# Structure of Cross-wired Lamplighter Groups

Benjamin Jeffers

###### Abstract

We answer a question posed in [1], proving that for a closed cocompact
subgroup $\Gamma$ of ${\rm Isom}({\rm DL}(n,n))$ not contained in ${\rm
Isom}^{+}({\rm DL}(n,n))$, the sequence $1\to H\to\Gamma\to D_{\infty}\to 1$
splits, where $H$ is the unique open normal subgroup such that $\Gamma/H\cong
D_{\infty}$.

##  1 Introduction

Cross-wired lamplighter groups were introduced in [1] as a generalization of
lamplighter groups. Traditional lamplighters are a heavily studied object, it
is defined as the restricted wreath product
$\mathbb{Z}_{2}\wr\mathbb{Z}=\left(\bigoplus_{\mathbb{Z}}\mathbb{Z}/2\mathbb{Z}\right)\rtimes\mathbb{Z}$
where the action of $\mathbb{Z}$ on
$\bigoplus_{\mathbb{Z}}\mathbb{Z}/2\mathbb{Z}$ is given by shifting the
coordinates in the direct sum. Geometrically, the lamplighter group can be
thought of as an infinite street with an infinite number of lamps, and a
person to light the lamps, called the lamplighter. The lamplighter can move
left and right, turning the lamps on or off, while only a finite number of
lamps are on. An element in $\mathbb{Z}_{2}\wr\mathbb{Z}$ corresponds to the
positions of the on lamps, and the integer $n\in\mathbb{Z}$ is the position of
the lamplighter. More generally, for any finite group $F$, the lamplighter
group of $F$ is the wreath product
$F\wr\mathbb{Z}=\left(\bigoplus_{\mathbb{Z}}F\right)\rtimes\mathbb{Z}$. This
group is step-2 solvable and it has exponential growth. Grigorchuk and Zuk
showed that $\mathbb{Z}/2\mathbb{Z}\wr\mathbb{Z}$ can be constructed as the
automata groupof a 2-state automaton. Yang further to show that many automata
groups arising from the Cayley machines defined by Steinberg and Silva, [2],
are also cross-wired lamplighters, [3]

For one particular choice of generators the Cayley graph of a lamplighter
group, is a Diestel-Leader graph [4]. For two positive integers, $m,n\geq 2$,
the vertex set of a Diestel-Leader graph is defined as

$DL(m,n)=\left\\{(x,y)\in T_{m}\times T_{n}\mid b(x)+b^{\prime}(y)=0\right\\}$

where $T_{m}$ and $T_{n}$ are the regular infinite tree of degrees $m$ and $n$
respectively, and $b,b^{\prime}$ are a Busemann functions on $T_{m}$ and
$T_{n}$ respectively. Two vertices $(x,y)$, $(x^{\prime},y^{\prime})$ are
connected by an edge if $x$ and $x^{\prime}$ are connected by an edge in
$T_{m}$ and $y$ and $y^{\prime}$ are connected by an edge in $T_{n}$. Eskin,
Fisher, and Whyte proved that a finitely generated group is quasi-isometric to
a lamplighter group $F\wr\mathbb{Z}$ if and only if it acts properly and
cocompactly by isometries on a Diestel-Leader graph $DL(n,n)$ [5]. Moreover,
they showed that if $m\neq n$, then $DL(m,n)$ is not quasi-isometric to the
Cayely graph of any finitely generated group, proving a conjecture of Diestel
and Leader, [6].

In [1], Cornulier, Fisher, and Kashyap study the quasi-isometric rigidity of
lamplighter groups. They do this by studying the cocompact lattices in the
isometry group of a Diestel-Leader graph, ${\rm Isom}({\rm DL}(n,n))$, and
closed, cocompact subgroups of ${\rm Isom}({\rm DL}(m,n))$ when $m\neq n$.
These lattices turn out to not be lamplighter groups in general, and instead
are called _cross-wired lamplighters_. The name is due to the fact that we can
view them as an infinite street with an infinite number of lamps and a
lamplighter. The difference is that when the lamplighter changes the state of
one lamp, it might change the state of other lamps as well.

In this paper we answer question 1.4(iii) posed in [1]. They ask, if given
$\Gamma$, a closed cocompact subgroup of ${\rm Isom}({\rm DL}(n,n))$ not
contained in ${\rm Isom}^{+}({\rm DL}(n,n))$ , does the extension

$1\to H\to\Gamma\to D_{\infty}\to 1$

necessarily split? Here $H$ is the unique open normal subgroup such that
$\Gamma/H\cong D_{\infty}$, and $D_{\infty}$ is the infinite dihedral group.
We prove that the answer is yes by giving an isomorphism between the groups
$(U\rtimes_{\phi}\mathbb{Z})\rtimes_{\psi}Z/2\mathbb{Z}$ and
$U\rtimes_{\phi}(\mathbb{Z}\rtimes_{\psi}\mathbb{Z}/2\mathbb{Z})$. At the end
we make a couple of comments on question 1.4(i) posed in [1]. They ask if
there exists a cross-wired lamplighter group that is not virtually symmetric.
Where a cross-wired lamplighter $\Gamma$ is symmetric if it admits an
automorphism $\alpha$ such that $\pi\circ\alpha=-\pi$ where $\pi$ is the
projection from $\Gamma$ to $\mathbb{Z}$. A cross-wired lamplighter is
virtually symmetric if it has a finite index symmetric subgroup. While we do
not prove the statement either way we demonstrate that the example given in
[1] of $\Gamma=\mathbb{H}(\mathbb{F}_{q}[X,X^{-1}]$ is symmetric, where
$\Gamma$ is the Heisenberg group of upper triangular matrices over
$\mathbb{F}_{q}[X,X^{-1}]$.

Acknowledgments I would like to thank Nataša Macura for the suggestion of the
problem and many helpful discussions and Ryan Daileda for many fruitful
conversations.

##  2 Preliminaries

In this section we present a general theorem that will be of use to us, as
well as construct the Diestel-Leader graph, and describe its isometry group,
and then we end with a brief discussion of cross-wired lamplighers including
the main structure theorem of [1]. Throughout we denote the regular infinite
tree of degree $n$ by $T_{n}$.

We will make use of the following theorem characterizing splittings of short
exact sequences of groups to prove our main theorem.

######  Definition 1.

We say that a sequence of groups $1\to
H\xrightarrow{\alpha}G\xrightarrow{\beta}K\to 1$ _splits_ if there is a
homomorphism $\varphi\colon K\to{\rm Aut}(H)$ and an

${1}$${H}$${G}$${K}$${1}$${1}$${H}$${H\rtimes_{\varphi}K}$${K}$${1}$id$\scriptstyle{\alpha}$$\scriptstyle{\theta}$$\scriptstyle{\beta}$id

commutes.

######  Definition 2.

The infinite dihedral group, denoted $D_{\infty}$ is given by the
presentation:

$D_{\infty}\cong\mathbb{Z}\rtimes\mathbb{Z}/2\mathbb{Z}=\left<r,s\mid
s^{2}=1,srs=r^{-1}\right>$

######  Remark.

The group $D_{\infty}$ can be thought of as the isometries of $\mathbb{Z}$. It
is generated by a shift to the right by one, and a flip around the origin.

Next, we construct the Diestel-Leader graph as in [4], except we only consider
the case when $m=n$. Given a $n$-regular infinite tree, fix a preferred end
$-\infty$. For any two vertices we define their confluent $\widehat{xy}$ to be
the point where the infinite ray from $x$ to $-\infty$ meets the infinite ray
from $y$ to $-\infty$. Now, fix a vertex $o$, this will be the origin. Define
the Busemann function or height function to be $b\colon V(T_{n})\to\mathbb{Z}$
given by

$b(x)=d(x,\widehat{xo})-d(o,\widehat{xo}).$

This will partition the vertices of $T_{n}$ into sets which we call the level
sets $H_{k}$. An example of the level sets with respect to an origin and
$-\infty$ in $T_{2}$ can be seen in figure 2. To define the Diestel-Leader
graph, we take two copies of the tree $T_{n}$ along with a Busemann functions,
$b$ and let

$V({\rm DL}(n,n))=\left\\{(x,y)\in V(T_{n}\times T_{n})\mid
b(x)+b(y)=0\right\\}.$

Two vertices $(a,b),(c,d)$ are connected in ${\rm DL}(n,n)$ if and only if $a$
is adjacent to $c$ in $T_{n}$ and $b$ is adjacent to $d$ in the second copy of
$T_{n}$.

######  Remark.

The Diestel-Leader graph is often depicted with the two trees side by side,
with one upside down, so that the level sets add up to zero, as can be seen in
figure 1.

Figure 1: The trees $T_{1}$ and $T_{2}$ side by side with the height functions
of the vertices.

######  Remark.

The resulting tree and level sets is independent of the choices of $-\infty$
and $o$ since a a tree with different choice of origin and boundary point can
be interchanged by an isometry

Figure 2: Level sets of $T_{2}$

Now we construct the group of isometries of ${\rm DL}(n,n)$. See [7] for more
details. Let $U_{0}$ denote the isometries of $T_{n}$ which fix $-\infty$ and
preserve the level sets of the Busemann function $b$. We can explicitly
describe this group. Take a vertex $v$, in a level set $H_{k}$, that is on the
main line with the origin $o$ as in figure 2. The children of $v$ will be the
adjacent vertices in level set $H_{k+1}$. An automorphism that fixes $-\infty$
and preserves the level set will be a rooted tree automorphisms with $v$ as
the root. We can compose also compose these automorphisms with other rooted
tree automorphisms taking different vertices on the main line as a root. These
compositions will also preserve $-\infty$ and the level sets. In fact, these
are all the possible automorphisms in $U_{0}$ as is shown in the next lemma.

######  Lemma 3.

The only isometries in $U_{0}$ are those that are a composition of rooted tree
automorphisms to each branch, as well as the isometries that swap the children
of a vertex on the main branch.

###### Proof.

It is well known that the only other types of infinite tree isometries are
those that push everything along one line in the tree, and those that rotate
around a vertex. If we rotate around a specific vertex, then this will not
preserve $-\infty$, thus they are not in $U_{0}$.

Let $A$ be an axis in the tree, and let $x$ be a vertex on $A$ such that
$b(x)=k$. Then when we push along $A$, the vertex $x$ must end up in the level
set $H_{k}$, however, looking at the child of $x$ that also lies in $A$, which
we will call $y$, we see that this vertex must end up as a parent of $x$ in
the final position, meaning $b(y)=k-1$, thus this isometry does not preserve
the level sets. ∎

Then the group $U=U_{0}\times U_{0}$ is contained in ${\rm Isom}({\rm
DL}(n,n))$. Let $\varphi$ be the hyperbolic isometry of $T_{n}$ that has
translation length 1, with $-\infty$ as the repelling end. This means that
$\varphi$ has no fixed points, and we can essentially think of it as moving
branch 0 to branch 1, branch 1 to branch 2 and so forth as seen in figure 3

.

Figure 3: The action of $\phi_{2}$ on $T_{2}$

Define the isometry $(x,y)\mapsto(\phi(x),\phi^{-1}(y))$ of $T_{n}\times
T_{n}$. This shifts one of the trees up a level, so a vertex of $T_{n}$ with
height $i$ will now have height $i+1$, while a vertex of the other copy of
$T_{n}$ with height $-i$ will now have height $-i-1$, so that the height still
sums to 0. Finally, there exists an isometry $\psi$ defined by
$\psi(x,y)=(y,x)$, it swaps the two trees.

A result by Bartholdi, Neuhauser, and Woess in [8] shows that ${\rm Isom}({\rm
DL}(n,n)$ contains the groups generated by $U,\phi$, and $\psi$, and nothing
else.

######  Proposition 4.

For $n\geq 2$,

${\rm Isom}({\rm
DL}(n,n))=(U\rtimes_{\phi}\mathbb{Z})\rtimes_{\psi}\mathbb{Z}/2\mathbb{Z}).$

Next we take a brief look at the actions used to define the isometry group.
Note that an element of the group comes in the form
$(((u_{0},u_{1}),a),\epsilon)$ where $u_{0},u_{1}$ are level set preserving
and $-\infty$ fixing isometries of $T_{n}$, $a$ is an integer and
$\epsilon\in\mathbb{Z}/2\mathbb{Z}$.

The action defined by $\phi$ is $\phi\colon\mathbb{Z}\to{\rm Aut}(U)$ given by
$\phi(a)(u_{0},u_{1})=\phi_{a}(u_{0},u_{1})=(\phi^{a}u_{0}\phi^{-a},\phi^{-a}u_{1}\phi^{a})$.
We also have an isometry where we flip the two trees, $(x,y)\mapsto(y,x)$.
Call this flip $\psi$. This action is given by
$\psi\colon\mathbb{Z}/2\mathbb{Z}\to{\rm Aut}(U\rtimes_{\phi}\mathbb{Z})$
where
$\psi(\epsilon)(u_{0},u_{1})=\psi_{\epsilon}((u_{0},u_{1}),a)=((u_{0+\epsilon},u_{1+\epsilon}),(-1)^{\epsilon}a)$
for $\epsilon\in\mathbb{Z}/2\mathbb{Z}$. Where $u_{1+\epsilon}$ is taken to be
modulo 2, this way if $\epsilon=1$, then we are flipping the two isometries.

Looking at the multiplication in the group
$(U\rtimes_{\phi}\mathbb{Z})\rtimes_{\psi}\mathbb{Z}/2\mathbb{Z}$, we have

$\displaystyle(((u_{0},u_{1}),a),\epsilon)\cdot(((v_{0},v_{1}),b),\delta)=$
$\displaystyle\;(((u_{0},u_{1}),a)\cdot\psi_{\epsilon}((v_{0},v_{1}),b),\epsilon+\delta)$
$\displaystyle=$
$\displaystyle\;(((u_{0},u_{1}),a)\cdot((v_{0+\epsilon},v_{1+\epsilon}),(-1)^{\epsilon}b),\epsilon+\delta)$
$\displaystyle=$
$\displaystyle\;(((u_{0},u_{1})\cdot\phi_{a}(v_{0},v_{1})),a+(-1)^{\epsilon}b),\epsilon+\delta)$
$\displaystyle=$
$\displaystyle\;((u_{0}\phi^{a}v_{0+\epsilon}\phi^{-a},u_{1}\phi^{a}v_{1+\epsilon}\phi^{-a}),a+(-1)^{\epsilon}\cdot
b),\epsilon+\delta).$

Geometrically, an element
$(((u_{0},u_{1}),a),\epsilon)\in(U\rtimes_{\phi}\mathbb{Z})\rtimes_{\psi}\mathbb{Z}/2\mathbb{Z}$
contains $u_{0},u_{1}$, the two level preserving isometries to perform on the
two trees that make up the Diestel-Leader graph, and an integer, $a$, which
tells us how many times we should perform $\phi$ on the first tree and how
many times we should perform $\phi^{-1}$ on the second tree, and finally
$\epsilon$ tells us whether or not we should flip the two trees. When we
multiply elements and get the conjugation action on the isometries
$(u_{0},u_{1})$ this moves the isometries up or down the number of times we
perform $\phi$.

