# Isotopisms of nilpotent
Leibniz algebras and Lie racks00footnotetext: Keywords: Leibniz algebra, Lie
rack, Coquegigrue problem, Isotopism. 00footnotetext: 2020 Mathematics Subject
Classification: 17A32, 17A36, 17B30, 20M99, 22A30.

###### Abstract

In this paper we study the isotopism classes of two-step nilpotent Leibniz
algebras. We show that every nilpotent Leibniz algebra with one-dimensional
commutator ideal is isotopic to the Heisenberg algebra $\mathfrak{h}_{2n+1}$
or to the Heisenberg Leibniz algebra $\mathfrak{l}_{2n+1}^{J_{1}}$, where
$J_{1}$ is the $n\times n$ Jordan block of eigenvalue $1$. We also prove that
two such algebras are isotopic if and only if the Lie racks integrating them
are isotopic. This gives the classification, up to isotopism, of Lie racks
whose tangent space at the unit element is a nilpotent Leibniz algebra with
commutator ideal of dimension one. Eventually we introduce new isotopism
invariants for Leibniz algebras and Lie racks.

Gianmarco La Rosa222Supported by University of Palermo and by the “National
Group for Algebraic and Geometric Structures, and their Applications” (GNSAGA
– INdAM)., Manuel Mancini22footnotemark: 2
Dipartimento di Matematica e Informatica, Università degli Studi di Palermo
Via Archirafi 34, 90123 Palermo, Italy
<EMAIL_ADDRESS>ORCID: 0000-0003-1047-5993
<EMAIL_ADDRESS>ORCID: 0000-0003-2142-6193

Gábor P. Nagy333Supported by the National Research, Development and Innovation
Fund of Hungary, NKFIH-OTKA Grant SNN 132625.
Department of Algebra, Budapest University of Technology and Economics
Műegyetem rkp 3, H-1111 Budapest, Hungary

Bolyai Institute, University of Szeged
Aradi Vértanúk Tere 1, H-6720 Szeged, Hungary
<EMAIL_ADDRESS>ORCID: 0000-0002-9558-4197

## Introduction

Leibniz algebras were first introduced by J.-L. Loday in [21] as a non-
antisymmetric version of Lie algebras. Earlier, such algebraic structures were
considered by A. Blokh, who called them D-algebras [7], for their strict
connection with the derivations. Leibniz algebras play a significant role in
different areas of mathematics and physics.

Many results of Lie algebras were also established in the frame of Leibniz
algebras. One of them is the _Levi decomposition_ (see [4]), which states that
every finite-dimensional Leibniz algebra $\mathfrak{g}$ over a field
$\mathbb{F}$, with $\operatorname{char}(\mathbb{F})=0$, is the semidirect
product of a solvable ideal (the _radical_ of $\mathfrak{g}$) and a semisimple
subalgebra, which is a Lie algebra. This makes clear the importance of the
problem of Lie and Leibniz algebras classification, which has been dealt with
since the early 20th century (see [5], [6], [3], [8], [23], [11] and [10],
just for giving some examples). However in general, given two Leibniz algebras
$\mathfrak{g}$ and $\mathfrak{h}$, it is hard to check if $\mathfrak{g}$ and
$\mathfrak{h}$ are isomorphic or not. As we will see, in some cases it it is
easier to verify if there exists an _isotopism_ between them, i.e. if there is
a triple of linear isomorphisms
$(f,g,h)\colon\mathfrak{g}\leftrightarrows\mathfrak{h}$ such that
$[f(x),g(y)]_{\mathfrak{h}}=h([x,y]_{\mathfrak{g}})$, for every
$x,y\in\mathfrak{g}$.

In [20] two-step nilpotent Leibniz algebras over a field $\mathbb{F}$, with
$\operatorname{char}(\mathbb{F})\neq 2$, were classified and integrated into
Lie racks. However, the problem of finding the isomorphism classes of these
algebras is still open, also in the case that $\mathbb{F}$ is the field of
complex numbers.

In this paper our goal is to use the notion of _isotopism_ , which was
introduced by A. A. Albert in 1942 (see [1], [2] and [17]) in order to
classify non-associative algebras, and then used in [15] and [16] for the
studying of some classes of Lie algebras, to find a complete classification of
nilpotent Leibniz algebras with one-dimensional commutator ideal. We prove
that two such algebras are isotopic if and only if the Lie racks integrating
them are isotopic. By using this result, we are able to find the isotopism
classes of Lie racks whose tangent space at the unit element is a nilpotent
Leibniz algebra with one-dimensional commutator ideal.

The first section is devoted to some background material on Leibniz algebras.
We address the reader to [4] and [21] for more details.

In Sections 2 and 3 we resume the main results about the classification and
the integration of two-step nilpotent Leibniz algebras (see [20]). We recall
that, up to isomorphism, there are only three classes of indecomposable
nilpotent Leibniz algebras with one-dimensional commutator ideal, namely the
_Heisenberg Leibniz algebras_ $\mathfrak{l}_{2n+1}^{A}$, parametrized by their
dimension $2n+1$ and a $n\times n$ matrix $A$ in canonical form, the
_Kronecker Leibniz algebras_ $\mathfrak{k}_{n}$ and the _Dieudonné Leibniz
algebras_ $\mathfrak{d}_{n}$, both parametrized by their dimension only. We
also recall the construction of the Lie rack integrating a two-step nilpotent
Leibniz algebra.

