# Representations of Plesken Lie algebras

P. G. Romeo and S. N. Arjun  Dept. of Mathematics, Cochin University of
Science and Technology, Kochi, Kerala, INDIA.
$romeo_{-}parackal<EMAIL_ADDRESS>
###### Abstract.

Cohen and Taylor introduced Plesken Lie algebras of finite groups and studied
their structural properties. As a further step, we will introduce Plesken Lie
algebra representations, Plesken Lie algebra modules and discuss its
irreducibility.

###### Key words and phrases:

Lie algebra, Group algebra, Plesken Lie algebra, Representations.

###### 1991 Mathematics Subject Classification:

22D20, 17B05, 17B10

Representations of Lie groups and Lie algebras is an active area of research
related to Lie theory which has its roots in the work of Sophus Lie, who
studied certain transformation groups that are now called Lie groups. At
present both Lie groups and Lie algebras have become essential to many parts
of mathematics and theoretical physics. The Lie algebra, being a linear
object, is more immediately accessible than the Liegroup. It was Wilhelm
Killing (1847–1923) who insisted that before one could classify all group
actions one should begin by classifying all (finite dimensional real) Lie
algebras. The gradual evolution of the ideas of Lie, Friedrich Engel
(1861–1941), and Killing, made it clear that determining all simple Lie
algebras was fundamental. Throughout this paper, unless otherwise stated, a
Lie algebra is a finite dimensional over a field of characteristic zero.

A group algebra of a finite group $G$ is a vector space having basis
consisting of group elements which also carries extra structure involving the
product operation on $G$. In a sense, group algebras are the source of all you
need to know about representation theory of finite groups. A representation of
a finite group $G$ on a finite dimensional complex vector space $V$ is a
homomorphism $\rho:G\to GL(V)$ of $G$ to the group of automorphisms of $V$. A
subrepresentation of a representation $V$ is a vector subspace $W$ of $V$
which is invariant under $G$. That is $\rho(g)(w)\in W$ for all $g\in G$ and
$w\in W$. A representation $\rho$ is irreducible if there is no proper
invariant subspace $W$ of $V$. The group algebra assumes the structure of a
Lie algebra via the bracket (commutator) operation.

W. Plseken and Arjeh M. Cohen described the construction of a Lie algebra of a
finite group and call it as Plesken Lie algebra. The question that they
encountered was for which groups the construction would produce a simple Lie
algebra? In [1], Arjeh M. Cohen and D. E. Taylor studied the structure of
Plesken Lie algebras in detail by explicitely determining the groups for which
the Plesken Lie algebra is simple or semisimple. In this paper we discuss the
representations of Plesken Lie algebras, Plesken Lie algebra modules and the
Plesken Lie algebra representations induced from the group representations.
Further we also determine the irreducible Plesken Lie algebra representations.

##  1\. Preliminaries

We will first go through preliminaries of Lie algebras, group algebras and
Plesken Lie algebras. Also we describes some structural properties of group
algebra. Further we will introduce the notion of representation of Plesken Lie
algebras. We also discuss some irreducible representations of Plesken Lie
algebras of some groups.

A Lie algebra $L$ over an arbitrary field $\mathbb{F}$ is a vector space over
$\mathbb{F}$ endowed with an operation called Lie bracket satisfying the
following properties:

  1. (1)

Bilinearity : For $x,y,z\in L,a,b\in\mathbb{F}$

$[ax+by,z]=a[x,z]+b[y,z]$

$[x,ay+bz]=a[x,y]+b[x,z]$

  2. (2)

$[x,x]=0$ for all $x\in L$

  3. (3)

Jacobi identity :
$[x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0$ for all $x,y,z\in L$

A vector subspace $K\subset L$ is a Lie subalgebra of $L$ if $[x,y]\in K$ for
all $x,y\in K$, and a subspace $I$ of a Lie algebra $L$ is called an ideal if
$[x,y]\in I$ for all $x\in L,y\in I$. A Lie algebra homomorphism from $L$ to
$L^{\prime}$ is a linear map $\phi:L\to L^{\prime}$ such that
$\phi([x,y])=[\phi(x),\phi(y)]$ for all $x,y\in L$.

A group algebra $\mathbb{F}G$ of a group $G$ over $\mathbb{F}$ is a vector
space with group elemensts as basis. That is,

$\mathbb{F}G=\\{\displaystyle\sum_{i=1}^{n}a_{i}g_{i}:a_{i}\in\mathbb{F}\
\text{for all i}\\}$

Here the multiplication is defined by

$\begin{pmatrix}\displaystyle\sum_{i=1}^{n}a_{i}g_{i}\end{pmatrix}\begin{pmatrix}\displaystyle\sum_{j=1}^{n}b_{j}g_{j}\end{pmatrix}=\displaystyle\sum_{i,j=1}^{n}a_{i}b_{j}(g_{i}g_{j}),\
\indent a_{i},b_{j}\in\mathbb{F}$

A linear map between two group algebras ${\mathbb{F}G}$ and ${\mathbb{F}H}$
which preserves the algebra multiplication is a homomorphism of group
algebras.