######  Definition 5.

Let ${\rm Isom}({\rm DL}(m,n))$ be the isometry group of ${\rm DL}(m,n)$. An
isometry $f\in{\rm Isom}({\rm DL}(n,n))$ is _positive_ if it is the
restriction to ${\rm DL}(n,n)$ of some product isometry $(\alpha,\beta)\in{\rm
Aut}(T_{n}\times T_{n})$ of $T_{n}\times T_{n}$. A non positive isometry is
the composition of a positive isometry and a flip $(x,y)\mapsto(y,x)$. Note
that this can only happen when $m=n$, otherwise we could not perform a flip
since the two trees making up the Diestel-Leader graph will have different
valences.

We let $G={\rm Isom}^{+}({\rm DL}(n,n))=U\rtimes_{\phi}\mathbb{Z}$ be the
index two subgroup of ${\rm Isom}({\rm DL}(n,n))$. Let $\pi\colon
G\to\mathbb{Z}$ be the projection onto $\mathbb{Z}$ so that $U=\ker\pi$.

The following is corollary 2.3 from [1].

######  Lemma 6.

If $\Gamma\leq{\rm Isom}(DL(n,n))$ is a closed cocompact subgroup,then there
is an element $t\in\Gamma$ such that $\Gamma=(\Gamma\cap
U)\rtimes\left<t\right>.$

Finally we end this statement with the main structure theorem from [1].

######  Theorem 7.

Suppose $n\geq 2$.

  1. 1.

Let $\Gamma$ be a closed, cocompact subgroup of ${\rm Isom}^{+}({\rm
DL}(n,n))$. Then $\Gamma$ has a unique open normal subgroup $H$ such that
$\Gamma/H$ is infinite cyclic. Moreover, if $t$ is any element of $\Gamma$
mapping to a generator of $\Gamma/H$, then $H$ has two open subgroups
$L,L^{\prime}$ such that:

     * •

$tLt^{-1}$ and $t^{-1}L^{\prime}t$ are subgroups of index $n$ in $L$ and
$L^{\prime}$ respectively.

     * •

$\bigcup_{k\in\mathbb{Z}}t^{-k}Kt^{k}=\bigcup_{k\in\mathbb{Z}}t^{k}L^{\prime}t^{-k}=H$,
the unions are increasing.

     * •

$L\cap L^{\prime}$ is a vertex stabilizer and thus is compact.

     * •

$LL^{\prime}=H$.

Moreover, $\Gamma$ has no nontriial compact normal subgroup and $\Gamma$ is
discrete. If the action of $\Gamma$ is simply transitive, then $L\cap
L^{\prime}=\left\\{1\right\\}$ and $LL^{\prime}=H$.

  2. 2.

Conversely, let $\Gamma$ be a locally compact group, with a semidirect product
decomposition $\Gamma=H\rtimes\left<t\right>$, with $H$ noncompact. Assume
that $H$ has open subgroups $L,L^{\prime}$ such that:

     * •

$tLt^{-1}$ and $t^{-1}Lt$ are finite index subgroups, of index $n$, in $L$ and
$L^{\prime}$ respectively.

     * •

$\bigcup_{k\in\mathbb{Z}}t^{-k}Lt^{k}=\bigcup_{k\in\mathbb{Z}}t^{k}L^{\prime}t^{-k}=H$
(the unions are increasing).

     * •

$L\cap L^{\prime}$ is compact.

     * •

the double coset space $L\backslash H/L^{\prime}$ is finite of cardinality
$d$.

Then $H$ is locally elliptic (i.e., every compact subset of $H$ is containd in
a compact subgroup), and $\Gamma$ has a proper, transitive action on ${\rm
DL}(m,n)$, for which $L\cap L^{\prime}$ is a vertex stabilizer, and whose
kernel is $\bigcap_{k\in\mathbb{Z}}t^{-k}(L\cap L^{\prime})t^{k}$. Moreover,
if $\Gamma$ is discrete, then it is finitely generated.

######  Remark.

Given a cross-wired lampligher $\Gamma$ contained in ${\rm Isom}({\rm
DL}(n,n))$, and not contained in ${\rm Isom}^{+}({\rm DL}(n,n))$, we can take
the index two subgroup of $\Gamma$, call it $\Gamma^{\prime}$, that is
contained in ${\rm Isom}^{+}({\rm DL}(n,n))$, and by Theorem 7, there is a
unique subgroup $H$ such that $\Gamma^{\prime}/H\cong\mathbb{Z}$. Thus
$\Gamma/H\cong D_{\infty}$.

##  3 Proof of the Main Theorem

Now, we look at the action in the group
$U\rtimes_{\phi}(\mathbb{Z}\rtimes_{\psi}\mathbb{Z}/2/\mathbb{Z})$. We first
see that the action of $\mathbb{Z}/2\mathbb{Z}$ on $\mathbb{Z}$ is defined in
the only possible way, by sending $\epsilon$ to the automorphism
$(-1)^{\epsilon}\cdot n$ for $n\in\mathbb{Z}$.

We define the action of
$U\rtimes_{\phi}(\mathbb{Z}\rtimes_{\psi}\mathbb{Z}/2\mathbb{Z})$ by the
homomorphism, $\phi\colon\mathbb{Z}\rtimes_{\phi}\mathbb{Z}/2\mathbb{Z}\to{\rm
Aut}(U)$ given by
$\phi(a,\epsilon).=\phi_{(a,\epsilon)}(u_{0}.u_{1})=(\phi^{a}u_{0+\epsilon}\phi^{-a},\phi^{-a}u_{1+\epsilon}\phi^{a})$
for $(a,\epsilon)\in\mathbb{Z}\rtimes_{\phi}\mathbb{Z}/2\mathbb{Z}$. We first
check that it is a homomorphism:

$\displaystyle\phi(a,\epsilon)\phi(b,\delta)(u_{0},u_{1})=$
$\displaystyle\;\phi(a,\epsilon)(\phi^{b}u_{0+\delta}\phi^{-b},\phi^{-b}u_{1+\delta}\phi^{b})$
$\displaystyle=$
$\displaystyle\;(\phi^{a}\phi^{(-1)^{\epsilon}b}u_{0+\epsilon+\delta}\phi^{(-1)^{\epsilon+1}b}\phi^{-a},\phi^{-a}\phi^{(-1)^{\epsilon+1}b}u_{1+\epsilon+\delta}\phi^{(-1)^{\epsilon}b}\phi^{a})$
$\displaystyle=$
$\displaystyle\;(\phi^{1+(-1)^{\epsilon}b}u_{0+\epsilon+\delta}\phi^{-a-(-1)^{\epsilon}b},\phi^{-a-(-1)^{\epsilon}b}u_{1+\epsilon+\delta}\phi^{1+(-1)^{\epsilon}b})$
$\displaystyle=$
$\displaystyle\;\phi(a+(-1)^{\epsilon}b,\epsilon+\delta)=\phi((a,\epsilon)(b,\delta))$

Next, we check that it is an automorphism. Let $(u_{0},u_{1}),(v_{0},v_{1})\in
U$ and $(a,\epsilon)\in\mathbb{Z}\rtimes_{\phi}\mathbb{Z}/2\mathbb{Z}$. Then

$\displaystyle\phi_{(a,\epsilon)}(u_{0},u_{1})\phi_{(a,\epsilon)}(v_{0},v_{1})=$
$\displaystyle\;(\phi^{a}u_{0+\epsilon}\phi^{-a},\phi^{-a}u_{1+\epsilon}\phi^{a})(\phi^{a}v_{0+\epsilon}\phi^{-a},\phi^{-a}v_{1+\epsilon}\phi^{a})$
$\displaystyle=$
$\displaystyle\;(\phi^{a}u_{0+\epsilon}v_{0+\epsilon}\phi^{-a},\phi^{-a}u_{1+\epsilon}v_{1+\epsilon}\phi^{a})$

and

$\phi_{(a,\epsilon)}(u_{0}v_{0},u_{1}v_{1})=(\phi^{a}u_{0+\epsilon}v_{0+\epsilon}\phi^{-a},\phi^{-a}u_{1+\epsilon}v_{1+\epsilon}\phi^{a})$

It is also clear that this map is bijective,and thus an isomorphism.

Then looking at the multiplication in the group we have

$((u_{0},u_{1}),(a,\epsilon))\cdot((v_{0},v_{1}),(b,\delta))=((u_{0}v_{0+\epsilon}\phi^{a},u_{1}v_{1+\epsilon}\phi^{-a}),(a+(-1)^{\epsilon}\cdot
b,\epsilon+\delta)).$

Since the action gives us the same triple of elements after multiplication, we
can conclude they are isomorphic by sending $(((u_{0},u_{1}),a),\epsilon)$ and
$((u_{0},u_{1}),(a,\epsilon))$ to $((u_{0},u_{1}),a,\epsilon)$.

Combining all of the above we get the following lemma.

######  Lemma 8.

$(U\rtimes_{\phi}\mathbb{Z})\rtimes_{\psi}\mathbb{Z}/2\mathbb{Z}\cong
U\rtimes_{\phi}(\mathbb{Z}\rtimes_{\psi}\mathbb{Z}/2\mathbb{Z})$.

Now we turn to proving the main theorem we are interested in by first stating
a lemma that is an extension of corollary 2.3 from [1].

######  Lemma 9.

Let $\Gamma\leq{\rm Isom}({\rm DL}(n,n))$ be a closed, cocompact subgroup that
is not contained in ${\rm Isom}^{+}({\rm DL}(n,n))$. Then there is an element
$t\in\Gamma$ such that $\Gamma=((\Gamma\cap U)\rtimes\langle
t\rangle)\rtimes_{\psi}\mathbb{Z}/2\mathbb{Z}$.

###### Proof.

First take $\Gamma^{\prime}=\Gamma\cap{\rm Isom}^{+}({\rm DL}(n,n))$, then by
corollary 2.3 from [1], there is an element $t\in\Gamma^{\prime}$ such that
$\Gamma^{\prime}=(\Gamma^{\prime}\cap U)\rtimes\left<t\right>$. Finally, since
${\rm Isom}^{+}({\rm DL}(n,n))$ is an index 2 subgroup in ${\rm Isom}({\rm
DL}(n,n))$, we see that $\Gamma^{\prime}$ is an index 2 subgroup in $\Gamma$,
and thus
$\Gamma=\Gamma^{\prime}\rtimes\mathbb{Z}/2\mathbb{Z}=(\Gamma^{\prime}\cap
U)\rtimes\left<t\right>$. But $\Gamma^{\prime}\cap U=\Gamma\cap U$, giving us
the result. ∎

Following the proof of the theorem 1.1 [1], we let $H=\Gamma\cap U$. Then
$\Gamma/H\cong D_{\infty}$. For clarity, we will write $\Gamma$ as
$(H\rtimes\mathbb{Z})\rtimes\mathbb{Z}/2\mathbb{Z}$. Now, we can rewrite our
initial short exact sequence as

$1\to
H\to(H\rtimes\mathbb{Z})\rtimes\mathbb{Z}/2\mathbb{Z}\to\mathbb{Z}\rtimes\mathbb{Z}/2\mathbb{Z}\to
1.$

However, by lemma 8 we get that $((\Gamma\cap
U)\rtimes\mathbb{Z})\rtimes\mathbb{Z}/2\mathbb{Z}\cong(\Gamma\cap
U)\rtimes(\mathbb{Z}\rtimes\mathbb{Z}/2\mathbb{Z})$. Lemma 8 holds for
subgroups of $U$ since we can just map both $((\Gamma\cap
U)\rtimes\mathbb{Z})\rtimes\mathbb{Z}/2\mathbb{Z}$ and $(\Gamma\cap
U)\rtimes(\mathbb{Z}\rtimes\mathbb{Z}/2\mathbb{Z})$ to $(\Gamma\cap
U)\rtimes_{\phi}\mathbb{Z}\rtimes_{\psi}\mathbb{Z}/2\mathbb{Z}$ the same way
we did above. So, replacing $H=\Gamma\cap U$ we get

$1\to H\to
H\rtimes(\mathbb{Z}\rtimes\mathbb{Z}/2\mathbb{Z})\to\mathbb{Z}\rtimes\mathbb{Z}/2\mathbb{Z}\to
1.$

which by lemma 1, tells us that our exact sequence splits. Thus our extension
of interest splits and we get $\Gamma\cong H\rtimes D_{\infty}$.

##  4 Symmetry

We make a few remarks about the symmetry question in [1].

######  Definition 10.

Let $\Gamma$ be a cross-wired lamplighter group. We say $\Gamma$ is symmetric
if $\Gamma$ admits an automorphism $\alpha$ such that $\pi\circ\alpha=-\pi$
where $\pi\colon\Gamma=H\rtimes_{\phi}\mathbb{Z}\to\mathbb{Z}$ is the
projection onto the second coordinate. We say that $\Gamma$ is _virtually
symmetric_ if $\Gamma$ has a symmetric subgroup of finite index.

In [1], Cournulier, Fisher and Kashyap ask whether or not there exists a non
virtually-symmetric cross-wired lamplighter.

We provide an example showing that the cross-wired lamplighter group
$\mathbb{H}(\mathbb{F}_{q}[X,X^{-1}])\rtimes_{\phi}\mathbb{Z}$ is symmetric
where $\mathbb{H}(\cdot)$ is the Heisenberg group of upper triangular matrices
over a ring.

