Heisenberg XXX spin model

Question

Let $\pi$ be the standard representation of $sl_2(\mathbb{C})$ on $\mathbb{C}^2$. Let $p_1,p_2,p_3$ the three Pauli matrices. Define $S^a:=\frac{1}{2}\pi(p_a)$. What does such matrices looks like?

Answer 1

Using

In[6]:= s1 = -I/2 PauliMatrix[1]; s2 = -I/2 PauliMatrix[2]; 
        s3 = -I/2 PauliMatrix[3];

We can verify that $S^1$, $S^2$ and $S^3$ satisfy commutation relations of $\mathfrak{sl}_2(\mathbb{C})$:

In[8]:= {s1.s2 - s2.s1 - s3, s3.s1 - s1.s3 - s2, s2.s3 - s3.s2 - s1}

Out[8]= {{{0, 0}, {0, 0}}, {{0, 0}, {0, 0}}, {{0, 0}, {0, 0}}}

Here is their explicit form as given by Mathematica:

$$ S^1 = \left( \begin{array}{cc} 0 & -\frac{i}{2} \\ -\frac{i}{2} & 0 \\ \end{array} \right) \quad S^2 = \left( \begin{array}{cc} 0 & -\frac{1}{2} \\ \frac{1}{2} & 0 \\ \end{array} \right) \quad S^3 = \left( \begin{array}{cc} -\frac{i}{2} & 0 \\ 0 & \frac{i}{2} \\ \end{array} \right) $$