######  Definition 1.

(cf.[1]) Plesken Lie algebra $\mathcal{L}(G)$ of a group $G$ over $\mathbb{F}$
is the linear span of elements $\hat{g}=g-{g}^{-1}\in\mathbb{F}G$ together
with the Lie bracket

$[\hat{g},\hat{h}]=\hat{g}\hat{h}-\hat{h}\hat{g}$

######  Example 1.

Consider the symmetric group $S_{3}$, then

$\begin{split}L(S_{3})&=\text{span}\\{\sigma-{\sigma}^{-1}:\sigma\in
S_{3}\\}\\\ &=\\{a_{1}((1)-(1))+a_{2}((1\ 2)-(1\ 2))+a_{3}((1\ 3)-(1\
3))++a_{4}((2\ 3)-(2\ 3))\\\ &\ \indent\quad+a_{5}((1\ 2\ 3)-(1\ 3\
2))+a_{6}((1\ 3\ 2)-(1\ 2\ 3)):a_{i}\in\mathbb{C}\\}\\\ &=\\{a((1\ 2\ 3)-(1\
3\ 2)):a\in\mathbb{C}\\}\end{split}$

is a one dimensional Plesken Lie algebra over $\mathbb{C}$ with Lie bracket

$[a\widehat{(1\ 2\ 3)},b\widehat{(1\ 2\ 3)}]=0$

##  2\. Group algebra homomorphisms and Plesken Lie algebra homomorphisms

Here we go through some structural properties of group algebras and Plesken
Lie algebras such as group algebra homomorphisms, group algebra
representations and Plesken Lie algebra homomorphisms.

######  Lemma 1.

Let $f:G\rightarrow H$ be a group homomorphism. Then
$\bar{f}:{\mathbb{F}G}\rightarrow{\mathbb{F}H}$ defined by

$\bar{f}(\displaystyle\sum_{i}^{n}a_{i}g_{i})=\displaystyle\sum_{i}^{n}a_{i}f(g_{i})$

is a homomorphism between group algebras.

###### Proof.

Let $G=\\{g_{1},g_{2},...,g_{n}\\}$, then
${\mathbb{F}G}=\\{\displaystyle\sum_{i=1}^{n}a_{i}g_{i}:a_{i}\in\mathbb{F}\
\text{for all i}\\}$. Clearly $\bar{f}$ is linear and for
$\alpha=\displaystyle\sum_{i=1}^{n}a_{i}g_{i},\,\beta=\displaystyle\sum_{j=1}^{n}b_{j}g_{j}\in{\mathbb{F}G}$,

$\begin{split}\bar{f}(\displaystyle\sum_{i=1}^{n}a_{i}g_{i}\displaystyle\sum_{j}b_{j}g_{j})=\bar{f}(\displaystyle\sum_{i,j=1}^{n}a_{i}b_{j}g_{i}g_{j})=\displaystyle\sum_{i,j=1}^{n}a_{i}b_{j}f(g_{i})f(g_{j})=\displaystyle\sum_{i=1}^{n}a_{i}f(g_{i})\displaystyle\sum_{j=1}^{n}b_{j}f(g_{j})\end{split}$

hence, $\bar{f}$ is a homomorphism between group algebras. ∎

However, it is not necessary that any group algebra homomorphisms induced by a
group homomorphism as seen in the following example.

######  Example 2.

Consider the group $G=\\{e,\sigma\\}$ where $\sigma^{2}=e$. Define
$f:\mathbb{C}G\to\mathbb{C}G$ by

$f(e)=e$ and $f(\sigma)=-\sigma$ with $f$ is linear

Then $f$ is a group algebra homomorphism. But $f$ is not induced from any
group homomorphism.

A representation of a group algebra $\mathbb{F}G$ is a linear map
$\phi:\mathbb{F}G\to\mathfrak{gl}(V)$ ( where $\mathfrak{gl}(V)$ is the set of
all endomorphisms on $V$) such that $\phi$ is an algebra homomorphism. A
representation of a group $G$ gives rise to a representation of the group
algebra $\mathbb{F}G$ as follows.

######  Definition 2.

Given a representation $\rho:G\to GL(V)$ of a group $G$. Then
$\phi:\mathbb{F}G\to\mathfrak{gl}(V)$ defined by

$\phi(\displaystyle\sum_{i=1}^{n}a_{i}g_{i})=\displaystyle\sum_{i=1}^{n}a_{i}\rho(g_{i})$

is a representation of $\mathbb{F}G$.

A subrepresentation of a representation $\phi:\mathbb{F}G\to\mathfrak{gl}(V)$
is a map $\phi^{\prime}:\mathbb{F}G\to\mathfrak{gl}(W)$ where $W$ is a vector
subspace $V$ which is invariant under $\mathbb{F}G$. A representation $\phi$
is irreducible if there is no proper $\mathbb{F}G$-invariant subspace $W$ of
$V$. That is, there is no proper invariant subspace $W$ such that
$\phi(x)(w)\in W$ for all $w\in W$ and $x\in\mathbb{F}G$.

######  Theorem 1.

If $\rho:G\to GL(V)$ is an irreducible representation of $G$, then the
representation $\phi:\mathbb{F}G\to\mathfrak{gl}(V)$ defined by

$\phi(\displaystyle\sum_{i=1}^{n}a_{i}g_{i})=\displaystyle\sum_{i=1}^{n}a_{i}\rho(g_{i})$

is also irreducible.