######  Proposition 11.

The group $\mathbb{H}(\mathbb{F}_{q}[X,X^{-1}])\rtimes_{\phi}\mathbb{Z}$ is
symmetric.

###### Proof.

Let $\\\ sigma\colon\Gamma\to\Gamma$ be given by

$\sigma\left(\begin{pmatrix}1&f(X,X^{-1})&h(X,X^{-1})\\\ 0&1&g(X,X^{-1})\\\
0&0&1\end{pmatrix},a\right)=\left(\begin{pmatrix}1&f(X^{-1},X)&h(X^{-1},X)\\\
0&1&g(X^{-1},X)\\\ 0&0&1\end{pmatrix},-a\right).$

So $\sigma$ restricts to an automorphism of $\mathbb{F}_{q}[X,X^{-1}]$ that
sends $X$ to $X^{-1}$. It is clear the $\sigma$ is bijective. We check that it
is a homomorphism:

$\displaystyle\sigma\left(\begin{pmatrix}1&f(X,X^{-1})&h(X,X^{-1})\\\
0&1&g(X,X^{-1})\\\
0&0&1\end{pmatrix},a\right)\sigma\left(\begin{pmatrix}1&i(X,X^{-1})&k(X,X^{-1})\\\
0&1&j(X,X^{-1})\\\ 0&0&1\end{pmatrix},b\right)$ $\displaystyle=$
$\displaystyle\;\left(\begin{pmatrix}1&f(X^{-1},X)&h(X^{-1},X)\\\
0&1&g(X^{-1},X)\\\
0&0&1\end{pmatrix},-a\right)\left(\begin{pmatrix}1&i(X^{-1},X)&k(X^{-1},X)\\\
0&1&j(X^{-1},X)\\\ 0&0&1\end{pmatrix},-b\right)$ $\displaystyle=$
$\displaystyle\;\left(\begin{pmatrix}1&f(X^{-1}X)+X^{-a}i(X^{-1},X)&X^{-2a}h(X^{-1},X)+X^{-a}f(X^{-1},X)j(X^{-1},X)+h(X^{-1},X)\\\
0&1&g(X^{-1},X)+X^{-a}j(X^{-1},X)\\\ 0&0&1\end{pmatrix},-a-b\right)$

and

$\displaystyle\sigma\left(\left(\begin{pmatrix}1&f(X,X^{-1})&h(X,X^{-1})\\\
0&1&g(X,X^{-1})\\\
0&0&1\end{pmatrix},a\right)\left(\begin{pmatrix}1&i(X,X^{-1})&k(X,X^{-1})\\\
0&1&j(X,X^{-1})\\\ 0&0&1\end{pmatrix},b\right)\right)$ $\displaystyle=$
$\displaystyle\sigma\left(\begin{pmatrix}1&f(X,X^{-1})+X^{a}i(X,X^{-1})&x^{2a}k(X,X^{-1})+X^{a}f(X,X^{-1})j(X,X^{-1})+h(X,X^{-1})\\\
0&1&g(X,X^{-1})+X^{a}j(X,X^{-1})\\\ 0&0&1\end{pmatrix},a+b\right)$
$\displaystyle=$
$\displaystyle\;\left(\begin{pmatrix}1&f(X^{-1}X)+X^{-a}i(X^{-1},X)&X^{-2a}h(X^{-1},X)+X^{-a}f(X^{-1},X)j(X^{-1},X)+h(X^{-1},X)\\\
0&1&g(X^{-1},X)+X^{-a}j(X^{-1},X)\\\ 0&0&1\end{pmatrix}\right)$

Thus $\sigma$ is an automorphism and $\Gamma$ is symmetric. ∎

We end by stating a few open questions about cross-wired lamplighters.

  1. 1.

Is there a cross-wired lamplighter that is not virtually symmetrc?

  2. 2.

Given a cross-wired lamplighter, is the pair $\left\\{L,L^{\prime}\right\\}$
unique up to commensurability and automorphisms? Suppose that there are two
embeddings of $\Gamma$ as a cross-wired lamplighter group
$(L_{1},L_{1}^{\prime})$ and $(L_{2},L_{2}^{\prime})$. Is there an
automorphism $\beta$ of $\Gamma$ such that $\beta(L_{1})$ is commensurable to
$L_{1}^{\prime}$ and $\beta(L_{2})$ is commensurable to $L_{2}^{\prime}$. This
is question 1.4(ii) from [1].

  3. 3.

Is there a non-linear cross-wired lamplighter? All of the example of cross-
wired lamplighters from [1], and [3] are linear, so finding a non-linear one
or proving that they are all linear will give great insight into their
structure.

## References

  * [1] Y. De Cornulier, F. David, and N. Kashyap, “Cross-wired lamplighter groups,” New York J. Math., vol. 18, pp. 667–677, 2012.
  * [2] P. Silva and B. Steinberg, “On a class of automata groups generalizing lamplighter groups,” International Journal of Algebra and Computation, vol. 15, pp. 1213–1234, 2005.
  * [3] N. Yang, “Cross-wired lamplighter groups and linearity of automata groups,” New York J. Math., vol. 22, pp. 379–404, 2016.
  * [4] R. Diestel and I. Leader, “A conjecture concerning a limit of non-cayley graphs,” J. Algebraic Comb., vol. 14, p. 17–25, jul 2001.
  * [5] A. Eskin, D. Fisher, and K. Whyte, “Quasi-isometries and rigidity of solvable groups,” Pure Appl. Math. Q., vol. 3, pp. 927–947, 2007.
  * [6] A. Eskin, D. Fisher, and K. Whyte, “Coarse differentiation of quasi isometries. i: Spaces not quasi-isometric to cayley graphs,” Ann. Math., vol. 176-1, pp. 221–260, 2012.
  * [7] W. Woess, “Diestel-leader graphs, random walks, and harmonic functions.,” Combin. Probab. Comput., vol. 14, pp. 415–433, 2005.
  * [8] M. W. W. Bartholdi, Laurent; Neuhauser, “Horocyclic products of trees,” J. Eur. Math.Soc. (JEMS), vol. 10, pp. 771–816, 1008.


# Some three-dimensional non-nilpotent Leibniz algebras: automorphism groups

L. A. Kurdachenko, O. O. Pypka
Oles Honchar Dnipro National University, Dnipro, Ukraine
e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
I. Ya. Subbotin
National University, Los Angeles, USA
e-mail<EMAIL_ADDRESS>
###### Abstract

The article presents the structure of the automorphism groups of two types of
non-nilpotent Leibniz algebras with a dimension of 3.

Key Words: Leibniz algebra, automorphism group.

2020 MSC: 17A32, 17A36.

##  1 Introduction.

A Leibniz algebra is defined as a vector space $L$ over a field $F$ with a
bilinear binary operation $[,]$ that satisfies the following condition:

$[x,[y,z]]=[[x,y],z]+[y,[x,z]]$

for all $x,y,z\in L$ [2]. The term “Leibniz algebra” first appeared in [14]
and [15]. We will not provide all the definitions related to Leibniz algebras
here. They can be found, for example, in the monographs [1] or [10].

Despite the many analogies between Leibniz algebras and Lie algebras, these
types of algebras differ significantly from each other (see, for example,
survey articles [3, 4, 12, 17]).

A bijective linear transformation $f$ of a Leibniz algebra $L$ is called an
automorphism of the algebra $L$ if it satisfies the following condition:

$f([x,y])=[f(x),f(y)]$

for all $x,y\in L$. It is easy to prove that the set $Aut_{[,]}(L)$ of all
automorphisms of $L$ is a group by a multiplication (see, for example, [9]).

Describing the automorphism groups of algebraic structures is a fundamental
and essential problem in algebra. Leibniz algebras are no exception. It is
prudent to begin by exploring the automorphism groups of those Leibniz
algebras whose structures are already well understood.

As of today, the structure of the automorphism groups of one-generated Leibniz
algebras [9, 13], two-dimensional Leibniz algebras [11], and some three-
dimensional Leibniz algebras [5, 6, 7, 11, 16] is known.

This article extends our research to two other types of three-dimensional non-
nilpotent Leibniz algebras. It should be noted that the most comprehensive
description of three-dimensional Leibniz algebras over arbitrary fields can be
found in [8].

##  2 The automorphism group of $L_{1}$.

Let us consider the following type of non-nilpotent Leibniz algebras:

$\displaystyle L_{1}=Fe_{1}\oplus Fe_{2}\oplus Fe_{3}\ \mbox{where
}[e_{1},e_{1}]=[e_{1},e_{3}]=e_{3},\ [e_{1},e_{2}]=e_{2},$
$\displaystyle[e_{2},e_{1}]=[e_{2},e_{2}]=[e_{2},e_{3}]=[e_{3},e_{1}]=[e_{3},e_{2}]=[e_{3},e_{3}]=0.$

In other words, $L_{1}$ is the sum of the ideal $A_{2}=Fe_{2}$ and the non-
nilpotent one-generated Leibniz algebra $A_{1}=Fe_{1}\oplus Fe_{3}$, which has
dimension 2, $[A_{1},A_{2}]=A_{2}$, $Leib(L_{1})=Fe_{2}\oplus
Fe_{3}=[L_{1},L_{1}]=\zeta^{\mathrm{left}}(L_{1})$,
$\zeta^{\mathrm{right}}(L_{1})=\zeta(L_{1})=\langle 0\rangle$.

The following theorem illustrates the structure of the automorphism group for
these Leibniz algebras.

Denote by $\Xi$ the canonical monomorphism of $Aut_{[,]}(L)$ in $M_{3}(F)$.

######  Theorem 2.1.

Let $G$ be an automorphism group of the Leibniz algebra $L_{1}$. Then $G$ is
isomorphic to a subgroup of $GL_{3}(F)$ that consists of matrices of the form:

$\left(\begin{array}[]{ccc}1&0&0\\\ \alpha_{2}&\beta_{2}&\alpha_{2}\\\
\alpha_{3}&\beta_{3}&1+\alpha_{3}\end{array}\right),$

$\alpha_{2},\alpha_{3},\beta_{2},\beta_{3}\in F$. Furthermore, $G$ is
isomorphic to the group $GL_{2}(F)$. Moreover, it is a product of two
subgroups $C_{1}=C_{G}(e_{1})$, $C_{2}=C_{G}(e_{2})$, and $\Xi(C_{1})$ is the
set of matrices of the form

$\left(\begin{array}[]{ccc}1&0&0\\\ 0&\beta_{2}&0\\\
0&\beta_{3}&1\end{array}\right),$

$\beta_{2},\beta_{3}\in F$, and $\Xi(C_{2})$ is the set of matrices of the
form

$\left(\begin{array}[]{ccc}1&0&0\\\ \alpha_{2}&1&\alpha_{2}\\\
\alpha_{3}&0&1+\alpha_{3}\end{array}\right),$

$\alpha_{2},\alpha_{3}\in F$. A subgroup $C_{1}$ is a semidirect product of a
normal subgroup $C_{3}$, which is isomorphic to the additive group of a field
$F$, and a subgroup $C_{4}$, which is isomorphic to the multiplicative group
of a field $F$; a subgroup $C_{2}$ is a semidirect product of a normal
subgroup, which is isomorphic to the additive group of a field $F$, and a
subgroup that is isomorphic to the multiplicative group of a field $F$.

Proof. Let $L=L_{1}$ and let $f\in Aut_{[,]}(L)$. By [9, Lemma 2.1],
$f([L,L])=[L,L]$, so that

$\displaystyle f(e_{1})$
$\displaystyle=\alpha_{1}e_{1}+\alpha_{2}e_{2}+\alpha_{3}e_{3},$
$\displaystyle f(e_{2})$ $\displaystyle=\beta_{2}e_{2}+\beta_{3}e_{3},$
$\displaystyle f(e_{3})$ $\displaystyle=\gamma_{2}e_{2}+\gamma_{3}e_{3}.$

Then

$\displaystyle f(e_{3})$ $\displaystyle=f([e_{1},e_{1}])=[f(e_{1}),f(e_{1})]$
$\displaystyle=[\alpha_{1}e_{1}+\alpha_{2}e_{2}+\alpha_{3}e_{3},\alpha_{1}e_{1}+\alpha_{2}e_{2}+\alpha_{3}e_{3}]$
$\displaystyle=\alpha_{1}^{2}[e_{1},e_{1}]+\alpha_{1}\alpha_{2}[e_{1},e_{2}]+\alpha_{1}\alpha_{3}[e_{1},e_{3}]$
$\displaystyle=\alpha_{1}^{2}e_{3}+\alpha_{1}\alpha_{2}e_{2}+\alpha_{1}\alpha_{3}e_{3}$
$\displaystyle=\alpha_{1}\alpha_{2}e_{2}+(\alpha_{1}^{2}+\alpha_{1}\alpha_{3})e_{3},$
$\displaystyle f(e_{3})$ $\displaystyle=f([e_{1},e_{3}])=[f(e_{1}),f(e_{3})]$
$\displaystyle=[\alpha_{1}e_{1}+\alpha_{2}e_{2}+\alpha_{3}e_{3},\gamma_{2}e_{2}+\gamma_{3}e_{3}]$
$\displaystyle=\alpha_{1}\gamma_{2}[e_{1},e_{2}]+\alpha_{1}\gamma_{3}[e_{1},e_{3}]$
$\displaystyle=\alpha_{1}\gamma_{2}e_{2}+\alpha_{1}\gamma_{3}e_{3},$
$\displaystyle f(e_{2})$ $\displaystyle=f([e_{1},e_{2}])=[f(e_{1}),f(e_{2})]$
$\displaystyle=[\alpha_{1}e_{1}+\alpha_{2}e_{2}+\alpha_{3}e_{3},\beta_{2}e_{2}+\beta_{3}e_{3}]$
$\displaystyle=\alpha_{1}\beta_{2}[e_{1},e_{2}]+\alpha_{1}\beta_{3}[e_{1},e_{3}]$
$\displaystyle=\alpha_{1}\beta_{2}e_{2}+\alpha_{1}\beta_{3}e_{3}.$

Thus, we can see that

$\displaystyle\alpha_{1}\alpha_{2}e_{2}+(\alpha_{1}^{2}+\alpha_{1}\alpha_{3})e_{3}=\alpha_{1}\gamma_{2}e_{2}+\alpha_{1}\gamma_{3}e_{3}=\gamma_{2}e_{2}+\gamma_{3}e_{3},$
$\displaystyle\alpha_{1}\beta_{2}e_{2}+\alpha_{1}\beta_{3}e_{3}=\beta_{2}e_{2}+\beta_{3}e_{3}.$

It follows that

$\alpha_{1}\alpha_{2}=\alpha_{1}\gamma_{2}=\gamma_{2},\
\alpha_{1}^{2}+\alpha_{1}\alpha_{3}=\alpha_{1}\gamma_{3}=\gamma_{3},\
\alpha_{1}\beta_{2}=\beta_{2},\ \alpha_{1}\beta_{3}=\beta_{3}.$

If we suppose that $\alpha_{1}=0$, then $f(e_{1})\in Fe_{2}\oplus Fe_{3}$.
Since $f(e_{2}),f(e_{3})\in Fe_{2}\oplus Fe_{3}$, we obtain that $f(L)\leq
Fe_{2}\oplus Fe_{3}$ – in particular, $f(L)\neq L$ – and we obtain a
contradiction with the fact that $f$ is an automorphism of $L$. Hence,
$\alpha_{1}\neq 0$. It follows that $\alpha_{2}=\gamma_{2}$.

