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Let $ \mathbb{H} $ be the ring of quaternions and make the vector space $A = \mathbb{H} \otimes \mathbb{C}$ into a ring by defining $$(a \otimes w)(b \otimes z) = (ab \otimes wz) $$ for $a,b \in \mathbb{H}$ and $w,z \in \mathbb{C}$

Show that $A \simeq \mathrm{M}_2(\mathbb{C})$ the ring of complex $2\times 2$ matrices.

I would normally post my solution attempt but i haven't really gotten anywhere with this problem.

Update

Haven't managed to solve this yet, could use some more help. I admit, i don't quite understand the quaternions or the tensor product. Does any of this make sense?

$ \mathbb{H} \otimes \mathbb{C}$ = ($ \mathbb{R}1 \oplus \mathbb{R}i \oplus \mathbb{R}j \oplus \mathbb{R}k$) $\otimes$ $\mathbb{C} $ = ($ \mathbb{R}1 \otimes \mathbb{C} $) $\oplus$ ($\mathbb{R}i \otimes \mathbb{C} $) $\oplus$ ($\mathbb{R}j \otimes \mathbb{C} $) $\oplus$ ($\mathbb{R}k \otimes \mathbb{C} $) $\simeq $ $\mathbb{C} \oplus \mathbb{C} \oplus \mathbb{C} \oplus \mathbb{C}$ $ \simeq \mathrm{M}_2(\mathbb{C})$. The last isomorphism is as additive groups by sending (a,b,c,d) to \begin{pmatrix} a & b \\ c & d \\ \end{pmatrix} a,b,c,d $\in \mathbb{C}$

But I don't know how to make it into a ring isomorphism.

Perhaps I could use that $ \mathbb{C} \simeq \begin{pmatrix} a & -b \\ b & a \\ \end{pmatrix} $ $ a,b \in \mathbb{R}$

roslavets
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  • seems like i failed with the code, if someone could help me edit this it'd be great. – roslavets Jun 05 '14 at 00:35
  • thanks Keenan, that's exactly what i wanted. – roslavets Jun 05 '14 at 00:49
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    You should mention that you're tensoring over $\Bbb R$. – anon Jun 05 '14 at 00:49
  • yes, missed to write that out. – roslavets Jun 05 '14 at 00:51
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    @sea turtles: you're assuming that $x$ and $y$ commute. – Qiaochu Yuan Jun 05 '14 at 01:07
  • @QiaochuYuan Ah, thanks. Phew. – anon Jun 05 '14 at 01:07
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    One approach is to use the fact that tensoring preserves presentations; that is, if you write down a presentation of $\mathbb{H}$ as an $\mathbb{R}$-algebra, then the tensor product with $\mathbb{C}$ over $\mathbb{R}$ has the same presentation, but as a $\mathbb{C}$-algebra. – Qiaochu Yuan Jun 05 '14 at 01:20
  • We haven't really covered that material in class so i am not sure. – roslavets Jun 05 '14 at 01:31
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    I would expect $1 \otimes 1$ should map to $I \in M_2 ( \mathbb{C})$ and then we just need three matrices which produce the quaternion algebra. Perhaps the Pauli matrices will do. Nope, see http://en.wikipedia.org/wiki/Pauli_matrices. We'll want $i$ times the Pauli matrices: $$ i \otimes 1 \mapsto \left[ \begin{array}{cc} 0 & i \ i & 0 \end{array} \right] $$ $$ j \otimes 1 \mapsto \left[ \begin{array}{cc} 0 & 1 \ -1 & 0 \end{array} \right] $$ $$ k \otimes 1 \mapsto \left[ \begin{array}{cc} i & 0 \ 0 & -i \end{array} \right] $$ But, at this point, I merely conjecture these may be useful. – James S. Cook Jun 05 '14 at 02:29

1 Answers1

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The canonical isomorphism $\Phi:\mathbb{C}\otimes_{\mathbb{R}}\mathbb{H}\to \mathrm{M}_2(\mathbb{C})$ is constructed as follows:

  1. $\mathbb{H}\simeq \mathbb{C}^2$ as vector spaces over $\mathbb{C}$. Explicitly, $\gamma: \mathbb{C}^2\to \mathbb{H}$ is $\gamma(z_1,z_2)=z_1 + z_2j$ [Here $i$ is the imaginary unit of $\mathbb{C}$, and $i,j,ij=k$ are the three imaginary units of $\mathbb{H}$]. Given $\mathbb{H}\ni u =z+wj$, we have $$\bar{u}=\bar{z}+\overline{wj}=\bar{z}+\bar{j}\bar{w}=\bar{z}-j\bar{w}=\bar{z}-wj$$

  2. We define the $\mathbb{C}$-algebra homomorphism $\Phi:\mathbb{C}\otimes_{\mathbb{R}}\mathbb{H}\to \mathrm{M}_2(\mathbb{C})$ via $$\Phi(1\otimes u)V=\gamma^{-1}(\gamma(V)\bar{u}), \qquad (z\in \mathbb{C}, u\in \mathbb{H}, V\in \mathbb{C}^2)$$ Explicitly, if $u=z+w j$, then $$ \phi(u):=\Phi(1\otimes u)=\begin{pmatrix} \bar{z} & \bar{w}\\ -w& z \end{pmatrix} $$ To finish, note that $\dim_{\mathbb{C}}\mathbb{C}\otimes_{\mathbb{R}}\mathbb{H}=\dim \mathrm{M}_2(\mathbb{C})=4$, so it suffices to check that $\Phi$ is surjective. Define $I=\phi(i)$, $J=\phi(j)$, $K=\phi(ij)$, then it is easy to check that $$\mathrm{M}_2(\mathbb{C})\ni \begin{pmatrix} a & b\\ c & d \end{pmatrix}= \frac{a+d}{2}+\frac{i(a-d)}{2}I+\frac{b-c}{2}J+\frac{i(b+c)}{2}K $$ I want to also mention that the isomorphism $\phi$ is the natural isomorphism $\mathrm{Sp}(1)\to \mathrm{SU}(2)$, where $\mathrm{Sp}(1)$ is the group of unit quaternions.

Hamed
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