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I am confused about the notation.

What's the differences between $(\frac{1}{2},\frac{1}{2})$ and $(\frac{1}{2},0)\bigoplus (0,\frac{1}{2})$, or maybe $(\frac{1}{2},0)\bigoplus (\frac{1}{2},0)$ ?

(1) $so(1,3)=su(2) \bigoplus su(2)$ has different representations, I don't understand why $(\frac{1}{2},\frac{1}{2})$ is irreducible, since we already have direct sum $\bigoplus$ between two $su(2)$, and direct sum usually means reducible.

The Schwartz book (page 163) says $(\frac{n}{2},\frac{n}{2})$ These are each irreducible representations of the Lorentz algebra, but reducible representations of the $su(2)$ subalgebra corresponding to spin.

(2) The group elements are acting on some space, then $\bigoplus$ means the direct sum of these spaces, like $(\frac{1}{2},0)\bigoplus (0,\frac{1}{2})$. Then what does $(\frac{1}{2},\frac{1}{2})$ means, a direction product $(\frac{1}{2},0)\bigotimes (0,\frac{1}{2})$ ?

Qmechanic
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    One should distinguish between a Lie algebra and a Lie group. A direct sum of Lie algebras corresponds to a direct product of Lie groups. – higgsss Dec 02 '15 at 22:02
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    This question (v3) is explained in http://physics.stackexchange.com/q/149455/2451 and links therein. See also http://physics.stackexchange.com/q/28505/2451 – Qmechanic Dec 02 '15 at 22:34

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