If we suppose that both $\beta_{2}=\gamma_{2}=0$, then $f(e_{2})\in Fe_{3}$
and $f(e_{3})\in Fe_{3}$, so that $f([L,L])\leq Fe_{3}$. On the other hand,
$dim_{F}([L,L])=2$, and we, again, obtain a contradiction with the fact that
$f$ is an automorphism of $L$. Hence, $(\beta_{2},\gamma_{2})\neq(0,0)$. It
follows that $\alpha_{1}=1$ and then $1+\alpha_{3}=\gamma_{3}$.

Denote by $\Xi$ the canonical monomorphism of $Aut_{[,]}(L)$ in $M_{3}(F)$.
Then $\Xi(f)$ is the following matrix:

$\left(\begin{array}[]{ccc}1&0&0\\\ \alpha_{2}&\beta_{2}&\alpha_{2}\\\
\alpha_{3}&\beta_{3}&1+\alpha_{3}\end{array}\right),$

$\alpha_{2},\alpha_{3},\beta_{2},\beta_{3}\in F$.

Let $f$ be a linear transformation of $L$, having in a basis
$\\{e_{1},e_{2},e_{3}\\}$ the matrix above. Let $x,y$ be the arbitrary
elements of $L$, $x=\xi_{1}e_{1}+\xi_{2}e_{2}+\xi_{3}e_{3}$,
$y=\eta_{1}e_{1}+\eta_{2}e_{2}+\eta_{3}e_{3}$, where
$\xi_{1},\xi_{2},\xi_{3},\eta_{1},\eta_{2},\eta_{3}\in F$. Then

$\displaystyle[x,y]$
$\displaystyle=[\xi_{1}e_{1}+\xi_{2}e_{2}+\xi_{3}e_{3},\eta_{1}e_{1}+\eta_{2}e_{2}+\eta_{3}e_{3}]$
$\displaystyle=\xi_{1}\eta_{1}[e_{1},e_{1}]+\xi_{1}\eta_{2}[e_{1},e_{2}]+\xi_{1}\eta_{3}[e_{1},e_{3}]$
$\displaystyle=\xi_{1}\eta_{1}e_{3}+\xi_{1}\eta_{2}e_{2}+\xi_{1}\eta_{3}e_{3}$
$\displaystyle=\xi_{1}\eta_{2}e_{2}+(\xi_{1}\eta_{1}+\xi_{1}\eta_{3})e_{3}=\xi_{1}\eta_{2}e_{2}+\xi_{1}(\eta_{1}+\eta_{3})e_{3},$
$\displaystyle f(x)$
$\displaystyle=f(\xi_{1}e_{1}+\xi_{2}e_{2}+\xi_{3}e_{3})=\xi_{1}f(e_{1})+\xi_{2}f(e_{2})+\xi_{3}f(e_{3})$
$\displaystyle=\xi_{1}(e_{1}+\alpha_{2}e_{2}+\alpha_{3}e_{3})+\xi_{2}(\beta_{2}e_{2}+\beta_{3}e_{3})+\xi_{3}(\alpha_{2}e_{2}+(1+\alpha_{3})e_{3})$
$\displaystyle=\xi_{1}e_{1}+(\xi_{1}\alpha_{2}+\xi_{2}\beta_{2}+\xi_{3}\alpha_{2})e_{2}+(\xi_{1}\alpha_{3}+\xi_{2}\beta_{3}+\xi_{3}+\xi_{3}\alpha_{3})e_{3},$
$\displaystyle f(y)$
$\displaystyle=\eta_{1}e_{1}+(\eta_{1}\alpha_{2}+\eta_{2}\beta_{2}+\eta_{3}\alpha_{2})e_{2}+(\eta_{1}\alpha_{3}+\eta_{2}\beta_{3}+\eta_{3}+\eta_{3}\alpha_{3})e_{3},$
$\displaystyle f([x,y])$
$\displaystyle=f(\xi_{1}\eta_{2}e_{2}+\xi_{1}(\eta_{1}+\eta_{3})e_{3})=\xi_{1}\eta_{2}f(e_{2})+\xi_{1}(\eta_{1}+\eta_{3})f(e_{3})$
$\displaystyle=\xi_{1}\eta_{2}(\beta_{2}e_{2}+\beta_{3}e_{3})+\xi_{1}(\eta_{1}+\eta_{3})(\alpha_{2}e_{2}+(1+\alpha_{3})e_{3})$
$\displaystyle=(\xi_{1}\eta_{2}\beta_{2}+\xi_{1}(\eta_{1}+\eta_{3})\alpha_{2})e_{2}+(\xi_{1}\eta_{2}\beta_{3}+\xi_{1}(\eta_{1}+\eta_{3})(1+\alpha_{3}))e_{3},$
$\displaystyle[f(x),f(y)]$
$\displaystyle=[\xi_{1}e_{1}+(\xi_{1}\alpha_{2}+\xi_{2}\beta_{2}+\xi_{3}\alpha_{2})e_{2}+(\xi_{1}\alpha_{3}+\xi_{2}\beta_{3}+\xi_{3}+\xi_{3}\alpha_{3})e_{3},$
$\displaystyle\ \ \ \
\eta_{1}e_{1}+(\eta_{1}\alpha_{2}+\eta_{2}\beta_{2}+\eta_{3}\alpha_{2})e_{2}+(\eta_{1}\alpha_{3}+\eta_{2}\beta_{3}+\eta_{3}+\eta_{3}\alpha_{3})e_{3}]$
$\displaystyle=\xi_{1}\eta_{1}[e_{1},e_{1}]+\xi_{1}(\eta_{1}\alpha_{2}+\eta_{2}\beta_{2}+\eta_{3}\alpha_{2})[e_{1},e_{2}]$
$\displaystyle+\xi_{1}(\eta_{1}\alpha_{3}+\eta_{2}\beta_{3}+\eta_{3}+\eta_{3}\alpha_{3})[e_{1},e_{3}]$
$\displaystyle=\xi_{1}\eta_{1}e_{3}+\xi_{1}(\eta_{1}\alpha_{2}+\eta_{2}\beta_{2}+\eta_{3}\alpha_{2})e_{2}+\xi_{1}(\eta_{1}\alpha_{3}+\eta_{2}\beta_{3}+\eta_{3}+\eta_{3}\alpha_{3})e_{3}$
$\displaystyle=\xi_{1}(\eta_{1}\alpha_{2}+\eta_{2}\beta_{2}+\eta_{3}\alpha_{2})e_{2}+\xi_{1}(\eta_{1}+\eta_{1}\alpha_{3}+\eta_{2}\beta_{3}+\eta_{3}+\eta_{3}\alpha_{3})e_{3}.$

We can see that $f([x,y])=[f(x),f(y)]$. It follows that a linear
transformation $f$ having in a basis $\\{e_{1},e_{2},e_{3}\\}$ the above
matrix is an automorphism of a Leibniz algebra.

Denote by $\Phi$ the mapping of $\Xi(G)$ in $GL_{2}(F)$ defined by the rule:

$\left(\begin{array}[]{ccc}1&0&0\\\ \alpha_{2}&\beta_{2}&\alpha_{2}\\\
\alpha_{3}&\beta_{3}&1+\alpha_{3}\end{array}\right)\rightarrow\left(\begin{array}[]{cc}\beta_{2}&\alpha_{2}\\\
\beta_{3}&1+\alpha_{3}\end{array}\right).$

We have:

$\displaystyle\left(\begin{array}[]{ccc}1&0&0\\\
\alpha_{2}&\beta_{2}&\alpha_{2}\\\
\alpha_{3}&\beta_{3}&1+\alpha_{3}\end{array}\right)\left(\begin{array}[]{ccc}1&0&0\\\
\gamma_{2}&\kappa_{2}&\gamma_{2}\\\
\gamma_{3}&\kappa_{3}&1+\gamma_{3}\end{array}\right)=$

$\left(\begin{array}[]{ccc}1&0&0\\\
\alpha_{2}+\beta_{2}\gamma_{2}+\alpha_{2}\gamma_{3}&\beta_{2}\kappa_{2}+\alpha_{2}\kappa_{3}&\beta_{2}\gamma_{2}+\alpha_{2}(1+\gamma_{3})\\\
\alpha_{3}+\beta_{3}\gamma_{2}+(1+\alpha_{3})\gamma_{3}&\beta_{3}\kappa_{2}+(1+\alpha_{3})\kappa_{3}&\beta_{3}\gamma_{2}+(1+\alpha_{3})(1+\gamma_{3})\end{array}\right)$

and

$\displaystyle\left(\begin{array}[]{cc}\beta_{2}&\alpha_{2}\\\
\beta_{3}&1+\alpha_{3}\end{array}\right)\left(\begin{array}[]{cc}\kappa_{2}&\gamma_{2}\\\
\kappa_{3}&1+\gamma_{3}\end{array}\right)$
$\displaystyle=\left(\begin{array}[]{cc}\beta_{2}\kappa_{2}+\alpha_{2}\kappa_{3}&\beta_{2}\gamma_{2}+\alpha_{2}(1+\gamma_{3})\\\
\beta_{3}\kappa_{2}+(1+\alpha_{3})\kappa_{3}&\beta_{3}\gamma_{2}+(1+\alpha_{3})(1+\gamma_{3})\end{array}\right)$

These equalities show that a mapping $\Phi$ is a homomorphism. Clearly,
$Ker(\Phi)$ consists only of the identity matrix. Finally, let

$\left(\begin{array}[]{cc}\sigma_{11}&\sigma_{12}\\\
\sigma_{21}&\sigma_{22}\end{array}\right)$

be the arbitrary element of $GL_{2}(F)$. Consider the matrix

$\left(\begin{array}[]{ccc}1&0&0\\\ \sigma_{12}&\sigma_{11}&\sigma_{12}\\\
\sigma_{22}-1&\sigma_{21}&\sigma_{22}\end{array}\right).$

This matrix is a preimage of the matrix above by the mapping $\Phi$. Moreover,
both these matrices have the same determinants. It follows that the last
matrix belongs to $\Xi(G)$, so that a mapping $\Phi$ is an isomorphism.

Let $C_{1}=C_{G}(e_{1})$, $C_{2}=C_{G}(e_{2})$. Then $\Xi(C_{1})$ is the set
of matrices of the form

$\left(\begin{array}[]{ccc}1&0&0\\\ 0&\beta_{2}&0\\\
0&\beta_{3}&1\end{array}\right),$

$\beta_{2},\beta_{3}\in F$, and $\Xi(C_{2})$ is the set of matrices of the
form

$\left(\begin{array}[]{ccc}1&0&0\\\ \alpha_{2}&1&\alpha_{2}\\\
\alpha_{3}&0&1+\alpha_{3}\end{array}\right),$

$\alpha_{2},\alpha_{3}\in F$. We have:

$\displaystyle\left(\begin{array}[]{ccc}1&0&0\\\ \sigma_{2}&1&\sigma_{2}\\\
\sigma_{3}&0&1+\sigma_{3}\end{array}\right)\left(\begin{array}[]{ccc}1&0&0\\\
0&\kappa_{2}&0\\\
0&\kappa_{3}&1\end{array}\right)=\left(\begin{array}[]{ccc}1&0&0\\\
\sigma_{2}&\kappa_{2}+\sigma_{2}\kappa_{3}&\sigma_{2}\\\
\sigma_{3}&(1+\sigma_{3})\kappa_{3}&1+\sigma_{3}\end{array}\right).$

If

$\left(\begin{array}[]{ccc}1&0&0\\\ \alpha_{2}&\beta_{2}&\alpha_{2}\\\
\alpha_{3}&\beta_{3}&1+\alpha_{3}\end{array}\right)$

is an arbitrary matrix of $\Xi(G)$, then we obtain

$\displaystyle\left(\begin{array}[]{ccc}1&0&0\\\
\alpha_{2}&\beta_{2}&\alpha_{2}\\\
\alpha_{3}&\beta_{3}&1+\alpha_{3}\end{array}\right)=\left(\begin{array}[]{ccc}1&0&0\\\
\sigma_{2}&1&\sigma_{2}\\\
\sigma_{3}&0&1+\sigma_{3}\end{array}\right)\left(\begin{array}[]{ccc}1&0&0\\\
0&\kappa_{2}&0\\\ 0&\kappa_{3}&1\end{array}\right)$

where $\sigma_{2}=\alpha_{2}$, $\sigma_{3}=\alpha_{3}$,
$\kappa_{3}=\beta_{3}(1+\alpha_{3})^{-1}$,
$\kappa_{2}=\beta_{2}-\alpha_{2}\beta_{3}(1+\alpha_{3})^{-1}$. Thus, we can
see that a group $G$ is a product of the subgroups $C_{1}$ and $C_{2}$.

