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spaghetti3451

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##[T^{a}_{\bf R},T^{b}_{\bf R}]=i\epsilon^{abc}T^{c}_{\bf R},##

where ##T^{a}_{\bf R}## are the ##3## generators of the algebra.

In ##2## dimensions, these generators are the Pauli matrices

##T^{1}_{\bf 1} = \frac{1}{2}\begin{pmatrix}0 & 1\\ 1 & 0 \end{pmatrix}, \qquad T^{2}_{\bf 1} = \frac{1}{2}\begin{pmatrix}0 & -i\\ i & 0 \end{pmatrix}, \qquad

T^{3}_{\bf 1} = \frac{1}{2}\begin{pmatrix}1 & 0\\ 0 & -1 \end{pmatrix}.##

In ##3## dimensions, these generators are

##T^{1}_{\bf 2} = \frac{1}{\sqrt{2}}\begin{pmatrix}0 & 1 & 0\\ 1 & 0 & 1\\ 0 & 1 & 0 \end{pmatrix}, \qquad T^{2}_{\bf 2} = \frac{1}{\sqrt{2}}\begin{pmatrix}0 & -i & 0\\ i & 0 & -i\\ 0 & i & 0 \end{pmatrix}, \qquad

T^{3}_{\bf 2} = \begin{pmatrix}1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix}.##

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1. How can you derive the generators in ##2## and ##3## dimensions?

2. What are the generators in ##1## dimension?