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In quantum mechanics we learn about the Pauli spin matrices:

$$ \sigma_1 = \left( \begin{array}{cc} 0 & -1 \\ 1 & 0\end{array} \right) \hspace{0.25in} \sigma_2 = \left( \begin{array}{cc} i & 0 \\ 0 & -i\end{array} \right)\hspace{0.25in} \sigma_3 = \left( \begin{array}{cc} 0 & i \\ i & 0\end{array} \right)$$

Then in quantum mechanics we lump them together into $\vec{\sigma}= (\sigma_1, \sigma_2, \sigma_3)$ - what is the name for that?

It is a vector of matrices and it transforms in a particular way under $SU(2)$ (or possibly larger group) What representation is that? What type of geometric object is that?

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It is a simple (subjectively) basis of the Lie algebra $su(2)$ of $SU(2)$. The latter is a Lie group, acting by adjoint representation on the Lie algebra (this is a general fact).

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