In Section 4 we give the definitions of isotopism of Leibniz algebras and Lie
racks. We recall that every isotopism is isomorphic to a so-called _principal
isotopism_ and we give the explicit construction of an isotopism between the
Heisenberg algebra $\mathfrak{h}_{3}$ and a nilpotent Leibniz non-Lie algebra.

In the final section we introduce new isotopism invariants for Leibniz
algebras and Lie racks and we prove that every indecomposable nilpotent
Leibniz algebra $\mathfrak{g}$ with
$\dim_{\mathbb{F}}[\mathfrak{g},\mathfrak{g}]=1$ is either isomorphic to the
Heisenberg Leibniz algebra $\mathfrak{l}_{2n+1}^{J_{1}}$, where $J_{1}$ is the
$n\times n$ Jordan block of eigenvalue $1$, or it is isotopic to the
$(2n+1)-$dimensional Heisenberg Lie algebra $\mathfrak{h}_{2n+1}$. Finally we
show that, if $\mathfrak{g}$ and $\mathfrak{h}$ are real Leibniz algebra with
$\dim_{\mathbb{R}}[\mathfrak{g},\mathfrak{g}]=\dim_{\mathbb{R}}[\mathfrak{h},\mathfrak{h}]=1$
and $X,Y$ are the Lie racks integrating them, then $\mathfrak{g}$ is isotopic
to $\mathfrak{h}$ if and only if $X$ is isotopic to $Y$. This implies that a
Lie rack $X$ integrating a Leibniz algebra $\mathfrak{g}$, with
$\dim_{\mathbb{R}}[\mathfrak{g},\mathfrak{g}]=1$, is either isomorphic to the
Heisenberg rack $R_{2n+1}^{J_{1}}$, or it is isotopic to the conjugation
$\operatorname{Conj}(H_{2n+1})$ of the Heisenberg Lie group.

##  1 Preliminaries

We assume that $\mathbb{F}$ is a field with
$\operatorname{char}(\mathbb{F})\neq 2$. For the general theory we refer to
[4] and [21].

######  Definition 1.1.

A _left Leibniz algebra_ over $\mathbb{F}$ is a vector space $\mathfrak{g}$
over $\mathbb{F}$ endowed of a bilinear map (called $commutator$ or $bracket$)
$\left[-,-\right]\colon\mathfrak{g}\times\mathfrak{g}\rightarrow\mathfrak{g}$
which satisfies the _left Leibniz identity_

$\left[x,\left[y,z\right]\right]=\left[\left[x,y\right],z\right]+\left[y,\left[x,z\right]\right],\;\;\forall
x,y,z\in\mathfrak{g}.$

In the same way one can define a right Leibniz algebra, using the right
Leibniz identity

$\left[\left[x,y\right],z\right]=\left[[x,z],y\right]+\left[x,\left[y,z\right]\right],\,\,\forall
x,y,z\in\mathfrak{g}.$

A Leibniz algebra that is both left and right is called _symmetric Leibniz
algebra_. From now on we assume that $\dim_{\mathbb{F}}\mathfrak{g}<\infty$.

Every Lie algebra is a Leibniz algebra and every Leibniz algebra with skew-
symmetric commutator is a Lie algebra. The _Leibniz kernel_ of $\mathfrak{g}$

$\operatorname{Leib}(\mathfrak{g})=\langle\left[x,x\right]|\,x\in\mathfrak{g}\rangle$

is the smallest bilateral ideal of $\mathfrak{g}$ such that
$\mathfrak{g}/\operatorname{Leib}(\mathfrak{g})$ is a Lie algebra. This
defines an adjunction (see [22]) between the category
$\textbf{LieAlg}_{\mathbb{F}}$ of Lie algebras over $\mathbb{F}$ and the
category $\textbf{LeibAlg}_{\mathbb{F}}$ of Leibniz algebras over
$\mathbb{F}$. The left adjoint of the full inclusion
$i\colon\textbf{LieAlg}_{\mathbb{F}}\rightarrow\textbf{LeibAlg}_{\mathbb{F}}$
is
$\pi\colon\textbf{LeibAlg}_{\mathbb{F}}\rightarrow\textbf{LieAlg}_{\mathbb{F}}$,
defined by $\pi(\mathfrak{g})=\mathfrak{g}/\operatorname{Leib}(\mathfrak{g})$.

For a Leibniz algebra $\mathfrak{g}$, the left center and the right center are
respectively

$\operatorname{Z}_{l}(\mathfrak{g})=\left\\{x\in\mathfrak{g}\,|\,\left[x,\mathfrak{g}\right]=0\right\\},\,\,\,\operatorname{Z}_{r}(\mathfrak{g})=\left\\{x\in\mathfrak{g}\,|\,\left[\mathfrak{g},x\right]=0\right\\}$

and they coincide when $\mathfrak{g}$ is a Lie algebra. The _center_ of
$\mathfrak{g}$ is
$\operatorname{Z}(\mathfrak{g})=\operatorname{Z}_{l}(\mathfrak{g})\cap\operatorname{Z}_{r}(\mathfrak{g})$.
We recall that in general $\operatorname{Z}_{l}(\mathfrak{g})$ is an ideal of
the left Leibniz algebra $\mathfrak{g}$, meanwhile the right center is not
even a subalgebra. Moreover
$\operatorname{Leib}(\mathfrak{g})\subseteq\operatorname{Z}_{l}(\mathfrak{g})$,
since

$[[x,x],y]=[x,[x,y]]-[x,[x,y]]=0,$

for every $x,y\in\mathfrak{g}$.

