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Oct 17, 2012 at 5:05 comment added Mariano Suárez-Álvarez The algebra $\mathbb RQ_8$ is semisimple. Since there are $|Q_8/Q_8'|=4$ modules of degree $1$ and the tautological representation on $\mathbb H$ is irreducible and has $\mathbb H$ as its endomorphism ring, as one can see at once, the wedderburn factorization must be $\mathbb R\times\mathbb R\times\mathbb R\times\mathbb R\times\times\mathbb H$, for there is no room for anything else.
Oct 17, 2012 at 4:33 history edited darij grinberg CC BY-SA 3.0
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Oct 17, 2012 at 4:25 history answered darij grinberg CC BY-SA 3.0