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I have here the following rules for representations of the Lorentz algebra including spinors

- The representations of the Lorentz algebra can be labeled by two half-integers [itex](j_{-},j_{+})[/itex].
- The dimension of the representation [itex](j_{-},j_{+})[/itex] is [itex](2j_{-}+1)(2j_{+}+1)[/itex].
- The generator of rotations [itex]\mathbf{J}[/itex] is related to [itex]\mathbf{J}^{+}[/itex] and [itex]\mathbf{J}^{-}[/itex] by [itex]\mathbf{J}=\mathbf{J}^{+}+\mathbf{J}^{-}[/itex]; therefore, by the usual addition of angular momenta in quantum mechanics, in the representation [itex](j_{-},j_{+})[/itex] we have states with all possible spin [itex]j[/itex] in integer steps between the values [itex]|j_{+}-j_{-}|[/itex] and [itex]j_{+}+j_{-}[/itex].

My question is that given a representation [itex](1,\frac{1}{2})[/itex], the dimension for the plus and minus generators are different (3 and 2), yet I am suppose to add these matrices together for the generators [itex]\mathbf{J}[/itex]. Would it be a direct product [itex]A\oplus B[/itex]? I don't see how you can add matrices together with different dimension.

Thanks,

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# Addition for generators of rotation

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