Now we recall the definitions of solvable and nilpotent Leibniz algebra.

######  Definition 1.2.

Let $\mathfrak{g}$ be a left Leibniz algebra over $\mathbb{F}$ and let

$\mathfrak{g}^{0}=\mathfrak{g},\,\,\mathfrak{g}^{k+1}=[\mathfrak{g}^{k},\mathfrak{g}^{k}],\;\;\forall
k\geq 0$

be the _derived series of $\mathfrak{g}$_. $\mathfrak{g}$ is _$n-$ step
solvable_ if $\mathfrak{g}^{n-1}\neq 0$ and $\mathfrak{g}^{n}=0$.

######  Definition 1.3.

Let $\mathfrak{g}$ be a left Leibniz algebra over $\mathbb{F}$ and let

$\mathfrak{g}^{(0)}=\mathfrak{g},\,\,\mathfrak{g}^{(k+1)}=[\mathfrak{g},\mathfrak{g}^{(k)}],\;\;\forall
k\geq 0$

be the _lower central series of $\mathfrak{g}$_. $\mathfrak{g}$ is _$n-$ step
nilpotent_ if $\mathfrak{g}^{(n-1)}\neq 0$ and $\mathfrak{g}^{(n)}=0$.

When $n=2$, we have the following.

######  Proposition 1.4.

[20] Let $\mathfrak{g}$ be a $n-$step nilpotent Leibniz algebra.

  * (i)

If $n=2$, then
$\mathfrak{g}^{(1)}=[\mathfrak{g},\mathfrak{g}]\subseteq\operatorname{Z}(\mathfrak{g})$
and $\mathfrak{g}$ is a symmetric Leibniz algebra;

  * (ii)

If $\dim_{\mathbb{F}}[\mathfrak{g},\mathfrak{g}]=1$, then $n=2$.

##  2 Nilpotent Leibniz algebras with one-dimensional commutator ideal

Let $\mathfrak{g}$ be a two-step nilpotent Leibniz algebra and let
$\\{h_{1},\ldots,h_{t}\\}$ be a basis of the commutator ideal
$[\mathfrak{g},\mathfrak{g}]$. Thus

$[x,y]=\phi_{1}(x,y)h_{1}+\cdots+\phi_{t}(x,y)h_{t},\;\;\forall
x,y\in\mathfrak{g},$

where $\phi_{i}\colon\mathfrak{g}\times\mathfrak{g}\rightarrow\mathbb{F}$ is a
bilinear form, for every $i=1,\ldots,t$. In this section we recall the main
steps to reduce such a given bilinear form to a suitable canonical
representation. For more deteails, we refer the reader to [20].

Let $\mathfrak{g}$ be a nilpotent Leibniz algebra with one-dimensional
commutator ideal $\left[\mathfrak{g},\mathfrak{g}\right]=\mathbb{F}h$, where
$h\in\mathfrak{g}$ is fixed, and let $\left[x,y\right]=\phi(x,y)h$, where
$\phi\colon\mathfrak{g}\times\mathfrak{g}\rightarrow\mathbb{F}$ is a bilinear
form. We observe that, if $\mathfrak{g}$ is not a Lie algebra, then
$\operatorname{Leib}(\mathfrak{g})=[\mathfrak{g},\mathfrak{g}]\subseteq\operatorname{Z}(\mathfrak{g})$.

We say that $\mathfrak{g}$ is _decomposable_ if there exists a subspace $U$ of
$\mathfrak{g}$ such that $\mathfrak{g}=U\oplus U^{\perp}$, where $U^{\perp}$
is the orthogonal space with respect to the bilinear forms

$\phi^{-}=\dfrac{\phi-\phi^{T}}{2}\;\;\text{and}\;\;\phi^{+}=\dfrac{\phi+\phi^{T}}{2},$

which are respectively the skew-symmetric and the symmetric part of $\phi$.
Since $\dim_{\mathbb{F}}\mathfrak{g}<\infty$, we can can assume that
$\mathfrak{g}$ is indecomposable. In this case, by using the simultaneous
reduction to canonical form of a pair of bilinear forms (see [19] and [12]),
we have that, up to isomorphisms, the matrix associated with $\phi$ turns out
to be one of the following

$\begin{pmatrix}0&I_{n}+A\\\ -I_{n}+A^{T}&0\end{pmatrix},$ (1)
$\begin{pmatrix}0&J_{0}+I_{n}\\\ -J_{0}^{T}+I_{n}&0\end{pmatrix},$ (2)
$\begin{pmatrix}0&M_{1}+M_{2}\\\ -M_{1}^{T}+M_{2}^{T}&0\end{pmatrix},$ (3)

where $A\in\operatorname{M}_{n}(\mathbb{F})$ is the companion matrix of the
power of a monic irreducible polynomial $f(x)\in\mathbb{F}\left[x\right]$,
$J_{0}$ is the $n\times n$ Jordan block of eigenvalue zero and $M_{1},M_{2}$
are the matrices associated with the linear applications
$\varphi_{1},\varphi_{2}\colon\mathbb{F}^{n}\rightarrow\mathbb{F}^{n+1}$
defined by $\varphi_{1}(x_{1},\ldots,x_{n})=(x_{1},\ldots,x_{n},0)$ and
$\varphi_{2}(x_{1},\ldots,x_{n})=(0,x_{1},\ldots,x_{n})$, for every
$(x_{1},\ldots,x_{n})\in\mathbb{F}^{n}$.

