Special unitary group

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In mathematics, the special unitary group of degree <math>n<math> is the group of <math>n<math> by <math>n<math> unitary matrices with determinant <math>1<math> and entries from the field <math>\mathbb{C}<math> of complex numbers, with the group operation that of matrix multiplication. It is written as <math>\mathrm{SU}(n)<math>. This is a subgroup of the unitary group <math>\mathrm{U}(n)<math>, itself a subgroup of the general linear group <math>\mathrm{GL}(n, \mathbb{C})<math>.

From now on, we will assume <math>n > 2<math>.

The special unitary group <math>\mathrm{SU}(n)<math> is a real Lie group of dimension <math>n^2 - 1<math>. It is compact, connected, simply connected, and (for <math>n > 2<math>) simple and semisimple. Its center is the cyclic group <math>\mathbb{Z}_n<math>. Its outer automorphism group for <math>n > 3<math> is <math>\mathbb{Z}_2<math>. The outer automorphism group of <math>\mathrm{SU}(2)<math> is the trivial group.

The group <math>\mathrm{SU}(2)<math> is isomorphic to the group of quaternions of absolute value <math>1<math>, 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 of Lie groups <math>\mathrm{SU}(2)<math> and the special orthogonal group <math>\mathrm{SO}(3, \mathbb{R})<math> whose kernel is <math>\{+I, -I\}<math>.

The Lie algebra corresponding to <math>\mathrm{SU}(n)<math> is denoted by <math>\mathfrak{su}(n)<math>. It consists of the traceless antihermitian <math>n \times n<math> complex matrices, with the regular commutator as Lie bracket. Note that this is a real and not a complex Lie algebra.

For example, the following matrices form a 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 factor arises because physicists like to include a factor of <math>i<math> in their real Lie algebras, which is a different convention from mathematicians).

This representation is often used in quantum mechanics (see Pauli 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 <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>.

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{u}(2)<math> with commutator brackets instead.

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

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

Let's complexify the Lie algebra now, so that any traceless <math>n \times n<math> 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 <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>\mathfrak{su}(n)<math> has a 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 roots is

<math>(1, -1, 0, \dots, 0)<math>,
<math>(0, 1, -1, \dots, 0)<math>,
<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.


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