Special Unitary Group

The simplest case, SU(1), is a Trivial Group , having only a single element. The group SU(2) is isomorphic to the group of Quaternion s 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\}

.

PROPERTIES

The special unitary group SU(''n'') is a real Matrix Lie Group of dimension ''n''² − 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 .

LIE ALGEBRA

The Lie Algebra corresponding to

\mathrm{SU}(n)

is denoted by

\mathfrak{su}(n)

. It consists of the Traceless Antihermitian

n 	imes n

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

\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 factor arises because Physicist s like to include a factor of

i

in their real Lie algebras, which is a different convention from
Mathematician s).

This representation is often used in Quantum Mechanics (see '' Pauli Matrices '' and '' Gell-Mann Matrices ''), to represent the Spin of Fundamental Particle s such as Electron s. They also serve as Unit Vector s 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

\mathrm{Cl}_3

, whereas you generate the Lie algebra

\mathfrak{u}(2)

with commutator brackets instead.''

Back to general

\mathrm{SU}(n)

:

If we choose an (arbitrary) particular basis, then the Subspace of traceless Diagonal

n 	imes n

matrices with imaginary entries forms an

n - 1

dimensional Cartan Subalgebra .

Complexify the Lie algebra, so that any traceless

n 	imes n

matrix is now allowed. The Weight Eigenvector s are the Cartan subalgebra itself and the matrices with only one nonzero entry which is off diagonal. Even though the Cartan subalgebra

\mathrm{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 Root s 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 \ dots & dots & dots & \ddots & dots \ 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 .

EXTERNAL LINKS

Physics 558 - Lecture 1, Winter 2003

	APPAREL
	BABY
	BEAUTY
	BOOKS
	CAR TOYS
	CELL PHONES
	DVD'S
	ELECTRONICS
	GOURMET FOOD
	GROCERIES
	HEALTH & PERSONAL
	HOME & GARDEN
	JEWELRY
	MUSIC
	MUSIC INSTRUMENTS
	OFFICE PRODUCTS
	SOFTWARE
	SPORTING GOODS
	TOOLS & HARDWARE
	TOYS
	VIDEO GAMES
	SHOPPING HOME

	MORE SHOPPING...

	CATEGORIES ABOUT SPECIAL UNITARY GROUP

	lie groups

Information About

Special Unitary Group