Thus there are only three classes of indecomposable nilpotent Leibniz algebras
with one-dimensional commutator ideal.

######  Definition 2.1.

[20] Let $f(x)\in\mathbb{F}\left[x\right]$ be a monic irreducible polynomial.
Let $k\in\mathbb{N}$, $n=k\deg f(x)$ and let $A$ be the companion matrix of
$f(x)^{k}$. The _Heisenberg_ Leibniz algebra $\mathfrak{l}_{2n+1}^{A}$ is the
$(2n+1)$-dimensional indecomposable Leibniz algebra with basis
$\left\\{e_{1},\ldots,e_{n},f_{1},\ldots,f_{n},h\right\\}$ and associated
matrix of type $(\ref{modulo1})$.

If $A$ is the zero matrix, then we obtain the Heisenberg algebra
$\mathfrak{h}_{2n+1}$, i.e. the $(2n+1)-$dimensional Lie algebra with
$[e_{i},f_{j}]=\delta_{ij}h$, for any $i,j=1,\ldots,n$, where $\delta_{ij}$ is
the _Kronecker delta_.

######  Definition 2.2.

[20] Let $n\in\mathbb{N}$. The _Kronecker_ Leibniz algebra $\mathfrak{k}_{n}$
is the $(2n+1)$-dimensional indecomposable Leibniz algebra with basis
$\left\\{e_{1},\ldots,e_{n},f_{1},\ldots,f_{n},h\right\\}$ and associated
matrix $(\ref{modulo2})$.

######  Definition 2.3.

[20] The _Dieudonné_ Leibniz algebra $\mathfrak{d}_{n}$ is the
$(2n+2)-$dimensional Leibniz algebra with basis
$\left\\{e_{1},\ldots,e_{2n+1},h\right\\}$ and associated matrix
$(\ref{modulo3})$.

For every $n\in\mathbb{N}$, the Kronecker Leibniz algebra $\mathfrak{k}_{n}$
and the Dieudonné Leibniz algebra $\mathfrak{d}_{n}$ are not Lie algebras and
they are unique up to isomorphisms.

Now we recall the description of the Heisenberg Leibniz algebras in the case
the field $\mathbb{F}$ is $\mathbb{C}$ or $\mathbb{R}$.

###  2.1 Complex Heisenberg Leibniz algebras

Let $n\in\mathbb{N}$ and let $f(x)=x-a\in\mathbb{C}[x]$. Then the companion
matrix $A$ of $f(x)^{n}$ is similar to the $n\times n$ Jordan block $J_{a}$ of
eigenvalue $a$. Thus $\mathfrak{l}_{2n+1}^{A}\cong\mathfrak{l}_{2n+1}^{J_{a}}$
and the non-trivial commutators are

$\left[e_{i},f_{i}\right]=(1+a)h,\;\;\left[f_{i},e_{i}\right]=(-1+a)h,\;\;\forall
i=1,\ldots,n,$
$\left[e_{i},f_{i-1}\right]=\left[f_{i-1},e_{i}\right]=h,\;\;\forall
i=2,\ldots,n,$

where $\left\\{e_{1},\ldots,e_{n},f_{1},\ldots,f_{n},h\right\\}$ is a basis of
$\mathfrak{l}_{2n+1}^{J_{a}}$. One can check that

$\operatorname{Z}_{l}(\mathfrak{l}_{2n+1}^{J_{a}}))=\operatorname{Z}_{r}(\mathfrak{l}_{2n+1}^{J_{a}}))=[\mathfrak{l}_{2n+1}^{J_{a}},\mathfrak{l}_{2n+1}^{J_{a}}]=\mathbb{F}h,$

for any $a\neq\pm 1$,

$\operatorname{Z}_{l}(\mathfrak{l}_{2n+1}^{J_{1}})=\operatorname{Z}_{r}(\mathfrak{l}_{2n+1}^{J_{-1}})=\langle
f_{n},h\rangle,$
$\operatorname{Z}_{r}(\mathfrak{l}_{2n+1}^{J_{1}})=\operatorname{Z}_{l}(\mathfrak{l}_{2n+1}^{J_{-1}})=\langle
e_{1},h\rangle.$

and we have the following.

######  Proposition 2.4.

[20] Let $a\in\mathbb{C}$. The Heisenberg Leibniz algebras
$\mathfrak{l}_{2n+1}^{J_{a}}$ and $\mathfrak{l}_{2n+1}^{J_{-a}}$ are
isomorphic.

We observe that the problem of finding the isomorphism classes of the complex
Heisenberg Leibniz algebras is still open, but when $n=1$ the converse result
of Proposition 2.4 is also true.

######  Proposition 2.5.

[20] Let $a,a^{\prime}\in\mathbb{C}$. Then $\mathfrak{l}_{3}^{a}$ and
$\mathfrak{l}_{3}^{a^{\prime}}$ are isomorphic if and only if $a^{\prime}=\pm
a$.

