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I am confused by conjugation, and the action of group elements on themselves. If I have a rotation acting on the generators of $su(2)$, i.e.

\begin{align} R_\theta (L_1, L_2, L_3) \end{align}

where $R_\theta$ is some rotation matrix. The elements of the vector $(L_1, L_2, L_3)$ are themselves the $su(2)$ generators, in an $N$-dimensional representation. Is it true that this action can be equivalently stated as a conjugation of the elements themselves? i.e.

\begin{align} R_\theta (L_1, L_2, L_3) \leftrightarrow (\Omega L_1 \Omega^\dagger, \Omega L_2 \Omega^\dagger, \Omega L_3 \Omega^\dagger) \end{align}

where $\Omega$ is defined by, for the example of a rotation around the $z$-axis,

$$\Omega = e^{i\theta L_3}$$

Qmechanic
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1 Answers1

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I think that you are groping for the equation $$ U(R) \sigma_i U^{-1}(R)= \sigma_j {R^j}_i, $$ where ${R^j}_i$ is an ${\rm SO}(3)$ rotation matrix, the $\sigma_i$ are the matrix generators of $\mathfrak{su}(2)$ in some representation, and the $U(R)$ are the corresponding ${\rm SU}(2)$ matrices in the same representation.

This is saying that the the adjoint action of the group ${\rm SU}(2)$ on its Lie algebra implements the ${\rm SO}(3)$ rotation group.

mike stone
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  • Thanks mike. Can you recommend a reference on this that has physicists as its target audience? –  Jul 16 '20 at 18:47
  • I have a follow-up question about the language used here. When we refer to the "adjoint action of the group SU(2)" we are talking about the conjugation, the LHS of your expression, but does the $\sigma_i$ have to be also an element of SU(2)? Here that would appear to me to not be the case, since the $\sigma_i$ being acted on is an element of the Lie algebra $\mathfrak{su}2$, rather than the group SU(2). Is that understanding correct? –  Jul 16 '20 at 19:01
  • The $\sigma_i$ are elements of the Lie algebra of ${SU}(2)$. Any Lie group $G$ acts on its algebra $\mathfrak g$ via a map ${\rm Adj}(g): \lambda_i \mapsto g \lambda_i g^{-1} = \lambda_j {[{\rm Adj}(g)]^j}_i$. This is indeed a conjugation, but is called the adjoint action in Lie theory. – mike stone Jul 16 '20 at 19:07