- #1

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I have a couple questions.

In the literature, they say the algebra satisfied by the three generators, [itex]T_1, T_2, T_3[/itex] is

[tex][T_1,\,T_2]=-i T_3[/tex]

[tex][T_2,\,T_3]=i T_1[/tex]

[tex][T_3,\,T_1]=i T_2[/tex]

[tex][T_2,\,T_3]=i T_1[/tex]

[tex][T_3,\,T_1]=i T_2[/tex]

Is this correct?

And, secondly, given that the group is non-compact, I should expect the generators to be represented by infinite-dimensional matrices only. But, in the literature, I found the following:

[tex]T_1 = \frac{1}{2}\begin{pmatrix}0&1\\-1&0\end{pmatrix}\qquad T_2=\frac{1}{2}\begin{pmatrix}0&-i\\-i&0\end{pmatrix}\qquad

T_3=\frac{1}{2}\begin{pmatrix}1&0\\0&-1\end{pmatrix}[/tex]

T_3=\frac{1}{2}\begin{pmatrix}1&0\\0&-1\end{pmatrix}[/tex]

and these appear to satisfy the algebra above. So why aren't they infinite-dimensional? What's the deal?