###### Proof.

Since $\rho:G\to GL(V)$ is an irreducible representation, $V$ has no proper
$G$-invariant subspace. Let $\phi:\mathbb{F}G\to\mathfrak{gl}(V)$ be the
corresponding representation of $\mathbb{F}G$. Also let $W$ be an invariant
subspace of $V$ under the action $\phi$. That is $\phi(x)(w)\in W$ for all
$w\in W$ and $x\in\mathbb{F}G$.

$\begin{split}\phi(x)(w)&=\phi(\displaystyle\sum_{i}^{n}a_{i}g_{i})(w)\\\
&=\displaystyle\sum_{i}^{n}a_{i}\rho(g_{i})(w)\end{split}$

Inparticular, $\rho(g_{i})(w)\in W$ for all $w\in W$ and $g_{i}\in G$. That is
$W$ is an invariant subspace of $V$ under the action $\rho$, which is a
contradiction since $\rho$ is irreducible. Hence there is no invariant
subspace $W$ of $V$ under the action $\phi$. ∎

The following example shows that the group algebra representations of the
irreducible representations of the symmetric group $S_{3}$ are also
irreducible.

######  Example 3.

Consider the symmetric group $S_{3}$ and its irreducible representations
$\rho_{1},\rho_{2}$ and $\rho_{3}$ where $\rho_{1}$ is the trivial
representation with degree 1, $\rho_{2}$ is the alternating representation
with degree 1 and $\rho_{3}$ is the standard representation with degree 2.
Also,

$\begin{split}\rho_{1}&:\sigma\mapsto[1]\text{ for all }\sigma\in S_{3}\\\
\rho_{2}&:\text{even permutations maps to [1] and odd permutations maps to
[-1]}\\\ \rho_{3}&:(1\ 2\
3)\mapsto\begin{bmatrix}\frac{-1}{2}&\frac{\sqrt{3}}{2}\\\
\frac{\sqrt{3}}{2}&\frac{-1}{2}\end{bmatrix},(1\
2)\mapsto\begin{bmatrix}1&0\\\ 0&-1\end{bmatrix}\end{split}$

Then from Theorem 2, the group algebra representations are :
$\phi_{1}:\mathbb{R}S_{3}\to\mathfrak{gl}(\mathbb{R})$ given by

$\phi_{1}(a_{1}(1)+a_{2}(1\ 2)+a_{3}(2\ 3)+a_{4}(1\ 3)+a_{5}(1\ 2\ 3)+a_{6}(1\
3\ 2))=(a_{1}+a_{2}+a_{3}+a_{4}+a_{5}+a_{6})\begin{bmatrix}1\end{bmatrix}$

$\phi_{1}:\mathbb{R}S_{3}\to\mathfrak{gl}(\mathbb{R})$ given by

$\phi_{2}(a_{1}(1)+a_{2}(1\ 2)+a_{3}(2\ 3)+a_{4}(1\ 3)+a_{5}(1\ 2\ 3)+a_{6}(1\
3\ 2))=(a_{1}+a_{5}+a_{6})[1]+(a_{2}+a_{3}+a_{4})[-1]$

$\phi_{3}:\mathbb{R}S_{3}\to\mathfrak{gl}(\mathbb{R}^{2})$ given by

$\phi_{2}(a_{1}(1)+a_{2}(1\ 2)+a_{3}(2\ 3)+a_{4}(1\ 3)+a_{5}(1\ 2\ 3)+a_{6}(1\
3\ 2))=a_{1}\begin{bmatrix}1&0\\\ 0&1\end{bmatrix}+a_{2}\begin{bmatrix}1&0\\\
0&-1\end{bmatrix}+a_{3}\begin{bmatrix}\frac{-1}{2}&\frac{-\sqrt{3}}{2}\\\
\frac{-\sqrt{3}}{2}&\frac{1}{2}\end{bmatrix}+a_{4}\begin{bmatrix}\frac{-1}{2}&\frac{\sqrt{3}}{2}\\\
\frac{\sqrt{3}}{2}&\frac{1}{2}\end{bmatrix}+a_{5}\begin{bmatrix}\frac{-1}{2}&\frac{-\sqrt{3}}{2}\\\
\frac{\sqrt{3}}{2}&\frac{-1}{2}\end{bmatrix}+a_{6}\begin{bmatrix}\frac{-1}{2}&\frac{\sqrt{3}}{2}\\\
\frac{-\sqrt{3}}{2}&\frac{-1}{2}\end{bmatrix}$

Since $\phi_{1}$ amd $\phi_{2}$ are of degree 1, they are irreducible. To
prove $\phi_{3}$ is irreducible, let $W$ be a proper invariant subspace of
$\mathbb{R}^{2}$. Then $W=span\\{(\alpha,\beta)\\}$ for some
$(\alpha,\beta)\in\mathbb{R}^{2}$. Since $W$ is invariant, $\phi_{3}(x)(w)\in
W$ for all $x\in\mathbb{R}S_{3}$ and $w\in W$. That is,
$\phi(\displaystyle\sum_{i=1}^{n}a_{i}\sigma_{i})(w)=k(\alpha,\beta)$ for some
$k\in\mathbb{R}$. From computations we obtained,