Furthermore, a subgroup $C_{1}$ has a normal subgroup $C_{3}$ such that
$\Xi(C_{3})$ is the subset of matrices of the form

$\left(\begin{array}[]{ccc}1&0&0\\\ 0&1&0\\\ 0&\beta_{3}&1\end{array}\right),$

which is isomorphic to the additive group of a field $F$, and a subgroup
$C_{4}$ such that $\Xi(C_{4})$ is the subset of matrices of the form

$\left(\begin{array}[]{ccc}1&0&0\\\ 0&\beta_{2}&0\\\ 0&0&1\end{array}\right),$

which is isomorphic to the multiplicative group of a field $F$. Moreover,
$C_{1}$ is clearly a semidirect product of $C_{3}$ and $C_{4}$.

Now, consider the group $\Xi(C_{2})$. It is the set of matrices of the form

$\left(\begin{array}[]{ccc}1&0&0\\\ \alpha_{2}&1&\alpha_{2}\\\
\alpha_{3}&0&1+\alpha_{3}\end{array}\right),$

$\alpha_{2},\alpha_{3}\in F$. Denote by $\Phi$ the mapping of $\Xi(C_{2})$ in
$GL_{2}(F)$ defined by the rule:

$\left(\begin{array}[]{ccc}1&0&0\\\ \alpha_{2}&1&\alpha_{2}\\\
\alpha_{3}&0&1+\alpha_{3}\end{array}\right)\rightarrow\left(\begin{array}[]{cc}1&\alpha_{2}\\\
0&1+\alpha_{3}\end{array}\right).$

We have:

$\displaystyle\left(\begin{array}[]{ccc}1&0&0\\\ \alpha_{2}&1&\alpha_{2}\\\
\alpha_{3}&0&1+\alpha_{3}\end{array}\right)\left(\begin{array}[]{ccc}1&0&0\\\
\gamma_{2}&1&\gamma_{2}\\\ \gamma_{3}&0&1+\gamma_{3}\end{array}\right)$
$\displaystyle=\left(\begin{array}[]{ccc}1&0&0\\\
\alpha_{2}+\gamma_{2}+\alpha_{2}\gamma_{3}&1&\gamma_{2}+\alpha_{2}(1+\gamma_{3})\\\
\alpha_{3}+(1+\alpha_{3})\gamma_{3}&0&(1+\alpha_{3})(1+\gamma_{3})\end{array}\right)$

and

$\displaystyle\left(\begin{array}[]{cc}1&\alpha_{2}\\\
0&1+\alpha_{3}\end{array}\right)\left(\begin{array}[]{cc}1&\gamma_{2}\\\
0&1+\gamma_{3}\end{array}\right)=\left(\begin{array}[]{cc}1&\gamma_{2}+\alpha_{2}(1+\gamma_{3})\\\
0&(1+\alpha_{3})(1+\gamma_{3})\end{array}\right)$

These equalities show that a mapping $\Phi$ is a homomorphism. Clearly,
$Ker(\Phi)$ consists only of identity matrix. Finally, let

$\left(\begin{array}[]{cc}1&\sigma_{12}\\\ 0&\sigma_{22}\end{array}\right)$

be the arbitrary element of $GL_{2}(F)$. Consider the matrix

$\left(\begin{array}[]{ccc}1&0&0\\\ \sigma_{12}&1&\sigma_{12}\\\
\sigma_{22}-1&0&\sigma_{22}\end{array}\right).$

This matrix is a preimage of the matrix above by $\Phi$. Moreover, both these
matrices have the same determinants. The last matrix belongs to $\Xi(G)$, so a
mapping $\Phi$ is an isomorphism.

In turn, clearly $\Xi(C_{2})$ is a semidirect product of a normal subgroup
that consists of matrices of the form

$\left(\begin{array}[]{cc}1&\sigma_{12}\\\ 0&1\end{array}\right),$

(i.e., a subgroup $UT_{2}(F)$) that is isomorphic to the additive group of a
field $F$ and a subgroup that consists of matrices of the form

$\left(\begin{array}[]{cc}1&0\\\ 0&\sigma_{22}\end{array}\right),$

which is isomorphic to the multiplicative group of a field $F$. $\Box$

##  3 The automorphism group of $L_{2}$.

Let us consider the second type of non-nilpotent Leibniz algebras:

$\displaystyle L_{2}=Fe_{1}\oplus Fe_{2}\oplus Fe_{3}\ \mbox{where
}[e_{1},e_{1}]=[e_{1},e_{3}]=e_{3},$ $\displaystyle[e_{1},e_{2}]=e_{2}+\lambda
e_{3},\ 0\neq\lambda\in F,$
$\displaystyle[e_{2},e_{1}]=[e_{2},e_{2}]=[e_{2},e_{3}]=[e_{3},e_{1}]=[e_{3},e_{2}]=[e_{3},e_{3}]=0.$

In other words, $L_{2}$ is the sum of the ideal $Leib(L_{2})=Fe_{2}\oplus
Fe_{3}$ and the non-nilpotent one-generated Leibniz algebra
$A_{1}=Fe_{1}\oplus Fe_{3}$ of dimension 2, $[A_{1},Leib(L_{2})]=Leib(L_{2})$,
$Leib(L_{2})=[L_{2},L_{2}]=\zeta^{\mathrm{left}}(L_{2})$,
$\zeta^{\mathrm{right}}(L_{2})=\zeta(L_{2})=\langle 0\rangle$.

The following theorem illustrates the structure of the automorphism group for
these Leibniz algebras.

######  Theorem 3.1.

Let $G$ be an automorphism group of the Leibniz algebra $L_{2}$. Then $G$ is
isomorphic to a subgroup of $GL_{3}(F)$ that consists of matrices of the form

$\left(\begin{array}[]{ccc}1&0&0\\\ 0&\beta_{1}&0\\\
\beta_{1}-1&\beta_{2}&\beta_{1}\end{array}\right),$

$\beta_{1},\beta_{2}\in F$. Furthermore, $G$ is abelian, $G=C\times A$ such
that $\Xi(C)$ is the set of matrices of the form

$\left(\begin{array}[]{ccc}1&0&0\\\ 0&1&0\\\ 0&\sigma&1\end{array}\right),$

$\sigma\in F$, and $\Xi(A)$ is the set of matrices of the form

$\left(\begin{array}[]{ccc}1&0&0\\\ 0&\beta&0\\\
\beta-1&0&\beta\end{array}\right),$

$\beta\in F$. A subgroup $C$ is isomorphic to the additive group of a field
$F$, and a subgroup $A$ is isomorphic to the multiplicative group of a field
$F$.

Proof. Let $L=L_{2}$ and let $f\in Aut_{[,]}(L)$. By [9, Lemma 2.1],
$f([L,L])=[L,L]$, so that

$\displaystyle f(e_{1})$
$\displaystyle=\alpha_{1}e_{1}+\alpha_{2}e_{2}+\alpha_{3}e_{3},$
$\displaystyle f(e_{2})$ $\displaystyle=\beta_{2}e_{2}+\beta_{3}e_{3},$
$\displaystyle f(e_{3})$ $\displaystyle=\gamma_{2}e_{2}+\gamma_{3}e_{3}.$

Then

$\displaystyle f(e_{3})$ $\displaystyle=f([e_{1},e_{1}])=[f(e_{1}),f(e_{1})]$
$\displaystyle=[\alpha_{1}e_{1}+\alpha_{2}e_{2}+\alpha_{3}e_{3},\alpha_{1}e_{1}+\alpha_{2}e_{2}+\alpha_{3}e_{3}]$
$\displaystyle=\alpha_{1}^{2}[e_{1},e_{1}]+\alpha_{1}\alpha_{2}[e_{1},e_{2}]+\alpha_{1}\alpha_{3}[e_{1},e_{3}]$
$\displaystyle=\alpha_{1}^{2}e_{3}+\alpha_{1}\alpha_{2}(e_{2}+\lambda
e_{3})+\alpha_{1}\alpha_{3}e_{3}$
$\displaystyle=\alpha_{1}\alpha_{2}e_{2}+(\alpha_{1}^{2}+\lambda\alpha_{1}\alpha_{2}+\alpha_{1}\alpha_{3})e_{3},$
$\displaystyle f(e_{3})$ $\displaystyle=f([e_{1},e_{3}])=[f(e_{1}),f(e_{3})]$
$\displaystyle=[\alpha_{1}e_{1}+\alpha_{2}e_{2}+\alpha_{3}e_{3},\gamma_{2}e_{2}+\gamma_{3}e_{3}]$
$\displaystyle=\alpha_{1}\gamma_{2}[e_{1},e_{2}]+\alpha_{1}\gamma_{3}[e_{1},e_{3}]$
$\displaystyle=\alpha_{1}\gamma_{2}(e_{2}+\lambda
e_{3})+\alpha_{1}\gamma_{3}e_{3}$
$\displaystyle=\alpha_{1}\gamma_{2}e_{2}+(\lambda\alpha_{1}\gamma_{2}+\alpha_{1}\gamma_{3})e_{3},$
$\displaystyle f([e_{1},e_{2}])$
$\displaystyle=[f(e_{1}),f(e_{2})]=[\alpha_{1}e_{1}+\alpha_{2}e_{2}+\alpha_{3}e_{3},\beta_{2}e_{2}+\beta_{3}e_{3}]$
$\displaystyle=\alpha_{1}\beta_{2}[e_{1},e_{2}]+\alpha_{1}\beta_{3}[e_{1},e_{3}]$
$\displaystyle=\alpha_{1}\beta_{2}(e_{2}+\lambda
e_{3})+\alpha_{1}\beta_{3}e_{3}$
$\displaystyle=\alpha_{1}\beta_{2}e_{2}+(\lambda\alpha_{1}\beta_{2}+\alpha_{1}\beta_{3})e_{3},$
$\displaystyle f([e_{1},e_{2}])$ $\displaystyle=f(e_{2}+\lambda
e_{3})=f(e_{2})+\lambda f(e_{3})$
$\displaystyle=\beta_{2}e_{2}+\beta_{3}e_{3}+\lambda(\gamma_{2}e_{2}+\gamma_{3}e_{3})$
$\displaystyle=(\beta_{2}+\lambda\gamma_{2})e_{2}+(\beta_{3}+\lambda\gamma_{3})e_{3}.$

Thus, we can see that

$\displaystyle\alpha_{1}\alpha_{2}e_{2}+(\alpha_{1}^{2}+\lambda\alpha_{1}\alpha_{2}+\alpha_{1}\alpha_{3})e_{3}=$
$\displaystyle\alpha_{1}\gamma_{2}e_{2}+(\lambda\alpha_{1}\gamma_{2}+\alpha_{1}\gamma_{3})e_{3}=\gamma_{2}e_{2}+\gamma_{3}e_{3},$
$\displaystyle\alpha_{1}\beta_{2}e_{2}+(\lambda\alpha_{1}\beta_{2}+\alpha_{1}\beta_{3})e_{3}=(\beta_{2}+\lambda\gamma_{2})e_{2}+(\beta_{3}+\lambda\gamma_{3})e_{3}.$

It follows that

$\displaystyle\alpha_{1}\alpha_{2}=\alpha_{1}\gamma_{2}=\gamma_{2},\
\alpha_{1}^{2}+\lambda\alpha_{1}\alpha_{2}+\alpha_{1}\alpha_{3}=\lambda\alpha_{1}\gamma_{2}+\alpha_{1}\gamma_{3}=\gamma_{3},$
$\displaystyle\alpha_{1}\beta_{2}=\beta_{2}+\lambda\gamma_{2},\
\lambda\alpha_{1}\beta_{2}+\alpha_{1}\beta_{3}=\beta_{3}+\lambda\gamma_{3}.$

If we suppose that $\alpha_{1}=0$, then $f(e_{1})\in Fe_{2}\oplus Fe_{3}$.
Since $f(e_{2}),f(e_{3})\in Fe_{2}\oplus Fe_{3}$, we obtain that $f(L)\leq
Fe_{2}\oplus Fe_{3}$ – in particular, $f(L)\neq L$ – and we obtain a
contradiction with the fact that $f$ is an automorphism of $L$. Hence,
$\alpha_{1}\neq 0$. It follows that $\alpha_{2}=\gamma_{2}$.

If we suppose that both $\beta_{2}=\gamma_{2}=0$, then $f(e_{2})\in Fe_{3}$,
and $f(e_{3})\in Fe_{3}$. We obtain that $f([L,L])\leq Fe_{3}$. On the other
hand, $dim_{F}([L,L])=2$, and we, again, obtain a contradiction with the fact
that $f$ is an automorphism of $L$. Hence, $(\beta_{2},\gamma_{2})\neq(0,0)$.
Then, if we suppose that $\beta_{2}=0$, we will see that
$\lambda\gamma_{2}=0$. Since $\lambda\neq 0$, we obtain that $\gamma_{2}=0$,
which is impossible. It follows that $\beta_{2}\neq 0$. Suppose that
$\gamma_{2}=0$. Since $\beta_{2}\neq 0$, we obtain that $\alpha_{1}=1$. If
$\gamma_{2}\neq 0$, then from $\alpha_{1}\gamma_{2}=\gamma_{2}$, we obtain
that $\alpha_{1}=1$. Thus, in every case, $\alpha_{1}=1$. Then we obtain an
equation $\beta_{2}=\beta_{2}+\lambda\gamma_{2}$ or $\lambda\gamma_{2}=0$.
Since $\lambda\neq 0$, we obtain that $\gamma_{2}=0$. Then $\alpha_{2}=0$, and
we obtain $1+\alpha_{3}=\gamma_{3}$,
$\lambda\beta_{2}+\beta_{3}=\beta_{3}+\lambda\gamma_{3}$. The last equation
implies that $\lambda\beta_{2}=\lambda\gamma_{3}$. Since $\lambda\neq 0$, we
obtain that $\beta_{2}=\gamma_{3}$.

Denote by $\Xi$ the canonical monomorphism of $Aut_{[,]}(L)$ in $M_{3}(F)$.
Then $\Xi(f)$ is the following matrix

$\left(\begin{array}[]{ccc}1&0&0\\\ 0&\beta_{2}&0\\\
\beta_{2}-1&\beta_{3}&\beta_{2}\end{array}\right),$

$\beta_{2},\beta_{3}\in F$. Since this matrix must be non-degenerate,
$\beta_{2}\neq 0$.

