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I just began to study about algebras over rings and quickly came across the fact that the quaternions are not an algebra over the complex numbers. I would prefer an answer as elementary as possible.

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    $\begingroup$ What is your definition of "algebra over a ring"? The probably big problem is that $w\alpha\neq \alpha w$ for $w$ complex and $\alpha$ a quaternion. $\endgroup$ – Thomas Andrews Dec 28 '15 at 19:25
  • $\begingroup$ Possible duplicate of Quaternions as a counterexample to the Gelfand–Mazur theorem $\endgroup$ – Noah Schweber Dec 28 '15 at 20:36
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    $\begingroup$ The usual definition of an algebra $A$ over a commutative ring $R$ is: “$A$ is a ring and there is a ring homomorphism $R\to Z(A)$”, where $Z(A)$ is the center of $A$. $\endgroup$ – egreg Dec 28 '15 at 20:55
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    $\begingroup$ This question seems different from the suggested dupe. Over there, the OP was oblivious that the quaternions were not a complex algebra and needed help clearing up the resulting "contradiction" with Gelfand-Mazur. Here the poster is aware it's not a complex algebra and is looking for explanations why it is not the case. $\endgroup$ – rschwieb Dec 28 '15 at 22:05
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If it were a $\Bbb C$ algebra, it would have to be dimension $2$ and contain a copy of $\Bbb C$.

Taking any $x$ not in the copy of $\Bbb C$ situated in $\Bbb H$, the span of $\Bbb C$ and $x$ is the whole ring. But products between elements of this span commute with each other, and that means the span is a commutative ring. This contradicts the fact the quaternions aren't commutative.

At another level, the Artin-Wedderburn theorem says that the only possible simple Artinian $\Bbb C$-algebras are the square matrix rings over $\Bbb C$, but none of them have dimension $2$. ($\Bbb H$ is a simple Artinian ring.)

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  • $\begingroup$ How can I see that $\mathbb{C}Q_8$ is a two dimensional algebra over $\mathbb{C}$? Can we define this algebra in two different ways, one in which complex i interacts with the quaternion i and one in which they behave separately? $\endgroup$ – AgentSmith May 15 '18 at 18:15
  • $\begingroup$ The group algebra $\mathbb C[Q_8]$ where $Q_8$ is the quaternion group? By definition, that has dimension $8$ over $\mathbb C$. If you mean you want to show that $\mathbb H$ is two dimensional over the subring $\mathbb C$, then it's well known that you can represent $\mathbb H$ inside of $M_2(\mathbb C)$ as the set of matrices $\begin{bmatrix}a&b\\-\bar{b}&\bar a \end{bmatrix}$. See this $\endgroup$ – rschwieb May 15 '18 at 20:06
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For an algebra we have the following axiom for scalar multiplication $$\alpha x \centerdot \beta y = (\alpha \beta) x \centerdot y.$$ This does not hold for the quaternions with complex scalars. Check for $\alpha = \beta = i$ and $x=y=j$.

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