$\begin{split}\alpha(a_{1}+a_{2}-\frac{a_{3}}{2}-\frac{a_{5}}{2}-\frac{a_{6}}{2})+\frac{\sqrt{3}\beta}{2}(a_{3}+a_{4}-a_{5}-a_{6})&=k\alpha\\\
\frac{\sqrt{3}\alpha}{2}(a_{4}-a_{3}+a_{5}-a_{6})+\beta(a_{1}-a_{2}+\frac{a_{3}}{2}+\frac{a_{4}}{2}-\frac{a_{5}}{2}-\frac{a_{6}}{2})&=k\beta\end{split}$

This is true for every $x\in\mathbb{R}S_{3}$ and $w\in W$. Thus take
$x=\frac{2}{\sqrt{3}}(2\ 3)$, then it can be seen that $\alpha=\beta=0$. Thus
$W=0$ and hence $\phi_{3}$ is irreducible.

######  Definition 3.

Let $G$ be a finite group and $\mathbb{F}$ be a field. The group algebra
$\mathbb{F}G$ over $\mathbb{F}$ with the Lie bracket $[\ ,\
]:\mathbb{F}G\times\mathbb{F}G\rightarrow\mathbb{F}G$ defined by

$[\alpha,\beta]=\alpha\beta-\beta\alpha$ where
$\alpha=\displaystyle\sum_{i}a_{i}g_{i},\beta=\displaystyle\sum_{i}b_{i}g_{i}$
in $\mathbb{F}G$

is a Lie algebra, denote it by $L_{\mathbb{F}G}$ and is called the Group Lie
algebra.

A linear map between two group Lie algebras $L_{\mathbb{F}G}$ and
$L_{\mathbb{F}H}$ which preserves the Lie bracket is a homomorphism of group
Lie algebras.

The following lemmas are immediately follow from the definitions.

######  Lemma 2.

Let $f:G\rightarrow H$ be a group homomorphism. Then
$\bar{f}:L_{\mathbb{F}G}\rightarrow L_{\mathbb{F}H}$ defined by

$\bar{f}(\displaystyle\sum_{i=1}^{n}a_{i}g_{i})=\displaystyle\sum_{i=1}^{n}a_{i}f(g_{i})$

is a homomorphism between group Lie algebras.

###### Proof.

Let $G=\\{g_{1},g_{2},...,g_{n}\\}$, then
$L_{\mathbb{F}G}=\\{\displaystyle\sum_{i=1}^{n}a_{i}g_{i}:a_{i}\in\mathbb{F}\
\text{for all i}\\}$. Clearly $\bar{f}$ is linear and for
$\alpha=\displaystyle\sum_{i=1}^{n}a_{i}g_{i},\,\beta=\displaystyle\sum_{j=1}^{n}b_{j}g_{j}\in
L_{\mathbb{F}G}$,

$\begin{split}\bar{f}([\alpha,\beta])&=\bar{f}(\alpha\beta-\beta\alpha)\\\
&=\bar{f}(\displaystyle\sum_{i,j=1}^{n}a_{i}b_{j}g_{i}g_{j}-\displaystyle\sum_{j,i=1}^{n}b_{j}a_{i}g_{j}g_{i})\\\
&=\displaystyle\sum_{i,j=1}^{n}a_{i}b_{j}f(g_{i})f(g_{j})-\displaystyle\sum_{j,i=1}^{n}b_{j}a_{i}f(g_{j})f(g_{i})\\\
&=\displaystyle\sum_{i=1}^{n}a_{i}f(g_{i})\displaystyle\sum_{j=1}^{n}b_{j}f(g_{j})-\displaystyle\sum_{j=1}^{n}b_{j}f(g_{j})\displaystyle\sum_{i=1}^{n}a_{i}f(g_{i})\\\
&=[\bar{f}(\alpha),\bar{f}(\beta)]\end{split}$

hence, $\bar{f}$ is a homomorphism between Lie algebras of group algebras. ∎

Clearly the Plesken Lie algebra $\mathcal{L}(G)$ is a proper subset of
$L_{\mathbb{F}G}$ and moreover, it is a Lie subalgebra of $L_{\mathbb{F}G}$.

A linear map between two Plesken Lie algebras $\mathcal{L}(G)$ and
$\mathcal{L}(H)$ is a Plesken Lie algebra homomorphism if it preserves the Lie
bracket.

######  Example 4.

Consider the Plesken Lie algebras $\mathcal{L}(S_{3})=\\{\alpha((1\ 2\ 3)-(1\
3\ 2)):\alpha\in\mathbb{C}\\}$ of $S_{3}$ over $\mathbb{C}$ and
$\mathcal{L}(D_{4})=\\{\alpha(a-a^{3}):\alpha\in\mathbb{C}\\}$ of
$D_{4}=<a,b:a^{4}=b^{2}=e,aba=b^{-1}>$ over $\mathbb{C}$. Then
$\hat{f}:\mathcal{L}(S_{3})\to\mathcal{L}(D_{4})$ defined by

$\hat{f}(\alpha((1\ 2\ 3)-(1\ 3\ 2)))=\alpha(a-a^{3})$

is a Plesken Lie algebra homomorphism.