Let $f$ be a linear transformation of $L$, having in a basis
$\\{e_{1},e_{2},e_{3}\\}$ the matrix above. Let $x,y$ be the arbitrary
elements of $L$, $x=\xi_{1}e_{1}+\xi_{2}e_{2}+\xi_{3}e_{3}$,
$y=\eta_{1}e_{1}+\eta_{2}e_{2}+\eta_{3}e_{3}$ where
$\xi_{1},\xi_{2},\xi_{3},\eta_{1},\eta_{2},\eta_{3}\in F$. Then

$\displaystyle[x,y]$
$\displaystyle=[\xi_{1}e_{1}+\xi_{2}e_{2}+\xi_{3}e_{3},\eta_{1}e_{1}+\eta_{2}e_{2}+\eta_{3}e_{3}]$
$\displaystyle=\xi_{1}\eta_{1}[e_{1},e_{1}]+\xi_{1}\eta_{2}[e_{1},e_{2}]+\xi_{1}\eta_{3}[e_{1},e_{3}]$
$\displaystyle=\xi_{1}\eta_{1}e_{3}+\xi_{1}\eta_{2}(e_{2}+\lambda
e_{3})+\xi_{1}\eta_{3}e_{3}$
$\displaystyle=\xi_{1}\eta_{2}e_{2}+(\xi_{1}\eta_{1}+\lambda\xi_{1}\eta_{2}+\xi_{1}\eta_{3})e_{3},$
$\displaystyle f(x)$
$\displaystyle=f(\xi_{1}e_{1}+\xi_{2}e_{2}+\xi_{3}e_{3})=\xi_{1}f(e_{1})+\xi_{2}f(e_{2})+\xi_{3}f(e_{3})$
$\displaystyle=\xi_{1}(e_{1}+(\beta_{2}-1)e_{3})+\xi_{2}(\beta_{2}e_{2}+\beta_{3}e_{3})+\xi_{3}(\beta_{2}e_{3})$
$\displaystyle=\xi_{1}e_{1}+\xi_{2}\beta_{2}e_{2}+(\xi_{1}(\beta_{2}-1)+\xi_{2}\beta_{3}+\xi_{3}\beta_{2})e_{3},$
$\displaystyle f(y)$
$\displaystyle=\eta_{1}e_{1}+\eta_{2}\beta_{2}e_{2}+(\eta_{1}(\beta_{2}-1)+\eta_{2}\beta_{3}+\eta_{3}\beta_{2})e_{3}.$

Therefore,

$\displaystyle f([x,y])$
$\displaystyle=f(\xi_{1}\eta_{2}e_{2}+(\xi_{1}\eta_{1}+\lambda\xi_{1}\eta_{2}+\xi_{1}\eta_{3})e_{3})$
$\displaystyle=\xi_{1}\eta_{2}f(e_{2})+(\xi_{1}\eta_{1}+\lambda\xi_{1}\eta_{2}+\xi_{1}\eta_{3})f(e_{3})$
$\displaystyle=\xi_{1}\eta_{2}(\beta_{2}e_{2}+\beta_{3}e_{3})+(\xi_{1}\eta_{1}+\lambda\xi_{1}\eta_{2}+\xi_{1}\eta_{3})\beta_{2}e_{3}$
$\displaystyle=\xi_{1}\eta_{2}\beta_{2}e_{2}+(\xi_{1}\eta_{2}\beta_{3}+\beta_{2}\xi_{1}\eta_{1}+\lambda\xi_{1}\eta_{2}\beta_{2}+\xi_{1}\eta_{3}\beta_{2})e_{3},$
$\displaystyle[f(x),f(y)]$
$\displaystyle=[\xi_{1}e_{1}+\xi_{2}\beta_{2}e_{2}+(\xi_{1}(\beta_{2}-1)+\xi_{2}\beta_{3}+\xi_{3}\beta_{2})e_{3},$
$\displaystyle\ \ \ \
\eta_{1}e_{1}+\eta_{2}\beta_{2}e_{2}+(\eta_{1}(\beta_{2}-1)+\eta_{2}\beta_{3}+\eta_{3}\beta_{2})e_{3}]$
$\displaystyle=\xi_{1}\eta_{1}[e_{1},e_{1}]+\xi_{1}\eta_{2}\beta_{2}[e_{1},e_{2}]+\xi_{1}(\eta_{1}(\beta_{2}-1)+\eta_{2}\beta_{3}+\eta_{3}\beta_{2})[e_{1},e_{3}]$
$\displaystyle=\xi_{1}\eta_{1}e_{3}+\xi_{1}\eta_{2}\beta_{2}(e_{2}+\lambda
e_{3})+\xi_{1}(\eta_{1}(\beta_{2}-1)+\eta_{2}\beta_{3}+\eta_{3}\beta_{2})e_{3}$
$\displaystyle=\xi_{1}\eta_{2}\beta_{2}e_{2}+(\xi_{1}\eta_{1}+\xi_{1}\eta_{2}\beta_{2}\lambda+\xi_{1}\eta_{1}\beta_{2}-\xi_{1}\eta_{1}+\xi_{1}\eta_{2}\beta_{3}+\xi_{1}\eta_{3}\beta_{2})e_{3}$
$\displaystyle=\xi_{1}\eta_{2}\beta_{2}e_{2}+(\xi_{1}\eta_{2}\beta_{2}\lambda+\xi_{1}\eta_{1}\beta_{2}+\xi_{1}\eta_{2}\beta_{3}+\xi_{1}\eta_{3}\beta_{2})e_{3}.$

We can see that $f([x,y])=[f(x),f(y)]$. It follows that a linear
transformation $f$ having in a basis $\\{e_{1},e_{2},e_{3}\\}$ the matrix
above, is an automorphism of a Leibniz algebra $L$.

The equality

$\displaystyle\left(\begin{array}[]{ccc}1&0&0\\\ 0&\beta&0\\\
\beta-1&\gamma&\beta\end{array}\right)\left(\begin{array}[]{ccc}1&0&0\\\
0&\sigma&0\\\ \sigma-1&\kappa&\sigma\end{array}\right)$
$\displaystyle=\left(\begin{array}[]{ccc}1&0&0\\\ 0&\beta\sigma&0\\\
\beta-1+\beta(\sigma-1)&\gamma\sigma+\beta\kappa&\beta\sigma\end{array}\right)=\left(\begin{array}[]{ccc}1&0&0\\\
0&\beta\sigma&0\\\
\beta\sigma-1&\gamma\sigma+\beta\kappa&\beta\sigma\end{array}\right)$

shows that a group $G$ is abelian.

According to the proof above, a subspace $Fe_{3}$ is $G$-invariant. Let
$C=C_{G}(Fe_{3})$. Then we can see that $\Xi(C)$ is the set of matrices of the
form

$\left(\begin{array}[]{ccc}1&0&0\\\ 0&1&0\\\ 0&\beta&1\end{array}\right),$

$\beta\in F$. It is not hard to see that $C$ is isomorphic to the additive
group of a field $F$.

Denote by $A$ the preimage by $\Xi$ of the set of matrices of the form

$\left(\begin{array}[]{ccc}1&0&0\\\ 0&\beta&0\\\
\beta-1&0&\beta\end{array}\right),$

$\beta\in F$. We have:

$\displaystyle\left(\begin{array}[]{ccc}1&0&0\\\ 0&\beta&0\\\
\beta-1&0&\beta\end{array}\right)\left(\begin{array}[]{ccc}1&0&0\\\
0&\sigma&0\\\ \sigma-1&0&\sigma\end{array}\right)$
$\displaystyle=\left(\begin{array}[]{ccc}1&0&0\\\ 0&\beta\sigma&0\\\
\beta-1+\beta(\sigma-1)&0&\beta\sigma\end{array}\right)=\left(\begin{array}[]{ccc}1&0&0\\\
0&\beta\sigma&0\\\ \beta\sigma-1&0&\beta\sigma\end{array}\right).$

We can now see that $\Xi(A)$ is a subgroup of $GL_{3}(F)$, so that $A$ is a
subgroup of $G$.

Moreover, the mapping, defined by the rule

$\beta\rightarrow\left(\begin{array}[]{ccc}1&0&0\\\ 0&\beta&0\\\
\beta-1&0&\beta\end{array}\right),$

$\beta\in F$, is an isomorphism of the multiplicative group of $F$ on
$\Xi(A)$. Thus, we obtain that $A$ is isomorphic to the multiplicative group
of a field $F$.

Furthermore, the equality

$\displaystyle\left(\begin{array}[]{ccc}1&0&0\\\ 0&\beta&0\\\
\beta-1&\sigma&\beta\end{array}\right)=\left(\begin{array}[]{ccc}1&0&0\\\
0&\beta&0\\\
\beta-1&0&\beta\end{array}\right)\left(\begin{array}[]{ccc}1&0&0\\\ 0&1&0\\\
0&\beta^{-1}\sigma&1\end{array}\right)$

shows that $G$ is a product of the subgroups $C$ and $A$. Moreover, this
product is direct. $\Box$

## References

  * [1] Ayupov, Sh., Omirov, B., Rakhimov, I.: Leibniz algebras: structure and classification. CRC Press, Taylor & Francis Group, 2020.
  * [2] Blokh, A.: On a generalization of the concept of Lie algebra. Dokl. Akad. Nauk SSSR 1965; 165(3): pp. 471–473.
  * [3] Chupordia, V.A., Pypka, A.A., Semko, N.N., Yashchuk, V.S.: Leibniz algebras: a brief review of current results. Carpathian Math. Publ. 2019; 11(2): pp. 250–257. https://doi.org/10.15330/cmp.11.2.250-257
  * [4] Kirichenko, V.V., Kurdachenko, L.A., Pypka, A.A., Subbotin, I.Ya.: Some aspects of Leibniz algebra theory. Algebra Discrete Math. 2017; 24(1): pp. 1–33.
  * [5] Kurdachenko, L.A., Minaiev, P.Ye., Pypka, O.O.: Automorphism groups of some non-nilpotent Leibniz algebras. Res. Math., 2024; 32(1): pp. 101–109. https://doi.org/10.15421/242409
  * [6] Kurdachenko, L.A., Pypka, O.O., Semko, M.M.: Description of the automorphism groups of some Leibniz algebras. Res. Math. 2023; 31(1): pp. 52–61. https://doi.org/10.15421/242305
  * [7] Kurdachenko, L.A., Pypka, O.O., Semko, M.M.: On the structure of the automorphism groups of some low-dimensional Leibniz algebras. Ukrainian Math. J. 2024; 76(6): pp. 864–876. https://doi.org/10.3842/umzh.v76i6.7868
  * [8] Kurdachenko, L.A., Pypka, O.O., Subbotin, I.Ya.: On the structure of low-dimensional Leibniz algebras: some revision. Algebra Discrete Math. 2022; 34(1): pp. 68–104. https://doi.org/10.12958/adm2036
  * [9] Kurdachenko, L.A., Pypka, A.A., Subbotin, I.Ya.: On the automorphism groups of some Leibniz algebras. Int. J. Group Theory 2023; 12(1): pp. 1–20. https://doi.org/10.22108/IJGT.2021.130057.1735
  * [10] Kurdachenko, L.A., Pypka, O.O., Subbotin, I.Ya.: General Theory of Leibniz Algebras. Springer, Switzerland, 2024.
  * [11] Kurdachenko, L.A., Pypka, O.O., Velychko, T.V.: On the automorphism groups for some Leibniz algebras of low dimensions. Ukrainian Math. J. 2023; 74(10): pp. 1526–1546. https://doi.org/10.1007/s11253-023-02153-2
  * [12] Kurdachenko, L.A., Semko, N.N., Subbotin, I.Ya.: Applying group theory philosophy to Leibniz algebras: some new developments. Adv. Group Theory Appl. 2020; 9: pp. 71–121. https://doi.org/10.32037/agta-2020-004
  * [13] Kurdachenko, L.A., Subbotin, I.Ya., Yashchuk, V.S.: On the endomorphisms and derivations of some Leibniz algebras. J. Algebra Appl. 2024; 23(1): no. 2450002. https://doi.org/10.1142/S0219498824500026
  * [14] Loday, J.-L.: Cyclic homology. Springer Verlag, 1992. https://doi.org/10. 1007/978-3-662-11389-9
  * [15] Loday, J.-L.: Une version non commutative des algébres de Lie; les algébras de Leibniz. Enseign. Math. 1993; 39: pp. 269–293.
  * [16] Minaiev, P.Ye., Pypka, O.O., Semko, L.P.: Low-dimensional nilpotent Leibniz algebras and their automorphism groups. Algebra Discrete Math. 2024; 37(2): pp. 275–286. https://doi.org/10.12958/adm2264
  * [17] Subbotin, I.Ya.: Methods of group theory in Leibniz algebras: some compelling results. Res. Math. 2021; 29(2): pp. 44–54. https://doi.org/10.15421/ 242108


# Consumer Demand Modeling During COVID-19 Pandemic

Shaz Hoda  Emerging Technology
PwC
New York, USA
<EMAIL_ADDRESS>Amitoj Singh  Emerging Technology
PwC
Mumbai, India
<EMAIL_ADDRESS>Anand Rao  Emerging Technology
PwC
Boston, USA
<EMAIL_ADDRESS>Remzi Ural  Consumer Markets
PwC
Chicago, USA
<EMAIL_ADDRESS>Nicholas Hodson  Consumer Markets
PwC
Utah, USA
<EMAIL_ADDRESS>
###### Abstract

The current pandemic has introduced substantial uncertainty to traditional
methods for demand planning. These uncertainties stem from the disease
progression, government interventions, economy and consumer behavior. While
most of the emerging literature on the pandemic has focused on disease
progression, a few have focused on consequent regulations and their impact on
individual behavior. The contributions of this paper include a quantitative
behavior model of fear of COVID-19, impact of government interventions on
consumer behavior, and impact of consumer behavior on consumer choice and
hence demand for goods. It brings together multiple models for disease
progression, consumer behavior and demand estimation– thus bridging the gap
between disease progression and consumer demand. We use panel regression to
understand the drivers of demand during the pandemic and Bayesian inference to
simplify the regulation landscape that can help build scenarios for resilient
demand planning. We illustrate this resilient demand planning model using a
specific example of gas retailing. We find that demand is sensitive to fear of
COVID-19 – as the number of COVID-19 cases increase over the previous week,
the demand for gas decreases - though this dissipates over time. Further,
government regulations restrict access to different services, thereby reducing
mobility, which in itself reduces demand.

