As the title insinuates, in my readings I can across this isomorphism $\mathbb{H} \otimes \mathbb{C} \cong \text{End}_{\mathbb{C}} (\mathbb{H})$ and i cannot see why this is the case. Can someone help me see why this is an isomorphism. Here $\mathbb{H}$ is referring to the quaternions.
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1It matters what (ring or field of?) scalars you use in forming the tensor product. It is customary to indicate the scalar ring as a subscript on the circle cross symbol. – hardmath Mar 11 '21 at 17:29
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You should clarify certain what field you are tensoring over and how you view the quaternions as a complex vector space (left or right: they are not in the center of the quaternions). See Example 2.6 and Exercise 5 of Section 2 here for a starting point.
KCd
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I see. I understand the embedding of $\mathbb{H}$ into $M_2(\mathbb{C})$. Is there a explicit morphism I can consider. – An Isomorphic Teen Mar 09 '21 at 22:59
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If you want to view things as $\mathbf R$-algebras, first get an isomorphism of $\mathbf R$-vector spaces and then check it is an algebra map (multiplicative). Do you have experience building a linear map out of a tensor product and showing it is an isomorphism? – KCd Mar 10 '21 at 00:13
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$\Bbb{H=C+Cj}$ is a 2-dimensional complex vector space so $End_\Bbb{C}(\Bbb{H})\cong M_2(\Bbb{C})$ as complex algebras.
Then we are willing to check that $\Bbb{H\otimes_R C }\cong M_2(\Bbb{C})$ as complex algebras, where $\Bbb{H\otimes_R C }$ comes with its own complex algebra structure.
For this, send $1\otimes a+i\otimes b+j\otimes c+ij\otimes d$ to $a\pmatrix{1&0\\0&1}+b\pmatrix{i&0\\0&-i}+c\pmatrix{0&1\\-1&0}+d\pmatrix{0&i\\i&0}$
reuns
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The action of $\Bbb{H}$ on itself is not complex linear so I don't see anything else. But maybe I'm missing something. – reuns Mar 09 '21 at 23:12