######  Lemma 3.

Let $f:G\rightarrow H$ be a group homomorphism. Then
$\hat{f}:\mathcal{L}(G)\rightarrow\mathcal{L}(H)$ defined by

$\hat{f}(\displaystyle\sum_{i=1}^{n}a_{i}\hat{g_{i}})=\displaystyle\sum_{i=1}^{n}a_{i}\widehat{f(g_{i})}$

is a Plesken Lie algebra homomorphism. Further, $\hat{f}$ is actually the
restriction of the homomorphism of the group Lie algebras
$\bar{f}:L_{\mathbb{F}G}\to L_{\mathbb{F}H}$(defined in Lemma 3) to
$\mathcal{L}(G)$. That is,

$\hat{f}=\bar{f}\arrowvert_{\mathcal{L}(G)}$

###### Proof.

The proof is similar as in Lemma 3. ∎

##  3\. Plesken Lie algebra representations

Next we proceed to define representation of a Plesken Lie algebra and discuss
some of its properties. We have already described the representations of group
algebras and it is seen that when the representations of group is irreducible
so is the representation of group algebras.

######  Definition 4.

A representation of a Plesken Lie algeba $\mathcal{L}(G)$ is a linear map
$\phi:\mathcal{L}(G)\to\mathfrak{gl}(V)$ (where $V$ is a vector space over
$\mathbb{F}$) such that $\phi$ is a Lie algebra homomorphism.

The following theorem shows that if we have a representation of a group $G$,
then we can find a representation of the Plesken Lie algebra $\mathcal{L}(G)$.

######  Lemma 4.

If $\rho:G\to GL(V)$ is a representation of $G$ on $V$, then
$\psi:\mathcal{L}(G)\to\mathfrak{gl}(V)$ defined by

$\psi(\displaystyle\sum_{i=1}^{n}a_{i}\hat{g_{i}})=\displaystyle\sum_{i=1}^{n}a_{i}\widehat{\rho(g_{i})}$
for $\displaystyle\sum_{i=1}^{n}a_{i}\hat{g_{i}}\in\mathcal{L}(G)$

is a representation of the Plesken Lie algebra $\mathcal{L}(G)$.

###### Proof.

Since $\rho:G\to GL(V)$ is a repesentation of $G$, $\rho(g_{i})$ and
$\rho({g_{i}}^{-1})$ are automorphisms on $V$. Then

$\widehat{\rho(g_{i})}=(\rho(g_{i})-{\rho(g_{i})}^{-1})=\rho(g_{i})-\rho({g_{i}}^{-1})$.

is an endomorphism on $V$. Thus $\psi$ is defined.
Clearly $\psi$ is linear. By $Lemma\ 4$, $\psi$ is a homomorphism from
$\mathcal{L}(G)$ to $\mathfrak{gl}(V)$ and hence $\psi$ is a representation of
$\mathcal{L}(G)$.

∎

A Plesken Lie algebra representation $\psi:\mathcal{L}(G)\to\mathfrak{gl}(V)$
is irreducible if there is no proper $\mathcal{L}(G)$-invariant subspace $W$
of $V$.

The following theorem states that the Plesken Lie algebra representation
corresponding to a reducible group representation is reducible.

######  Theorem 2.

If $\rho:G\to GL(V)$ is a reducible representation of a group $G$, then the
Plesken Lie algebra representation $\psi:\mathcal{L}(G)\to\mathfrak{gl}(V)$ of
$\mathcal{L}(G)$ defined by

$\psi(\displaystyle\sum_{i=1}^{n}a_{i}\hat{g_{i}})=\displaystyle\sum_{i=1}^{n}a_{i}\widehat{\rho(g_{i})}$

is also reducible.

###### Proof.

Since $\rho:G\to GL(V)$ is a reducible representation, $V$ has a proper
invariant subspace. That is, there is a subspace $W$ of $V$ such that
$\rho(g)w\in W$ for all $g\in G$ and $w\in W$. We claim that $W$ is also
invariant under $\psi$. For let
$\hat{x}=\displaystyle\sum_{i=1}^{n}a_{i}\hat{g_{i}}\in\mathcal{L}(G),w\in W$,

$\begin{split}\psi(\hat{x})(w)=\psi(\displaystyle\sum_{i=1}^{n}a_{i}\hat{g_{i}})(w)=\displaystyle\sum_{i=1}^{n}a_{i}\widehat{\rho(g_{i})}(w)=\displaystyle\sum_{i=1}^{n}a_{i}(\widehat{\rho(g_{i})}(w)-\widehat{\rho({g_{i}}^{-1})}(w))\end{split}$

Since $\rho(g_{i})(w),\rho(g_{i}^{-1})(w)\in W$ and $W$ is a subspace,
$\widehat{\rho(g_{i})}(w)-\widehat{\rho({g_{i}}^{-1})}(w)\in W$. Thus
$\psi(\hat{x})w\in W$ for all $\hat{x}\in\mathcal{L}(G)$ and $w\in W$. ∎

Very often if $\rho$ is an irreducible representation of a group $G$, then
$\psi$ is also an irreducible representation as seen in the following
examples(where $\psi$ is defined as in $Lemma\ 5$).