###### Index Terms:

forecasting, scenario modeling, demand forecasting, panel regression,
segmentation, Bayesian inference

##  I Introduction

Understanding demand and its drivers in the pre-pandemic era was about
understanding underlying demographic characteristics of the customers and
short-run price and promotion decisions of the supplier [1]. Given that
consumer preferences and characteristics change slowly over time and price and
promotions have temporary effects on demand, it is simple to use machine
learning techniques to understand the drivers of demand and to make short (few
weeks), medium (few months) and even in some cases, long run (few years)
predictions [2]. These stable relationships, however, have all broken down
during the pandemic. Though traditional factors still determine demand to an
extent, new drivers become more important as individuals make decisions that
change rapidly with the changing environment. This makes demand planning and
consequently, planning for production and distribution difficult in consumer
markets. Given the lack of certainty in demand, it is important to 1)
understand the right demand drivers and 2) to be able to model them correctly
so as to get to an accurate understanding and forecast demand under different
potential scenarios. Further, these demand forecasts/scenarios need to be
available not just at a national level, but also at local levels.

A substantial body of research on the pandemic has focused on disease
progression [3][4][5][6] with a few looking at the consequent regulations and
their impact on individual behavior [7][8][9]. On demand forecasting during
the pandemic, researchers have demonstrated that reopening of businesses and
the return to normalcy will depend not only on regulations but also on the
actions of related businesses, and customers and suppliers as well [10].
Researchers have also identified drivers of household consumption behavior
during the pandemic ranging from demographic characteristics, political
orientation, geographic location and to a lesser extent, income [11]. Our
paper combines these strands of work by examining the interaction of these
common demand drivers with regulations and testing these drivers along with
others for their significance in predicting demand fluctuations.

The objective of our paper is to consolidate various factors that are
impacting consumer demand during the pandemic, as well as to provide an
empirical model to estimate demand at local level. We use a demand modeling
example of a gas retail company for this paper. For modeling purposes we use
gas demand data for 2019 and 2020 (from a gas retailer operating in USA), the
data is available at store level for 1000+ stores in 30+ states in the US. We
augment this data with multiple rich data sources and run different models
LSTM, SARIMAX and panel regression to arrive at variables that can
significantly explain the gas demand in 2020 as percentage of gas demand in
2019 for the same week. We finally choose panel regression for two main
reasons - 1) lack of data to properly tune other models for over-fitting and
2) for the high degree of interpretability that panel regression allows. We
then simplify the regulation landscape and determine how the simplified
regulation landscape influences these drivers to develop scenarios for
planning. This approach towards demand modeling for gas, with slight
modifications, can be extrapolated to modeling demand for other consumer goods
such as restaurants, apparel, groceries etc.

##  II DATA

As discussed in the previous section there are multiple factors that influence
demand and the objective of this paper is to create a well-founded framework
for determining demand drivers during the pandemic and testing the same
empirically. The drivers are then interplayed with regulations to develop
planning scenarios.

Figure 1 is a systems representation of how uncertainties from the disease,
government intervention, economy and customer behavior are all intricately
linked and cause uncertainty in supply and demand.

Figure 1: Systems view of the pandemic and its impact on society.

Source: Authors’ depiction.

We begin with laying multiple hypotheses for drivers of demand. Through
interviews conducted with various experts in the consumer goods industry, a
survey of literature and working with varied companies, we break down the
potential demand drivers into following categories of data that we collect:

###  II-A Macro Indicators

COVID-19 has led to socio-economic effects and behavioral changes because of
different policies and regulations adopted by governments across the globe,
and because of changes in macro-economic environment of countries – most
countries reporting sharp contraction of GDPs for 3rd quarter of FY20 [10].
These effects have impacted countries and regions, with each regime adopting
its own set of regulations . While dips in infections have shown signs of
recoveries in countries like Japan and South Korea, a second wave of pandemic
and stricter controls have been anticipated in most US states and EU. There
has also been a significant variability in the macro-economic response of
governments - US and EU countries responded by providing stimulus packages and
thereby boosting demand, while developing countries such as India, Mexico, are
more fiscally constrained [11]. These effects are essential for understanding
the state-specific impacts of the pandemic on various industries within the
USA and therefore, we have included the following set of key variables for the
purpose of our analysis.

  * •

GDP growth rate at national/state level for USA from IHS Markit - GDP growth
rate is a good indicator of overall demand/income in the economy and can
potentially inform demand for individual products

  * •

Unemployment rate at the state level for USA from IHS Markit - Unemployment
rate directly impacts consumer ability to spend. The job losses have been
skewed towards certain sectors – for example, leisure and hospitality being
one of the worst-hit sectors. The impact of unemployment was mitigated by the
stimulus-package in June, but as its effects wear down, the demand for retail-
products also falls significantly [12]. These disparities in job losses affect
demand for different goods across different states.

  * •

Personal Consumption Expenditures at state level for USA from IHS Markit -
Less disposable income or preference to spend during the uncertain time of a
pandemic can adversely impact demand [13].

  * •

COVID-19 Spread from Suspected, Infected, Recovered (SIR) models at USA state
level developed by the authors – The SIR model is an epidemiological model
indicating the number of infections in a population, based on infection-rates,
susceptibility rates and recovery-rates. The spread of infections captures the
fear of different individuals in an environment interacting with different
factors like strictness of lockdowns, going out and mobility norms [14].

  * •

Regulations data at state level for USA from Multistate - Regulations such as
closure of bars, restaurants and stores directly affect consumer choice and
hence demand for certain types of products.

###  II-B Micro Indicators

It is also important to look at how pandemic has specifically changed consumer
habits at an individual level. While the macro effects help explain region
specific variations, individual response to the pandemic, has also impacted
aggregate consumer demand– mainly via less mobility and more time spent at
home. With most white-collared offices moving towards a work from home based
approach [12] – significant reduction in mobility levels have been noted
across the world. Moreover, reduction in mobility has also, at occasions,
resulted in a preference for home-consumable products proved over outside-
consumption. For example: we noted that there has been a noticeable increase
in sales of salad dressing for consumer goods companies – a ’substitution
effect’ for visits to restaurants. To capture the mobility related effects, we
looked at various indicators of mobility for individuals collected through
Google Mobility and PlaceIQ. Different indicators of mobility at state/local
level can indicate change in behavior of consumer and shifting preferences

  * •

Work from Home Index (WFHI) at county level – We measure work-from-home as a
percentage change in time spent at work places compared to the median value in
the 5-week period: Jan 3 – Feb 6, 2020. A higher work from home results in
fewer trips and less urban mobility [12]. We source WFHI data from Google
mobility at a county level and map the WFHI of the county in which the store
is located against each store.

  * •

Stay at Home Index (SAHI) at county level – Similar to WFHI, SAHI captures the
percentage change in time spent at home as compared to the median value in the
5-week period: Jan 3 – Feb 6, 2020. We source SAHI data from Google mobility
at a county level and map the SAHI of the county in which the store is located
against each store

  * •

Visits to grocery stores at county level \- Higher visits to grocery stores
could be an indicator of consumer preferences shifting towards home cooked
meals as compared to restaurants [15][16]. In our case, for gas retail, visits
to grocery stores were particularly important since the gas stations were
generally attached to grocery store chains. We source visits to grocery stores
data from PlaceIQ at a county level and map the grocery visits of the county
in which the store is located against each store

Consumer habits and surveys at county levels - Different indicators of
consumer behavior captured through representative surveys

  * •

Comfort in going out during pandemic

  * •

Activities that consumers miss the most

###  II-C Company Data

We were provided with gas volumes for a company that owns gas-stations across
various states in the US. The stores were distributed across 30+ US states,
significantly present in rural sites (60% of total stores). The data was at a
weekly level for the 2 years – 2019 and 2020, and we modelled the gas-volumes
sold in 2020 as a percentage of sales in 2019 levels for the same period. This
was done in order to remove the seasonality-effects, to understand demand fall
relative to normal-levels and finally to forecast a recovery compared to 2019
levels. Overall the master data for analysis was created at store week level,
with dependent variable as 2020 sales as a proportion of 2019 sales for the
same week. For each store the drivers, discussed in previous two sections,
were mapped - based on their granularity- to either the county or the state in
which the store belonged.

##  III METHODOLOGY

With the above data, we wanted to explain and project demand for gas. We
followed a three step process to the problem:

###  III-A Testing hypotheses and choosing best fit model

We used the above set of explanatory variables to understand the impact of the
current pandemic on gas-sales. For our analysis, we considered all the macro
and micro indicators. However, we leave out regulations from hypotheses
testing of initial drivers because all other variables are impacted by state,
local and national regulations – we come back to regulations as an overarching
variable, once the initial driver model is laid out. For example, the macro
variable unemployment is caused due to states mandating closure of certain
businesses. Similarly the micro variable stay at home is impacted by closure
of business and firms responding to government regulations and guidelines.

For testing our hypothesis we tried three different machine learning methods:

  * •

Deep Learning based LSTM method \- Long short-term memory (LSTM) is an
artificial recurrent neural network (RNN) architecture [17]. LSTM networks are
well-suited to classifying, processing and making predictions based on time
series data.

  * •

SARIMAX models \- Seasonal Auto Regressive Integrated Moving Averages with
Exogenous regressors. Seasonal ARIMAX, is an extension of ARIMA that
explicitly supports univariate time series data with a seasonal component and
exogenous regressors. It adds three new hyperparameters to specify the
autoregression (AR), differencing (I) and moving average (MA) for the seasonal
component of the series, as well as additional parameters for the period of
the seasonality and exogenous variables.

  * •

Panel regression model with time series adjustments \- Panel regression tries
to overcome the limitations of or ordinary least-squares regressions (OLS) by
introducing fixed and random effects variables across individuals (states in
our case) and time. As we show in subsequent sections, this model gives us the
best fit and interpretability of the model.

###  III-B Overlaying impact of regulations

Once a best fit model is constructed – based on error rates and explainability
- we need to develop a model that can inform scenarios on the underlying
drivers. For instance, we are most interested in understanding how these
variables are interrelated and are determined by overarching factors. For
determining the movement of these variables and possible scenarios, we
introduce the variable of government regulations. With restrictions passed by
governments around mobility of individuals and functioning of businesses,
especially in the USA, different macro and micro indicators that we consider
are impacted.

Figure 2 shows how different micro indicators (sourced from Google mobility
data) behaved as different regulations and announcements were made in Texas,
as an example.

Given the importance of regulations in determining both macro and micro
indicators we procure regulations data in the form of severity of
closure/opening of certain businesses, defined with an openness at three
different levels. Table 2 (presented in Appendix) shows different types of
regulations we capture and how they are classified as Level 1 (mostly open) -
Level 4 (mostly closed) based on severity.

Figure 2: Different Regulations and Announcements in one of the US States and
associated mobility

Source: Author’s analysis based on Google Mobility and PlaceIQ data

###  III-C Building scenarios

After the models are built and the regulation landscape is set, we use
regulations as overarching variables for determining the scenarios and
estimating the demand given the regulation level. Through our approach,
detailed in a subsequent section, the models learn the dependence between the
regulation and our set of micro and macro indicators, taking into account both
time-based and location-based effects. Once this is achieved, all the end user
needs to do is to specify restrictions for a given time period and location to
generate forecasts for the underlying product. This gives the end user the
flexibility to plan for both easing and re-imposition of restrictions and
hence adopt a resilient strategy for the future. The scenario modeling
approach involves the following steps: 1) the user chooses expected regulation
levels in different US states across time or chooses from pre-built regulatory
scenarios (gradual easing/winter lockdown) 2) based on the chosen regulations,
we use the mathematical model to arrive at forecasts of drivers 3) we use the
forecasted drivers to understand how they can impact demand in the future and
create an estimate.

##  IV FINDINGS AND RESULTS

We present in this section of the paper, anonymized results for retail gas
demand projections to understand the demand for commonly consumed consumer
goods across multiple stores in different states. As discussed, in the
previous section we followed the three step approach to understand demand and
to build scenarios on it:

###  IV-A Testing hypotheses and choosing best fit model

  * •

LSTM \- though promising in terms of model error rates, the models generally
have a tendency to quickly overfit unless regularized properly, which can be a
tedious task. These models are also black box models which tend to overweight
variables that can have a high degree of correlation with the target variable,
even though a theoretical foundation may be missing. Hence, we avoid pursuing
this method further for demand scenario modeling. We leave LSTM models as a
topic to explore further as more data on the pandemic becomes available.

  * •

SARIMAX model \- This model gives promising results, however, requires careful
tuning of seasonality for which a long history of data is required to come up
with a reasonable estimation of seasonality and other components. The company
had about one and a half years’ worth of data, including the pandemic period.
This made it difficult to estimate reasonable models using SARIMAX and have
enough data points to validate the model.

  * •

Panel regression analysis \- We finally went ahead with this approach to
estimate demand. These models offer enough flexibility to fine tune variables
and to get parameter estimates to help ensure that model coefficients are as
expected. The panel time series model helps us capture both location-specific
and time-specific attributes, which the other models LSTM (lack of
interpretability) and SARIMAX (lack of location attributes) fail to capture.
As we were dealing with significant heterogeneity both in terms of locations
-1000+ stores spread across 30+ states – and in terms of time based effects in
COVID-19 responses and infection – for example: states differed by date of
first cases reported and periods of second-wave – controlling for such
unobserved factors was important, which a panel model enabled us do in a
simpler way. In this section we investigate the results of our panel
regression model further. We also overcome the issue of estimation of
seasonality by constructing our target variable as change in demand over
previous year. The final functional form of our panel regression is as
follows:

$Y_{isct}=\alpha_{i}+\beta_{1}X_{ist}+\beta_{2}V_{ic}+\beta_{3}Z_{ict}+\epsilon_{isct}$
(1)

Where the dependent variable $Y_{isct}$ is the ratio of gas demand in 2020 to
2019 in store i, state s, county c, and time period t. The explanatory
variables such as unemployment are either at state $X_{ist}$ or such as SAHI
at the county ($V_{i}c$) level.