In the last section we will show that, for any $a\neq\pm 1$, there exists an
_isotopism_ between $\mathfrak{l}_{2n+1}^{J_{a}}$ and the Heisenberg Lie
algebra $\mathfrak{h}_{2n+1}$.

###  2.2 Real Heisenberg Leibniz algebras

Let $f(x)\in\mathbb{R}[x]$ be an irreducible monic polynomial. If $f(x)=x-a$,
then $\mathfrak{l}_{2n+1}^{A}\cong\mathfrak{l}_{2n+1}^{J_{a}}$, as in the
complex case. So we suppose that $f(x)=x^{2}+bx+c$, with $b^{2}-4c<0$.

Let $z=\alpha+i\beta\in\mathbb{C}$ be a root of $f(x)$. Then
$f(x)=(x-z)(x-\bar{z})$ and the companion matrix $A$ of $f(x)^{n}$ in similar
to the $2n\times 2n$ real block matrix

$J_{R}=\begin{pmatrix}R&0&\cdots&0\\\ I_{2}&R&\cdots&0\\\
\vdots&\ddots&\ddots&\vdots\\\
0&\cdots&I_{2}&R\end{pmatrix},\;\;\text{where}\;\;R=\begin{pmatrix}\alpha&\beta\\\
-\beta&\alpha\end{pmatrix}$

is the realification of the complex number $z$. Thus
$\mathfrak{l}_{4n+1}^{A}\cong\mathfrak{l}_{4n+1}^{J_{R}}$ and we have the
following, when $n=1$.

######  Proposition 2.6.

[20] Let $f(x),g(x)\in\mathbb{R}[x]$ be irreducible monic polynomials of
degree two. Let $z,z^{\prime}\in\mathbb{C}$ be roots of $f(x)$ and $g(x)$
respectively and let $R,R^{\prime}$ be the realifications of $z$ and
$z^{\prime}$. Then $\mathfrak{l}_{5}^{R}\cong\mathfrak{l}_{5}^{R^{\prime}}$ if
and only if $R^{\prime}=\pm R$.

##  3 The _coquecigrue problem_ for nilpotent Leibniz algebras

In the case of the correspondence between a Lie group $G$ and its Lie algebra
$\mathfrak{g}=\operatorname{T}_{1}G$, the bracket

$[x,y]=\operatorname{ad}_{x}(y),$

where
$\operatorname{ad}\colon\mathfrak{g}\rightarrow\mathfrak{gl}(\mathfrak{g})$,
is the differential of the adjoint representation $\operatorname{Ad}\colon
G\rightarrow\operatorname{GL}(\mathfrak{g})$, which in turn is the
differential of the conjugation map $\gamma\colon x\mapsto\gamma_{x}$, where
$\gamma_{x}(y):=xyx^{-1}$, for every $x,y\in G$. In the case of Leibniz
algebras, there is no hope of finding such a map $\operatorname{Ad}$, but it
is still possible to define an algebraic structure $(X,\rhd,1)$, called a
_rack_ , whose operation, differentiated twice, defines on
$\operatorname{T}_{1}X$ a Leibniz algebra structure.

In this section, the underlying field of any vector space is
$\mathbb{F}=\mathbb{R}$.

######  Definition 3.1.

A _(left) rack_ is a set $X$ with a binary operation $\rhd\colon X\times
X\rightarrow X$ which is _(left) autodistributive_ ,

$x\rhd\left(y\rhd z\right)=\left(x\rhd y\right)\rhd\left(x\rhd
z\right),\;\;\forall x,y,z\in X$

and such that $x\rhd-\colon X\rightarrow X$ is a bijection for every $x\in X$.
A rack is _pointed_ if there exists an element $1\in X$, called the _unit_ ,
such that $1\rhd x=x$ and $x\rhd 1=1$, for every $x\in X$. A rack $(X,\rhd)$
is a _quandle_ if $x\rhd x=x$, for every $x\in X$ (i.e. $\rhd$ is idempotent).

Here we work with pointed racks.

######  Definition 3.2.

Let $(X,\rhd,1)$ and $(Y,\rhd_{Y},1_{Y})$ be pointed racks. A _pointed rack
homomorphism_ is a map $f\colon X\rightarrow Y$such that $f(1)=1_{Y}$ and
$f(x\rhd y)=f(x)\rhd f(y)$, for every $x,y\in X$.

######  Definition 3.3.

A non-empty subset $Z$ of a pointed rack $(X,\rhd,1)$ is a _subrack_ of $X$ if
$1\in Z$ and $x\rhd y\in Z$, for every $x,y\in Z$.

The basic example of a pointed rack is any group $G$ endowed with its
conjugation. We denote it by $\operatorname{Conj}(G)$. In this case the unit
element is the unit $1_{G}$ of the group $G$ and $\operatorname{Conj}(G)$ is a
quandle. More in general, it is defined a functor

$\operatorname{Conj}\colon\textbf{Grp}\rightarrow\textbf{PRack},$

where Grp is the category of groups and PRack is the category of pointed
racks, which has a left adjoint (see [22])

$\operatorname{As}_{p}\colon\textbf{PRack}\rightarrow\textbf{Grp}$

defined by

$\operatorname{As_{p}}(X)=\operatorname{As}(X)/\langle[1]\rangle,$

where

$\operatorname{As}(X)=\operatorname{F}(X)/\overline{\langle\\{xyx^{-1}\left(x\rhd
y^{-1}\right)\;|\;x,y\in X\\}\rangle},$

$\operatorname{F}(X)$ is the free group generated by $X$ and
$\langle[1]\rangle$ is the subgroup of $\operatorname{As}(X)$ generated by the
class $[1]$ of the unit element $1$ of the rack $X$.

