $H C 2$ , see [1]