7
$\begingroup$

I am trying to decompose $\mathbb{R}[Q]$ as a product of matrix rings ($Q$ is the group of quaternions)

By Maschke's theorem, $\mathbb{R}[Q]$ is semisimple. I will begin by decomposing it as a left $\mathbb{R}[Q]$-module.

I am looking for all simple $\mathbb{R}[Q]$-modules and how many times they occur in the decomposition.

Looking at the proof of Wedderburn's theorem, I see that any simple $\mathbb{R}[Q]$-module $M$ is isomorphic as an $\mathbb{R}$-algebra to $(\text{End}_{\mathbb{R}[Q]}(M)^{op})^n$ with $n$ being the number of times $M$ appears in the decomposition of $\mathbb{R}[Q]$ as an $\mathbb{R}[Q]$-module.

Is this right?

Looking at the quotient $Q/Z(Q)\cong V_4$, I already see 4 different ways to make $\mathbb{R}$ an $\mathbb{R}[Q]$-module. All are simple because they already are simple as $\mathbb{R}$-modules. I would like to know how many times each of them appears in the module decomposition. For that purpose, I look at $\text{End}_{\mathbb{R}[Q]}(M)^{op}$. It is a vector space over $\mathbb{R}$ and $\mathbb{R}$ is a vector space over it. So, as $\mathbb{R}$-vector spaces $\text{End}_{\mathbb{R}[Q]}(M)^{op} \cong \mathbb{R}$ . Thus, by dimension consideration over $\mathbb{R}$ I see that $n=1$ for each of these modules. That is - each of them appears once in the decomposition.

So, $\mathbb{R}\oplus\mathbb{R}\oplus\mathbb{R}\oplus\mathbb{R}$ is a summand of the module $\mathbb{R}[Q]$ and I have 4 more dimensions to "fill in".

For a simple $\mathbb{R}[Q]$-module $M$, what can $\text{End}_{\mathbb{R}[Q]}(M)$ be? It's a finite dimensional division algebra over $\mathbb{R}$, so it must be one of $\mathbb{R},\mathbb{C},\mathbb{H}$.

Let's try $\text{End}_{\mathbb{R}[Q]}(M)\cong\mathbb{H}$. Since we only have 4 dimensions left, it this is the case then surely this $M$ is isomorphic as an $\mathbb{R}[Q]$-module to $\mathbb{H}$. Trying $M=\mathbb{H}$ with the natural $\mathbb{R}[Q]$-action I really get a simple module.

So, I get an $\mathbb{R}[Q]$-module decomposition $\mathbb{R}[Q]\cong \mathbb{R}\oplus\mathbb{R}\oplus\mathbb{R}\oplus\mathbb{R}\oplus\mathbb{H}$.

Looking at this again together with Wedderburn's theorem I see that, as $\mathbb{R}[Q]$-algebras:

$\mathbb{R}[Q]\cong M_1(\mathbb{R})\times M_1(\mathbb{R})\times M_1(\mathbb{R})\times M_1(\mathbb{R})\times M_1(\mathbb{H})\cong \mathbb{R}\times \mathbb{R}\times \mathbb{R}\times \mathbb{R}\times \mathbb{H}$.

  1. Did I get it right?
  2. Did I explain it right?
  3. Is there a simpler way?
  4. Is there a text with examples of performing such decompositions for other group algebras?
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.