Special Unitary Lie Algebra of Degree 2: su(2) 📂Representation Theory

Special Unitary Lie Algebra of Degree 2: su(2)

Definition¹

The set of all $2 \times 2$ complex matrices whose trace is $0$ and which satisfy $X^{\ast} = - X$ is the Lie algebra of the special unitary group $\operatorname{SU}(2)$.

$$ \mathfrak{su}(2) = \left\{ X \in M_{2}(\mathbb{C}) : \trace(X) = 0 \text{ and } X^{\ast} = - X \right\} $$

Here $X^{\ast}$ denotes the conjugate transpose of $X$. The property $X^{\ast} = - X$ is called skew-Hermitian.

Explanation

$\mathfrak{su}(2)$ is the case $n = 2$ of the special unitary Lie algebra $\mathfrak{su}(n)$, and it is a $3$-dimensional $\mathbb{R}$-vector space. Note that, although it is a set of complex matrices, it is a real vector space whose field is the real numbers. Multiplying a skew-Hermitian matrix by $i$ produces a Hermitian matrix as shown below, so scalar multiplication is allowed only over the real field $\mathbb{R}$. Let $X$ be a skew-Hermitian matrix.

$$ (iX)^{\ast} = \overline{i} X^{\ast} = (-i)(-X) = iX $$

Basis and Commutation Relations

Let us find out what the basis actually looks like by using the condition $X \in \mathfrak{su}(2)$. From the skew-Hermitian condition $X^{\ast} = -X$ we see that the diagonal entries must be purely imaginary. Also, $x_{21} = -\overline{x_{12}}$ must hold, and since the trace must be $0$, we have $x_{22} = -x_{11}$. Therefore an element of $\mathfrak{su}(2)$ is expressed as follows in terms of real numbers $a, b, c$.

$$ X = \begin{bmatrix} ia & b + ic \\ -b + ic & -ia \end{bmatrix} $$

Hence $\dim \mathfrak{su}(2) = 3$, and the above expression naturally decomposes once more as below.

$$ \begin{align*} &\begin{bmatrix} ia & b + ic \\ -b + ic & -ia \end{bmatrix} \\ &= \begin{bmatrix} 0 & b + ic \\ -b + ic & 0 \end{bmatrix} + \begin{bmatrix} ia & 0 \\ 0 & -ia \end{bmatrix} \\ &= c\begin{bmatrix} 0 & i \\ i & 0 \end{bmatrix} + b\begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} + a\begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix} \end{align*} $$

Therefore one may take $\left\{ E_{1}, E_{2}, E_{3} \right\}$, consisting of the matrices below, as a basis of $\mathfrak{su}(2)$.

$$ E_{1} = -\frac{i}{2}\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}, \qquad E_{2} = -\frac{i}{2}\begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix}, \qquad E_{3} = -\frac{i}{2}\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} \tag{1} $$

Here the constants were taken to be $a = b = c = -\frac{1}{2}$, because then the brackets and the structure constants come out cleanly as follows.

$$ [E_{1}, E_{2}] = E_{3}, \qquad [E_{2}, E_{3}] = E_{1}, \qquad [E_{3}, E_{1}] = E_{2} $$

The above computation amounts to saying that the structure constants with respect to this basis are the Levi-Civita symbol $\epsilon_{jk\ell}$.

$$ [E_{j}, E_{k}] = \sum_{\ell=1}^{3} \epsilon_{jk\ell} E_{\ell} $$

Pauli Matrices

Meanwhile, the matrices appearing in $(1)$ have a special name, the Pauli matrices.

$$ \sigma_{1} = \sigma_{x} = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}, \qquad \sigma_{2} = \sigma_{y} = \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix}, \qquad \sigma_{3} = \sigma_{z} = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} $$

Since the Pauli matrices are Hermitian matrices, the matrices obtained by multiplying them by the complex number $i$ are skew-Hermitian.

$$ E_{1} = -\dfrac{i}{2}\sigma_{1},\quad E_{2} = -\dfrac{i}{2}\sigma_{2},\quad E_{3} = -\dfrac{i}{2}\sigma_{3} $$

Therefore, since every $2 \times 2$ Hermitian matrix with trace $0$ is a real linear combination of the Pauli matrices, $\mathfrak{su}(2)$ can also be written as follows.

$$ \mathfrak{su}(2) = \left\{ i(a_{1}\sigma_{1} + a_{2}\sigma_{2} + a_{3}\sigma_{3}) : a_{1}, a_{2}, a_{3} \in \mathbb{R} \right\} $$

In physics, instead of skew-Hermitian matrices, the Hermitian $J_{a} = \frac{1}{2}\sigma_{a}$ are used as the generators. In a system of units where $\hbar = 1$, these satisfy the commutation relations of the angular momentum operators such as $[J_{1}, J_{2}] = i J_{3}$; if one sets $X_{a} := -i J_{a}$, then $X_{a} \in \mathfrak{su}(2)$ while $[X_{1}, X_{2}] = X_{3}$, which has the same form as the commutation relations above. In the end, the Hermitian generators of physics and the skew-Hermitian elements of mathematics differ only by multiplying and dividing by $i$.