######  Example 5.

Consider the irreducible representations of $S_{3}$ as in Example 3. Then
corresponding Plesken Lie algebra representations are:

$\begin{split}\psi_{1}:\mathcal{L}(S_{3})\to\mathfrak{gl}(\mathbb{R})\text{
given by }\psi_{1}(a((1\ 2\ 3)-(1\ 3\ 2)))&=0\\\
\psi_{2}:\mathcal{L}(S_{3})\to\mathfrak{gl}(\mathbb{R})\text{ given by
}\psi_{2}(a((1\ 2\ 3)-(1\ 3\ 2)))&=0\\\
\psi_{3}:\mathcal{L}(S_{3})\to\mathfrak{gl}(\mathbb{R}^{2})\text{ given by
}\psi_{3}(a((1\ 2\ 3)-(1\ 3\
2)))&=a(\begin{pmatrix}\frac{-1}{2}&\frac{\sqrt{3}}{2}\\\
\frac{\sqrt{3}}{2}&\frac{-1}{2}\end{pmatrix}-\begin{pmatrix}\frac{-1}{2}&\frac{\sqrt{3}}{2}\\\
\frac{\sqrt{3}}{2}&\frac{-1}{2}\end{pmatrix}\\\
&=a(\begin{pmatrix}0&-\sqrt{3}\\\ \sqrt{3}&0\end{pmatrix})\end{split}$

Clearly, $\psi_{1}$ and $\psi_{2}$ are irreducible representations since each
of which has degree 1. We will prove $\psi_{3}$ is irreducible. For $W$ be a
proper invariant subspace of $\mathbb{R}^{2}$. Then
$W=span\\{(\alpha,\beta)\\}$ for some $(\alpha,\beta)\in\mathbb{R}^{2}$. Since
$W$ is invariant, $\psi_{3}(\hat{x})(w)\in W$ for all
$\hat{x}\in\mathcal{L}(S_{3})$ and $w\in W$.

$\begin{split}\psi_{3}(\hat{x})(w)\in W&\Rightarrow
a(\begin{pmatrix}0&-\sqrt{3}\\\
\sqrt{3}&0\end{pmatrix})\begin{pmatrix}\alpha\\\ \beta\end{pmatrix}\in W\\\
&\Rightarrow a(\begin{pmatrix}-\sqrt{3}\beta\\\
\sqrt{3}\alpha\end{pmatrix})\in W\\\
&\Rightarrow(-\sqrt{3}a\beta,\sqrt{3}a\alpha)=k(\alpha,\beta)\text{ for some
$k\in\mathbb{R}$ }\\\ &\Rightarrow k=\frac{-\sqrt{3}a\beta}{\alpha}\text{ and
}k=\frac{\sqrt{3}a\alpha}{\beta}\end{split}$

By equating we get, $\sqrt{3}a(\alpha^{2}+\beta^{2})=0$ which implies
$\alpha=\beta=0$. That is, $W=0$, hence $\psi_{3}$ is an irreducible
representation.

######  Example 6.

Consider the dihedral group $D_{4}=<a,b:a^{4}=b^{2}=e,aba=b>$ and its
irreducible representations $\rho_{1},\rho_{2},\rho_{3},\rho_{4}$ and
$\rho_{5}$ where $\rho_{i}$’s are given by

$\begin{split}\rho_{1}&:a\mapsto(1),b\mapsto(1)\\\
\rho_{2}&:a\mapsto(-1),b\mapsto(1)\\\ \rho_{3}&:a\mapsto(1),b\mapsto(-1)\\\
\rho_{4}&:a\mapsto(-1),b\mapsto(-1)\\\
\rho_{5}&:a\mapsto\begin{pmatrix}0&-1\\\
1&0\end{pmatrix},b\mapsto\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}\end{split}$

Then corresponding Plesken Lie algebra representations are:

$\begin{split}\psi_{1}&:\mathcal{L}(D_{4})\to\mathfrak{gl}(\mathbb{R})\text{ \
given by \ }\psi(\alpha(a-a^{3}))=0\\\
\psi_{2}&:\mathcal{L}(D_{4})\to\mathfrak{gl}(\mathbb{R})\text{ \ given by \
}\psi(\alpha(a-a^{3}))=0\\\
\psi_{3}&:\mathcal{L}(D_{4})\to\mathfrak{gl}(\mathbb{R})\text{ \ given by \
}\psi(\alpha(a-a^{3}))=0\\\
\psi_{4}&:\mathcal{L}(D_{4})\to\mathfrak{gl}(\mathbb{R})\text{ \ given by \
}\psi(\alpha(a-a^{3}))=0\\\
\psi_{5}&:\mathcal{L}(D_{4})\to\mathfrak{gl}(\mathbb{R^{2}})\text{ \ given by
\ }\psi(\alpha(a-a^{3}))=\alpha(\begin{pmatrix}0&-2\\\
2&0\end{pmatrix})\end{split}$

Since $\psi_{1},\psi_{2},\psi_{3}$ and $\psi_{4}$ has degree 1, they are
irreducible. Next we will prove $\psi_{5}$ is also an irreducible
representation. For let $W$ be a proper invariant subspace of
$\mathbb{R}^{2}$. Then $W=span{(\alpha,\beta)}$ for some
$(\alpha,\beta)\in\mathbb{R}^{2}$. Since $W$ is invariant,
$\psi_{5}(\hat{x})(w)\in W$ for all $\hat{x}\in\mathcal{L}(D_{4})$ and $w\in
W$.

