Well I am trying to understand the adjoint representation of the(adsbygoogle = window.adsbygoogle || []).push({}); algebra.su(2)

We know that the algebra is given:

[itex] [X_{i}, X_{j}]= ε_{ij}^{k} X_{k} [/itex]

(maybe I forgot an [itex]i[/itex] but I am not sure).

The adjoint representation is then ( in the matrix representation) defined by the [itex]ε_{ijk}[/itex] structure constants, via the identification [itex]X_{i}= [ε_{i}]_{j}^{k}[/itex]. Correct? Because by that we have:

[itex] (adX_{i})^{k}_{j}= ad X_{i} X_{j}|_{X_{k}} = [X_{i}, X_{j}]|_{X_{k}}=[ε_{i}]_{j}^{k} [/itex]

Now begins my question/problem. The matrices of [itex][ε_{i}]_{j}^{k}[/itex] are of dimension [itex]j_{max}\times k_{max}[/itex] so equal to the number of generators [itex]X_{i}[/itex].

Thealgebra has [itex]n^{2}-1=4-1=3[/itex] generators, so the adjoint representation can be seen as [itex]3\times3[/itex] matrices, so it naturally acts on 3 component vectors. That means that the adjoint representation is 3 dimensional representation and so it is the spin J=1. (?)su(2)

Is there another way to prove the last sentence? Eg using the existing isomorphism betweenalgebras? And one more question, what happens with the spin-1/2? The spin 1/2 representation (if my logic was correct) could only exist if the number of generators are 2!su(2)-so(3)

Can someone help?

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# SU(2) adjoint repr

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