######  Definition 3.4.

A Lie rack is a pointed rack $\left(X,\rhd,1\right)$ such that $X$ is a smooth
manifold, $\rhd$ is a smooth map and such that $x\rhd-$ is a diffeomorphism,
for every $x\in X$.

In [18] M. K. Kinyon showed that the tangent $\operatorname{T}_{1}X$ space at
the unit element of a Lie rack $(X,\rhd,1)$, endowed with the bracket

$[x,y]=\operatorname{ad}_{x}(y),$

where
$\operatorname{ad}\colon\operatorname{T}_{1}X\rightarrow\mathfrak{gl}(\operatorname{T}_{1}X)$
is the differential of the map
$\Phi:X\mapsto\operatorname{GL}(\operatorname{T}_{1}X)$,
$\Phi(x)=\operatorname{T}_{1}(x\rhd-)$, for every $x\in X$, is a Leibniz
algebra.

######  Proposition 3.5.

[18] Let $(X,\rhd,1)$ be a Lie rack. Then the above bracket
$[x,y]=\operatorname{ad}_{x}(y)$, defines on $\operatorname{T}_{1}X$ a Leibniz
algebra structure.

In other words, for every $x,y\in\operatorname{T}_{1}X$, we have

$\frac{\partial^{2}}{\partial s\partial
t}\bigg{|}_{s,t=0}\gamma_{1}(s)\rhd\gamma_{2}(t)=[x,y],$

where $\gamma_{1},\gamma_{2}\colon[0,1]\rightarrow X$ are smooth paths such
that $\gamma_{1}(0)=\gamma_{2}(0)=1$, $\gamma_{1}^{\prime}(0)=x$ and
$\gamma_{2}^{\prime}(0)=y$.

The converse problem, that is to find a manifold endowed with a smooth
operation such that the tangent space at the distinguished point, endowed with
the differential of this operation, gives a Leibniz algebra isomorphic to the
given one, is known as the coquecigrue problem. M. K. Kinyon solved the
_coquecigrue problem_ for the class of _split Leibniz algebras_ (see [18]).
More recently S. Covez in [9] showed how to integrate every Leibniz algebra
into a local Lie rack. The general coquecigrue problem is still open. In [20]
the authors showed that nilpotent real Leibniz algebras admit always a global
integration. They also integrated explicitly the indecomposable nilpotent
Leibniz algebras with one-dimensional commutator ideal classified in the
previous section.

Every left Leibniz algebra $\mathfrak{g}$ can be seen as an abelian extension
of a Lie subalgebra $\mathfrak{g}_{0}\subseteq\mathfrak{gl}(\mathfrak{g})$ by
a $\mathfrak{g}_{0}-$module $\mathfrak{a}$ (see [9]). For instance, one can
take $\mathfrak{a}=\operatorname{Z}_{l}(\mathfrak{g})$ and
$\mathfrak{g}_{0}=\operatorname{ad}(\mathfrak{g})\cong\mathfrak{g}/\operatorname{Z}_{l}(\mathfrak{g})$,
where
$\operatorname{ad}\colon\mathfrak{g}\rightarrow\operatorname{gl}(\mathfrak{g})$,
defined by $\operatorname{ad}_{x}=[x,-]$ for any $x\in\mathfrak{g}$, is the
_left adjoint map_. We associate with $\mathfrak{g}$ the short exact sequence

$0\rightarrow\mathfrak{a}\rightarrowtail\mathfrak{g}\twoheadrightarrow\mathfrak{g}_{0}\rightarrow
0.$

and we have a Leibniz algebras $2-$cocycle
$\omega\colon\mathfrak{g}_{0}\times\mathfrak{g}_{0}\rightarrow\mathfrak{a}$
such that $\mathfrak{g}=\mathfrak{g}_{0}\oplus_{\omega}\mathfrak{a}$. Moreover
the Leibniz algebra structure of $\mathfrak{g}$ is given by

$\left[\left(x,a\right),\left(y,b\right)\right]=\left(\left[x,y\right]_{\mathfrak{g}_{0}},\rho_{x}(b)+\omega(x,y)\right),$

where $\rho\colon\mathfrak{g}_{0}\times\mathfrak{a}\rightarrow\mathfrak{a}$ is
the action of $\mathfrak{g}_{0}$ on $\mathfrak{a}$.