$\begin{split}\psi_{5}(\hat{x})(w)\in
W&\Rightarrow\gamma(\begin{pmatrix}0&-2\\\
2&0\end{pmatrix})\begin{pmatrix}\alpha\\\ \beta\end{pmatrix}\in W\\\
&\Rightarrow\gamma(\begin{pmatrix}-2\beta\\\ 2\alpha\end{pmatrix})\in W\\\
&\Rightarrow(-2\gamma\beta,2\gamma\alpha)=k(\alpha,\beta)\text{ for some
$k\in\mathbb{R}$ }\\\ &\Rightarrow k=\frac{-2\gamma\beta}{\alpha}\text{ and
}k=\frac{2\gamma\alpha}{\beta}\end{split}$

By equating we get, $2\gamma(\alpha^{2}+\beta^{2})=0$ which implies
$\alpha=\beta=0$. That is, $W=0$, hence $\psi_{5}$ is an irreducible
representation.

Also we obtained that the representations of $\mathcal{L}(S_{4})$
corresponding to the irreducible representations of $S_{4}$ are irreducible.

However, the above situation is not a necessary and sufficient condition,
because it is possible to have $\psi$ may be $\mathcal{L}(G)$-reducible even
if $\rho$ is $\mathbb{F}G$-irreducible.

###  3.1. Plesken Lie algebra modules

Next we proceed to describes Plesken Lie algebra modules and obtain some
interesting theories such as Schur’s lemma.

######  Definition 5.

A vector space $V$, endowed with an operation $\mathcal{L}(G)\times V\to V$ is
an $\mathcal{L}(G)$-module if

  1. (1)

$(a\hat{x}+b\hat{y})v=a(\hat{x}v)+b(\hat{y}v)$

  2. (2)

$\hat{x}(av+bw)=a(\hat{x}v)+b(\hat{x}v)$

  3. (3)

$[\hat{x},\hat{y}]=\hat{x}\hat{y}v-\hat{y}\hat{x}v$

for all $\hat{x},\hat{y}\in\mathcal{L}(G),v,w\in V$ and $a,b\in\mathbb{F}$.

######  Remark 3.

Every $\mathbb{F}G$ is an $\mathcal{L}(G)$-module.

###### Proof.

Suppose $V$ is an $\mathbb{F}G$-module. Then for any
$\hat{x}\in\mathcal{L}(G)$ and $v\in V$,

$\hat{x}v=(\displaystyle\sum_{i=1}^{n}\hat{g_{i}})v=\displaystyle\sum_{i=1}^{n}a_{i}(\hat{g_{i}}v)$

Since $V$ is an $\mathbb{F}G$-module, $gv\in V$ for all $g\in G$ and $v\in V$.
Thus $\hat{g_{i}}v=g_{i}v-{g_{i}}^{-1}v\in V$ which implies $\hat{x}v\in V$
for all $\hat{x}\in\mathcal{L}(G)$ and $v\in V$ and this satisfies all the
axioms of an $\mathcal{L}(G)$-module. Thus $V$ is an $\mathcal{L}(G)$-module.
∎

Note that the converse of the remark need not be true.

######  Example 7.

Consider the group $G=<a:a^{3}=e>=\\{e,a,a^{2}\\}$ and the vector space
$V=span\\{v_{1},v_{2},v_{3}\\}$ over the field $\mathbb{F}$. define the
product $g_{i}v$ as follows :

$ev_{1}=v_{1},\quad ev_{2}=v_{2},\quad ev_{3}=v_{3}$

$av_{1}=v_{2},\quad av_{2}=v_{3},\quad av_{3}=v_{1}$

$a^{2}v_{1}=v_{3},\quad a^{2}v_{2}=-v_{1},\quad a^{2}v_{3}=v_{2}$

Then $V$ is an $\mathbb{F}G$-module and thus it is an $\mathcal{L}(G)$-module.
Define

$U=span\\{v_{1}+v_{2}\\}$

Then $U$ is a subspace of $V$. Since $a(v_{1}+v_{2})=v_{2}+v_{3}\notin U$, $U$
is not an $\mathbb{F}G$-submodule.
We have $\mathcal{L}(G)=\\{\alpha(a-a^{2}):\alpha\in\mathbb{F}\\}$. Then for
any $\hat{x}=\alpha(a-a^{2})\in\mathcal{L}(G)$, and $u=v_{1}+v_{2}\in U$,

$\begin{split}\alpha(a-a^{2})(v_{1}+v_{2})&=\alpha a(v_{1}+v_{2})-\alpha
a^{2}(v_{1}+v_{2})\\\ &=\alpha(v_{2}+v_{3}-(v_{3}-v_{1}))\\\
&=\alpha(v_{2}+v_{1})\in U\end{split}$

Thus $U$ is an $\mathcal{L}(G)$-submodule of $V$.

######  Theorem 4 (Schur’s lemma).

Let $V$ and $W$ are irreducible $\mathcal{L}(G)$-modules.

  1. (1)

If $\theta:V\to W$ is an $\mathcal{L}(G)$-homomorphism, then either $\theta$
is an $\mathcal{L}(G)$-isomorphism or $\theta(v)=0$ for all $v\in V$.

  2. (2)

If $\theta:V\to V$ is an $\mathcal{L}(G)$-isomorphism, then
$\theta(v)=\lambda_{v}I_{V}$.