As mentioned above, as a precursor to the panel model, we segmented the stores
to cluster similar stores together. Taking fall in demand during the first 16
weeks of the pandemic as our target variable, we ran a decision tree on using
characteristics underlying the store. These characteristics included multiple
demographic information pertaining to the neighborhood in which the stores
were located - e.g. average age, household income, ethnicity, rural/urbanness
and other such characteristics that are available from the US census. The
decision tree model segmented the stores into 3 distinct segments (the client
was present primarily in rural and semi-urban markets and not in major
cities):

  1. 1.

Primarily rural stores with low population density - These stores saw the
least fall in demand.

  2. 2.

Primarily urban stores with medium population density with low education
prevalence

  3. 3.

Primarily urban stores with medium population density with high education
prevalence - These stores saw a huge fall in demand, presumably due to higher
work from home of urban educated

For each of these segments we created a regression model to understand how
different macro and micro economic factors drive demand. There were four
factors that came out to be significant at 5% confidence levels across all
three segments:

  * •

Fear of COVID-19: COVID-19 infection rate damped exponentially with elapsed
time. This variable captures the fear of individuals to freely pursue
activities that influence demand of a product. For example, demand for
fashionable clothing is low, since people during the first few weeks of
pandemic largely stayed at home due to the lack of awareness and riskiness of
the virus. This fear, however, dissipated over time. We observed in our data
that demand for products goes up over time, even when all else remains the
same. That is, if a state was stuck at a certain level of regulation, it would
still see an increase in gas demand overtime as people got used to the new
normal and found alternate activities. We experimented with multiple
functional forms of this variable - linear dampening, log case dampened
linearly and exponential damping. The model with exponential dampening of fear
gave the best results in terms of meaningful and significant coefficients
along with high r-squared. Through our panel approach we test for different
dampening effects across different states, however, we don’t observe any
significant differences across states in the dampening rate or function. The
final fear of COVID-19 variable for each week (t) and each state (i) took the
following form:

$log\dfrac{COVID19infections_{ist}}{e^{0.12*t}}$

The factor of 0.12 for dampening effect came from running multiple iterations
to get the best fit model across states with the limited data. With increased
data on pandemic, we can better estimate this parameter.

  * •

Change in Stay-at-home index (SAHI) vs baseline (%) of pre-COVID-19 period.
This variable captures the impact on demand due to people staying at home.
This is different from the fear of COVID-19, as through this variable we try
to capture how different regulations affect mobility of people and
consequently demand.

  * •

Change in supermarket store visits vs year ago (%). This variable was
specifically introduced since the product was specifically linked to sales
through physical visits to supermarkets and could not be purchased online.

  * •

Change in unemployment rate (%, absolute from previous years). This variable
captures the effect on sales of the product due to macro-economic impact of
loss of employment.

The results of the regression and estimates of parameters are presented in
Table 1. Our target variable was % change in volume over the previous year and
coefficient values indicate meaningful values, confirming our hypothesis.

TABLE I: MODEL COEFFICIENTS FOR KEY VARIABLES PREDICTING DEMAND, PANEL DUMMIES
EXCLUDED/AVERAGED

Model variable | Segments
---|---
Rural | Urban - educated | Urban - less educated
Fear of COVID-19: COVID-19 infection rate damped with elapsed time in weeks | -0.468 | -0.411 | -0.058
Change in Stay-at-home index vs baseline (%) | -1.236 | -1.046 | -1.597
Change in Grocery store visits vs year ago (%) | 0.292 | 0.376 | 0.178
Change in unemployment rate (%, absolute) | -1.639 | -0.917 | -1.151
r-squared | 0.82 | 0.90 | 0.86

Source: Author’s estimation. All variables significant at 5% confidence

We find that the gas demand across segments decrease by 1 to 1.5 percent with
a 1 percent increase in SAHI. In contrast, demand is more sensitive to
unemployment changes in less-educated and rural segments as compared to the
educated-urban segments. Fear of COVID-19 matters relatively less in the
urban, less-educated segment vis-à-vis the others, as perhaps explained by
their higher engagement in retail sectors and other essential services. Once
the estimates are obtained, it is easy to apply these estimates to see how
demand moved during the peak of COVID-19 and gradually recovered as these
underlying variables changed. We pick as an example one of the major US
states, where regulations were significantly eased within eight weeks
following the first case of COVID-19, which helped increase overall mobility
and consequently demand (Figure 2).

###  IV-B Overlaying impact of regulations

It is important to recognize that most of these drivers themselves are driven
by the regulations in place and consumers adapting to those regulations. For
example, Stay-at-home Index (SAHI), is driven by what is open and closed
within the state such as retail stores, restaurants etc. Similarly, visits to
supermarkets depend on how restrictive the regulators are in allowing
businesses to operate and impose distancing norms. Given the overarching role
of regulation in determining these drivers, we develop a mathematical estimate
of impact of regulations on drivers.

Figure 3: Decline and recovery of Gas Demand for one of the major states
during COVID-19 - explained by the demand drivers

To incorporate regulation in our model, we need to classify each state into an
overall level of openness. However, as we can see in Table 2 (presented in
Appendix), the array of regulations that can be applied in a state and for
each of those regulations, the restrictiveness that can be imposed is very
complicated. There are a total of 9 different types of regulations and each of
these 9 regulations can have 4 levels of imposition. This means that each US
state can be in 262,144 ($4^{9}$) possible regulatory states at any point of
time. We simplify this complex array of regulations using Bayesian Inferencing
to find the hierarchy in which they are applied. For instance, when stay at
home order is in place, bars, retail and restaurants are shut as well. Hence
regulations are not

Figure 4: Bayesian network of hierarchy of regulations using NO TEARS
algorithm

randomly applied/removed as a suite of options between the 9 businesses
presented in table 2 (presented in Appendix), but more in a hierarchical
manner. We use the Non-combinatorial Optimization via Trace Exponential and
Augmented lagRangian for Structure learning (NO TEARS) algorithm to learn the
structure of regulations [18]. A major challenge of Bayesian Network Structure
Learning (BNSL) is enforcing the directed acyclic graph (DAG) constraint,
which is combinatorial. While existing approaches rely on local heuristics, NO
TEARS introduces a fundamentally different strategy: it formulates the problem
as a purely continuous optimization problem over real matrices that avoids
combinatorial constraint entirely. The hierarchy of key regulations across
businesses is shown through the Bayesian Network obtained in Figure 4.

Based on the Bayesian hierarchy, the regulations are classified into 6
different levels as presented in Figure 5. These levels are finally used to
understand how severity of regulation impacts the key drivers of demand data.

Figure 5: Classification of different types of regulations across states into
levels

For determining the impact of regulations on drivers, we create a simple
mathematical model that depends on regulation level and exponential decay. We
estimate the parameter value of the decay through optimization based on
available data.

The two demand drivers which are impacted by regulations are 1) SAHI 2)
Grocery Visits. For each of these drivers we fit a mathematical function to
determine future behavior based on regulations. The function takes the
following form:

$SAHI_{it}=\dfrac{SAHI_{i0}*regulationLevelAdjustment}{e^{-k*weeksSinceFirstRegulation}}$

Where each i represents a different state/county, t represents time in weeks
and ’regulationLevelAdjustment’ represents the regulation level multiplier for
each of the 6 different levels of regulation, its value ranging from .1 to 1.
We use optimization to estimate the value of k and ’regulationLevelAdjustment’
so as to get the best estimate of SAHI (and other demand drivers) based on
regulation.

###  IV-C Developing Scenarios

Finally using our regression model, overlaid with our regulation-oriented
framework for determining demand drivers, we build a scenario modeling tool
for the purposes of resiliency planning. The scenario modeling toolkit allows
us to play with different regulatory scenarios and understand the best/worst
case forecast along with an overall distribution of forecasts across multiple
months, as shown in Figure 6. A few interesting insights emerge:

  * •

As seen in Figure 6, the overall range and monthly forecast reduces overtime
(we hide the x-axis showing demand range to preserve client confidentiality).
This is to do with the fact that different states ease regulations overtime
and as the pandemic progresses, the initial uncertainty around regulation
imposition and compliance becomes more predictable. Further, as we had noted
in figure 2, demand gradually recovers overtime as people tend to find new
consumption avenues, though the rate of recovery depends on the product being
modeled.

Figure 6: Density plot of expected demand in different quarters as a
percentage of last year demand (x-axis decreasing towards right, range hidden
for confidentiality purposes)

  * •

The impact of another round of severe lockdown (figure 7), does not impact
demand as much as the initial round of severe lockdown observed in early April
(demand fell by 25-30% as compared to previous year). This is again to do with
the fact that individuals become resilient in planning and learning. Given the
experience in dealing with strict restrictions under a pandemic, consumers
find alternate channels and avenues to fulfil demand for goods.

Figure 7: Impact of another Lockdown in 2021 (at Level 6) on recovery of gas
demand in on one of the US states

##  V DISCUSSION

In this paper, we model consumer demand for a gas retailing company under the
uncertainties brought on by COVID-19. We provide a quantitative behavioral
model of fear around COVID-19 and analyze the impact of regulations on
consumer choices and consequently, consumer demand – thus bridging the gap
between disease progression and consumer demand. We test the performance of
several competing models to identify the drivers of demand and build forecasts
based on a range of potential scenarios.

According to the availability of data, the model is applicable to
disaggregated geographic levels. We model demand for a gas retail company,
though the same model with slight modifications can be applied (and has been)
to modeling demand for other consumer goods, such as restaurants, apparel etc.

Some limitations and possible extensions to our work would be to address: 1)
expanding the scope to include consumer durables (e.g. cars, houses etc.) 2)
applying/re-visiting other techniques (LSTM) as more data becomes available 3)
use agent based techniques to model long-term behavior changes that can shift
demand.

## References

  * Nijs et al. [2001] Vincent R Nijs, Marnik G Dekimpe, Jan-Benedict EM Steenkamps, and Dominique M Hanssens.  The category-demand effects of price promotions.  _Marketing science_ , 20(1):1–22, 2001.
  * Bajari et al. [2015] Patrick Bajari, Denis Nekipelov, Stephen P Ryan, and Miaoyu Yang.  Machine learning methods for demand estimation.  _American Economic Review_ , 105(5):481–85, 2015\.
  * Mollalo et al. [2020] Abolfazl Mollalo, Behzad Vahedi, and Kiara M Rivera.  Gis-based spatial modeling of covid-19 incidence rate in the continental united states.  _Science of The Total Environment_ , page 138884, 2020.
  * Wilder et al. [2020] Bryan Wilder, Marie Charpignon, Jackson A Killian, Han-Ching Ou, Aditya Mate, Shahin Jabbari, Andrew Perrault, Angel N Desai, Milind Tambe, and Maimuna S Majumder.  Modeling between-population variation in covid-19 dynamics in hubei, lombardy, and new york city.  _Proceedings of the National Academy of Sciences_ , 117(41):25904–25910, 2020.
  * Ji et al. [2020] Dong Ji, Dawei Zhang, Jing Xu, Zhu Chen, Tieniu Yang, Peng Zhao, Guofeng Chen, Gregory Cheng, Yudong Wang, Jingfeng Bi, et al.  Prediction for progression risk in patients with covid-19 pneumonia: the call score.  _Clinical Infectious Diseases_ , 2020.
  * Firth-Butterfield and Rao [2020] Kay Firth-Butterfield and Anand Rao.  Lessons from covid-19 modeling: The interplay of data, models, and behavior.  Technical report, World Economic Forum, 2020.
  * Lattanzio and Palumbo [2020] Salvatore Lattanzio and Dario Palumbo.  Lifting restrictions with changing mobility and the importance of soft containment measures: A seird model of covid-19 dynamics.  2020\.
  * Smyth [2020] Barry Smyth.  Lockdowns & rebounds: A data analysis of what happens next.  _Digital Government: Research and Practice_ , 1(4):1–7, 2020.
  * Silva et al. [2020] Petrônio CL Silva, Paulo VC Batista, Hélder S Lima, Marcos A Alves, Frederico G Guimarães, and Rodrigo CP Silva.  Covid-abs: An agent-based model of covid-19 epidemic to simulate health and economic effects of social distancing interventions.  _Chaos, Solitons & Fractals_, 139:110088, 2020.
  * Balla-Elliott et al. [2020] Dylan Balla-Elliott, Zoë B Cullen, Edward L Glaeser, Michael Luca, and Christopher T Stanton.  Business reopening decisions and demand forecasts during the covid-19 pandemic.  Technical report, National Bureau of Economic Research, 2020.
  * Baker et al. [2020] Scott R Baker, Nicholas Bloom, Steven J Davis, and Stephen J Terry.  Covid-induced economic uncertainty.  Technical report, National Bureau of Economic Research, 2020.
  * Coibion et al. [2020] Olivier Coibion, Yuriy Gorodnichenko, and Michael Weber.  The cost of the covid-19 crisis: Lockdowns, macroeconomic expectations, and consumer spending.  Technical report, National Bureau of Economic Research, 2020.
  * Elgin et al. [2020] Ceyhun Elgin, Gokce Basbug, and Abdullah Yalaman.  Economic policy responses to a pandemic: Developing the covid-19 economic stimulus index.  _Covid Economics_ , 1(3):40–53, 2020.
  * Chetty et al. [2020] Raj Chetty, John N Friedman, Nathaniel Hendren, Michael Stepner, et al.  How did covid-19 and stabilization policies affect spending and employment? a new real-time economic tracker based on private sector data.  Technical report, National Bureau of Economic Research, 2020.
  * Szymkowiak et al. [2020] Andrzej Szymkowiak, Piotr Kulawik, Kishokanth Jeganathan, and Paulina Guzik.  In-store epidemic behavior: scale development and validation.  _arXiv preprint arXiv:2005.02764_ , 2020.
  * Chang and Meyerhoefer [2020] Hung-Hao Chang and Chad D Meyerhoefer.  Covid-19 and the demand for online food shopping services: Empirical evidence from taiwan.  _American Journal of Agricultural Economics_ , 2020.
  * Hochreiter and Schmidhuber [1997] Sepp Hochreiter and Jürgen Schmidhuber.  Long short-term memory.  _Neural computation_ , 9(8):1735–1780, 1997.
  * Zheng et al. [2018] Xun Zheng, Bryon Aragam, Pradeep K Ravikumar, and Eric P Xing.  Dags with no tears: Continuous optimization for structure learning.  In _Advances in Neural Information Processing Systems_ , pages 9472–9483, 2018.