######  Theorem 3.6.

[9] Every Leibniz algebra
$\mathfrak{g}=\mathfrak{g}_{0}\oplus_{\omega}\mathfrak{a}$ can be integrated
into a local Lie rack of the form

$G_{0}\times_{f}\mathfrak{a}$

with operation defined by

$(g,a)\rhd(h,b)=\left(ghg^{-1},\phi_{g}(b)+f(g,h)\right)$

and unit $(1,0)$, where $G_{0}$ is a Lie group such that
$\operatorname{Lie}(G_{0})=\mathfrak{g}_{0}$, $\phi$ is the exponentiation of
the action $\rho$, $f\colon G_{0}\times G_{0}\rightarrow\mathfrak{a}$ is the
Lie local racks $2-cocycle$ defined by

$f(g,h)=\int_{\gamma_{h}}\left(\int_{\gamma_{g}}\tau^{2}(\omega)^{eq}\right)^{eq},\;\;\forall
g,h\in G_{0}$

and
$\tau^{2}(\omega)\in\operatorname{ZL}^{1}(\mathfrak{g}_{0},Hom(\mathfrak{g}_{0},\mathfrak{a}))$
is defined by $\tau^{2}(\omega)(x)(y)=\omega(x,y)$, for every
$x,y\in\mathfrak{g}_{0}$.

In the case that a Lie rack integrating a Leibniz algebra $\mathfrak{g}$ is a
quandle, we have the following.

######  Theorem 3.7.

[20] Let $X$ be a Lie rack integrating a Leibniz algebra $\mathfrak{g}$. $X$
is a quandle if and only if $\mathfrak{g}$ is a Lie algebra. Moreover
$X=\operatorname{Conj}(G)$, where $\operatorname{Lie}(G)=\mathfrak{g}$ and
$\operatorname{Lie}\colon\mathbf{LieGrp}_{\mathbb{R}}\rightarrow\mathbf{LieGrp}_{\mathbb{R}}$
is the functor which sends every real Lie group $G$ to its Lie algebra
$\operatorname{Lie}(G)$.

For the class of nilpotent Leibniz algebras, the integration proposed by S.
Covez is global.

######  Theorem 3.8.

[20] Every nilpotent real Leibniz algebra $\mathfrak{g}$ has a global
integration into a Lie rack.

When $\mathfrak{g}$ is a two-step nilpotent Leibniz algebra, a Lie rack
integrating $\mathfrak{g}$ can be defined without integrating the Leibniz
algebras $2-$cocycle associated with $\mathfrak{g}$, since
$[\mathfrak{g},\mathfrak{g}]\subseteq\operatorname{Z}_{l}(\mathfrak{g})$.

######  Theorem 3.9.

[20] Let $\mathfrak{g}$ be a two-step nilpotent Leibniz algebra, let
$\mathfrak{g}_{0}=\mathfrak{g}/\left[\mathfrak{g},\mathfrak{g}\right]$ and let
$\omega\colon\mathfrak{g}_{0}\times\mathfrak{g}_{0}\rightarrow\left[\mathfrak{g},\mathfrak{g}\right]$
be the Leibniz algebras $2-$cocycle associated with the short exact sequence

$0\rightarrow\left[\mathfrak{g},\mathfrak{g}\right]\hookrightarrow\mathfrak{g}\twoheadrightarrow\mathfrak{g}_{0}\rightarrow
0.$

Then the multiplication

$\left(x,a\right)\rhd\left(y,b\right):=(y,b)+[(x,a),(y,b)]=\left(y,b+\omega(x,y)\right)$

defines a Lie rack structure on the vector space
$\mathfrak{g}_{0}\times\left[\mathfrak{g},\mathfrak{g}\right]$, such that
$(0,0)$ is the unit element and
$\operatorname{T}_{(0,0)}(\mathfrak{g}_{0}\times\left[\mathfrak{g},\mathfrak{g}\right],\rhd)$
is a Leibniz algebra isomorphic to $\mathfrak{g}$.

Now we recall how to integrate the indecomposable nilpotent real Leibniz
algebras with one-dimensional commutator ideal listed in the previous section.
We suppose that $A\in\operatorname{M}_{n}(\mathbb{R})$ is the companion matrix
of the power of an irreducible monic polynomial $f(x)\in\mathbb{R}[x]$. Thus
$A$ is similar to the $n\times n$ Jordan block of eigenvalue $a\in\mathbb{R}$
or $A$ is similar to $J_{R}$, where $R\in\operatorname{M}_{2}(\mathbb{R})$ is
the realification of some complex number $z=\alpha+i\beta$.

######  Example 3.10.

Let $\mathfrak{g}=\mathfrak{l}_{2n+1}^{A}$ and let
$\left\\{e_{1},\ldots,e_{n},f_{1},\ldots,f_{n},h\right\\}$ be a basis of
$\mathfrak{g}$. The Lie global rack integrating $\mathfrak{g}$ is
$R_{2n+1}^{A}=(\mathfrak{g}_{0}\times\mathbb{R}h,\rhd)$ with multiplication

$\displaystyle(x_{1},\ldots,x_{n},y_{1},\ldots,y_{n},z)\rhd(x_{1}^{\prime},\ldots,x_{n}^{\prime},y_{1}^{\prime},\ldots,y_{n}^{\prime},z^{\prime})=$
$\displaystyle=\left(x_{1}^{\prime},\ldots,x_{n}^{\prime},y_{1}^{\prime},\ldots,y_{n}^{\prime},z^{\prime}+\sum_{i,j=1}^{n}\left[\left(\delta_{ij}+a_{ij}\right)x_{i}y_{j}^{\prime}+\left(-\delta_{ij}+a_{ij}\right)x_{i}^{\prime}y_{j}\right]\right).$

######  Definition 3.11.

[20] $(R_{2n+1}^{A},\rhd)$ is called the _Heisenberg rack_.