###### Proof.

  * (1)

Suppose $\theta(v)=0$ for some $v\in V$. Then $Im(\theta)\neq\\{0\\}$. Since
$Im(\theta)$ is a submodule of $W$ and $W$ is irreducible, $Im(\theta)=W$.
Thus $\theta$ is onto. Since $Ker(\theta)$ is a submodule of $V$, $V$ is
irreducible and $Ker(\theta)\subsetneq V$, $Ker(\theta)=\\{0\\}$. Thus
$\theta$ is one-one. Hence $\theta$ is an isomorphism.

  * (2)

Suppose $\theta:V\to V$ is an $\mathcal{L}(G)$-isomorphism. Then $\theta$ has
an eigen value $\lambda\in\mathbb{C}$. Thus $Ker(\theta-\lambda
I_{V})\neq\\{0\\}$. $Ker(\theta-\lambda I_{V})$ is a non-zero
$\mathcal{L}(G)$-submodule of $V$ and since $V$ is irreducible,
$Ker(\theta-\lambda_{v}I_{V})=V$. Hence, $\theta=\lambda I_{V}$.

∎

######  Theorem 5.

If $G$ is an abelian group, then every irreducible $\mathcal{L}(G)$-module has
dimension 1.

###### Proof.

Suppose $G$ is an abelian group. Then $\mathcal{L}(G)$ is an abelian Plesken
Lie algebra. That is, $[\hat{x},\hat{y}]=0$ for all
$\hat{x},\hat{y}\in\mathcal{L}(G)$ which implies that
$\hat{x}\hat{y}=\hat{y}\hat{x}$ for all $\hat{x},\hat{y}\in\mathcal{L}(G)$.
Let $V$ be an irreducible $\mathcal{L}(G)$-module. Define $\phi:V\to V$ by

$\phi(v)=v\hat{x}$

Then $\phi$ is linear and

$\phi(v\hat{y})=(v\hat{y})\hat{x}=v(\hat{y}\hat{x})=v(\hat{x}\hat{y})=(v\hat{x})\hat{y}=\phi(v)\hat{y}$

Thus $\phi$ is an $\mathcal{L}(G)$-module homomorphism on $V$. Then by Schur’s
lemma, $\phi=\lambda I_{V}$ for some $\lambda\in\mathbb{C}$. That is,
$v\hat{x}=\lambda v$ for all $v\in V$. That is, every subspace of $V$ is an
$\mathcal{L}(G)$-submodule of $V$. Since $V$ is irreducible, $dim(V)=1$. ∎

An $\mathcal{L}(G)$-module gives us many representations of Plesken Lie
algebras, all of the form

$\hat{x}\mapsto[\hat{x}]_{\mathfrak{B}}\quad(\hat{x}\in\mathcal{L}(G))$

for some $\mathfrak{B}$ of $V$. Also if
$\phi:\mathcal{L}(G)\to\mathfrak{gl}(V)$ is a representation of
$\mathcal{L}(G)$, then $V$ can be viewed as an $\mathcal{L}(G)$-module via the
action $\hat{x}v=\phi(\hat{x})(v)$. That is, there is a one-to-one
correspondence between $\mathcal{L}(G)$-modules and representations of the
Plesken Lie algebra $\mathcal{L}(G)$.

######  Theorem 6.

Let $\phi:\mathcal{L}(G)\to\mathfrak{gl}(V)$ be an irreducible representation.
Then the only endomorphisms of $V$ commuting with all
$\phi(\hat{x})$($\hat{x}\in\mathcal{L}(G)$) are the scalars.

###### Proof.

Suppose $y\in\mathfrak{gl}(V)$ commutes with matrices in
$\phi(\mathcal{L}(G))$. Let $\lambda$ be the eigen value of $y$ and
$V_{\lambda}$, the eigen space of $y$ relative to $\lambda$. For any
$\hat{x}\in\mathcal{L}(G)$,

$y(\phi(\hat{x})v)=\phi(\hat{x})yv=\phi(\hat{x})\lambda
v=\lambda(\phi(\hat{x})v)$

That is, $\phi(\hat{x})v\in V_{\lambda}$. We have $V_{\lambda}=Ker(y-\lambda
I_{V})$ is an $\mathcal{L}(G)$-submodule of $V$. Since $V$ is irreducible and
$0\neq\phi(\hat{x})v\in V_{\lambda}$, $V_{\lambda}=V$, which implies
$y=\lambda$, a scalar matrix. ∎

## References

  * [1] Arjeh M. Cohen D. E. Taylor, “On a Certain Lie algebra Defined by a Finite Group”, The American Mathematical Monthly, Aug. - Sep., 2007.
  * [2] John Cullinan Mona Merling, “On the Plesken Lie algebra defined over finite field”.
  * [3] William Fulton Joe Harris, "Representation theory, A first course",GTM-129, Springer.
  * [4] Brian C. Hall, “Introduction to Lie Algebras and Representation Theory”, Springer-Verlag, 1972.
  * [5] James E. Humphreys, ”Introduction to Lie Algebras and Representation Theory”, Springer - Verlag, 1972.
  * [6] Gordon James Martin Liebeck, ”Representations and Characters of Groups”, Second Edition, Cambridge University Press, 2001.
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