3 Quaternions and Associative Algebra

Representation of Quaternion Group

$\mathbb{H}=\mathbb{C}+\mathbb{C}j=\{\alpha +\beta i+\gamma j+\delta k:\alpha ,\beta ,\gamma ,\delta \in \mathbb{R}\}$ is a space $2$-dimensional over $\mathbb{C}$ and $4$-dimensional over $\mathbb{R}$.

Quaternion group is $Q=\{\pm 1,\pm i,\pm j,\pm k\}$ where $ijk=i^2=j^2=k^2=-1$ and $i^2=-1$. Equivalently, $Q=⟨x,y:x^4=1,x^2=y^2,x^y=x^{-1}⟩$.

Consider the representation $\rho :H\to {\mathbb{C}}^{\ast }$. Note that $ij{i}^{-1}{j}^{-1}=-1$, we have that $\rho (-1)=\rho (i)\rho (j)(\rho (i){)}^{-1}(\rho (j){)}^{-1}=1$. Since $\rho (i{)}^2=\rho (i^2)=1=\rho (j{)}^2=\rho (k{)}^2$, we have that $\{\rho (i),\rho (j)\}\in \{\pm 1\}$ and $\rho (k)=\rho (i)\rho (j)$. So there are $4$ one-dimensional representations and $|Q|=1^2\cdot 4+2^2\cdot 1$.

Note that $Z(Q)=\{1,-1\}$, then $Q/Z(Q)=C_2\times C_2$. Define $q:Q\to Q/Z(Q)$ and $\rho :C_2\times C_2\to {\mathbb{C}}^{\ast }$. Then $q\circ \rho :Q\to {\mathbb{C}}^{\ast }$ is one-dimensional representation of $Q$.

To get two dimensional representation of $H$, consider space of quaternions $\mathbb{H}=\mathbb{C}+\mathbb{C}j=\{(a+bi)+(c+di)j:a,b,c,d\in \mathbb{R}\}$. Since $\mathbb{H}$ is a $Q$-module where $\mathbb{H}$ is a $2$-dimensional vector space over $\mathbb{C}$, the we have the following matrices (also see Pauli matrices)

$$1\mapsto \left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],i\mapsto \left[\begin{array}{cc}i& \\ & -i\end{array}\right],j\mapsto \left[\begin{array}{cc}& 1\\ -1& \end{array}\right],k\mapsto \left[\begin{array}{cc}& i\\ i& \end{array}\right].$$

Representation of Associative Algebra

Definition

Associative algebra is vector space over $k$ with binary bilinear operation $A\times A\to A,(a,b)\mapsto ab$. Representation of $A$ is a homomorphism $\rho :A\to {\mathrm{End}}_k(V)$ satisfying $\rho (ab)=\rho (a)\rho (b)$.

Examples.

• $V=0$, then any $A\to {\mathrm{End}}_k(V)$ is a trivial representation.
• For an associative algebra $A$ and $V=A$, $\rho :A\to {\mathrm{End}}_k(A),\rho (a)b:=ab$ is a regular representation of $A$.
• For a group $G$, we get a group algebra $kG$ which can be seen as a vector space. Then $\rho :G\to {\mathrm{End}}_k(kG)$ is a representation.
• Direct sum of two representation or two $A$-modules: Let $V$ and $W$ be two $A$-module. Define $a(v\oplus w)=av\oplus aw$. Note that this representation is composable.

Definition

Non-zero representation $V$ is called indecomposable if it is not isomorphic to a direct sum of two non-zero representations.

Remark. There exists $A$-module which is indecomposable but not irreducible. For example, let $A=\{\left[\begin{array}{cc}a& b\\ & c\end{array}\right]:a,b,c\in k\}$. Then $(x,0):x\in k$ is a submodule, but $V$ is indecomposable.

Definition

Let $V$ and $W$ be two $A$-modules, and let $T:V\to W$ be a linear transformation. If the following diagram

$$\text{require}AMScd\begin{array}{ccc}V& \stackrel{T}{\to }& W\\ \rho (a)\downarrow & & \downarrow \sigma (a)\\ V& \underset{T}{\to }& W\end{array}$$

for all $a\in A$, then $T$ is a homomorphism of $A$-modules $V$ and $W$.

Example. $k=\mathbb{R}$, $A=V=\mathbb{C}$. Then $\phi :\mathbb{C}\to \mathbb{C},a+ib\mapsto a-ib$ is a field automorphism but not homomorphism of modules.

Schur lemma

Let $V,W$ be two irreducible $A$-modules. If $\phi :V\to W$ is a homomorphism, then either $\phi =0$ or $\phi$ is an isomorphism. If $k$ is algebraically closed and $V$ is finite-dimensional irreducible, then for any $\phi :V\to V$ there is $\lambda \in k$ such that $\phi =\lambda \mathrm{Id}$.

Examples.

• If $A$ is a commutative algebra and $\rho :A\to V$ is a representation, then $\rho (a):V\to V$ is a homomorphism by $\rho (a)\rho (b)=\rho (b)\rho (a)$ for any $b\in A$. Then by Schur lemma, $\rho (a)=\lambda \mathrm{id}$ and each irreducible module has dimension $1$. Therefore, if $A=\mathbb{C}$, then the only irreducible module is $V=\mathbb{C}$ and the only indecomposable module is $V=\mathbb{C}$.
• Let $A=\mathbb{C}[x]$. Define $\rho :A\to \mathbb{C},x\mapsto \alpha$. Then for any polynomial $p(x)$, we have $\rho (p(x))=\rho (\alpha )$. It is a one-dimensional representation of $A$.
• If $\rho :A\to {\mathrm{M}}_n(\mathbb{C})$ is indecomposable, then in dimension $n$ and $x\in A$ we have $x\mapsto \left[\begin{array}{ccc}\lambda & 1& \\ & \ddots & 1\\ & & \lambda \end{array}\right]$.