When $A$ is the zero matrix, i.e. when $\mathfrak{g}=\mathfrak{h}_{2n+1}$ is
the $(2n+1)-$dimensional Heisenberg Lie algebra, we have that
$R_{2n+1}^{0_{n\times n}}=\operatorname{Conj}(H_{2n+1})$.

In the same way, we can find the Lie racks integrating the _Kronecker_ Leibniz
algebra and the _Dieudonné_ Leibniz algebra.

######  Definition 3.12.

[20] Let $\mathfrak{g}=\mathfrak{k}_{n}$ and let
$\left\\{e_{1},\ldots,e_{n},f_{1},\ldots,f_{n},h\right\\}$ be a basis of
$\mathfrak{g}$. The _Kronecker_ rack
$K_{n}=(\mathfrak{g}_{0}\times\mathbb{R}h,\rhd)$ is the Lie rack with
multiplication

$\displaystyle x\rhd y=y+[x,y]_{\mathfrak{k}_{n}},\;\;\forall x,y\in K_{n}.$

######  Definition 3.13.

[20] Let $\mathfrak{g}=\mathfrak{d}_{n}$ and let
$\left\\{e_{1},\ldots,e_{2n+1},h\right\\}$ be a basis of $\mathfrak{g}$. The
_Dieudonné_ rack $D_{n}=(\mathfrak{g}_{0}\times\mathbb{R}h,\rhd)$ is the Lie
rack with multiplication

$\displaystyle x\rhd y=y+[x,y]_{\mathfrak{d}_{n}},\;\;\forall x,y\in D_{n}.$

One can check that $\operatorname{T}_{(0,0)}K_{n}\cong\mathfrak{k}_{n}$ and
$\operatorname{T}_{(0,0)}D_{n}\cong\mathfrak{d}_{n}$.

##  4 Isotopisms

The concept of isotopism between two algebraic structures was introduced in
[1] by A. A. Albert (see also [2] and [17]) in order to classify non-
associative algebras by generalizing the notion of isomorphism. We define here
isotopisms for Leibniz algebras and racks.

######  Definition 4.1.

Let $(\mathfrak{g},[-,-])$ and $(\mathfrak{h},[-,-]_{\mathfrak{h}})$ be left
Leibniz algebras over $\mathbb{F}$. An _isotopism_ between $\mathfrak{g}$ and
$\mathfrak{h}$ is a triple of linear isomorphisms

$(f,g,h)\colon\mathfrak{g}\leftrightarrows\mathfrak{h}$

such that

$[f(x),g(y)]_{\mathfrak{h}}=h([x,y]),\;\;\forall x,y\in\mathfrak{g}.$

We say that $\mathfrak{g}$ and $\mathfrak{h}$ are _isotopic_ Leibniz algebras.

An isotopism is said to be a _right isotopism_ if $f=h$ and a _left isotopism_
if $g=h$.

We observe that, if $f\colon\mathfrak{g}\leftrightarrows\mathfrak{h}$ is an
isomorphism of Leibniz algebras, then the triple $(f,f,f)$ is a (left and
right) isotopism.

######  Definition 4.2.

Let $(X,\rhd,1)$ and $(Y,\rhd_{Y},1_{Y})$ be left pointed racks. An
_isotopism_ between $X$ and $Y$ is a triple of bijections

$(f,g,h)\colon X\leftrightarrows Y$

such that $f(1)=g(1)=h(1)=1_{Y}$ and

$f(x)\rhd_{Y}g(y)=h(x\rhd y),\;\;\forall x,y\in X.$

Also in this case, every isomorphism of racks can be seen as a left and right
isotopism. Moreover, one can define the notion of _Lie rack isotopism_ by
asking that $f,g,h$ are diffeomorphisms.

The following is the generalization of a result proved by A. A. Albert in [2]
for isotopisms of _quasigroups_.

######  Remark 4.3.

Every isotopism $(f,g,h)$ between two algebraic structures is isomorphic to a
_principal isotopism_ , i.e. an isotopism of the form
$(\tilde{f},\tilde{g},\operatorname{id})$. To be more precise, let $(A,\cdot)$
and $(B,\circ)$ be two algebraic structures and let $(f,g,h)\colon
A\leftrightarrows B$ be an isotopism. We define a new binary operation
$*\colon B\times B\rightarrow B$ by

$h(x)*h(y)=h(xy),\;\;\forall x,y\in A.$

In this way $(A,\cdot)$ and $(B,*)$ are isomorphic and $h(x)*h(y)=f(x)\circ
g(y)$. Thus $(hf^{-1})(\bar{x})*(hg^{-1})(\bar{y})=\bar{x}\circ\bar{y}$, where
$\bar{x}=f(x)$, $\bar{y}=g(y)$ and we obtain the principal isotopism
$(fh^{-1},gh^{-1},\operatorname{id}_{B})\colon(B,\circ)\leftrightarrows(B,*)$.

${(B,\circ)}$${(B,*)}$${(A,\cdot)}$$\scriptstyle{(fh^{-1},gh^{-1},\operatorname{id}_{B})}$$\scriptstyle{(f,g,h)}$$\scriptstyle{h}$

In the case that $(A,\cdot)$ and $(B,\circ)$ are finite dimensional non-
associative algebras over a field $\mathbb{F}$, such as Leibniz algebras, we
can identify the underlying vector space of $A$ and $B$ is $\mathbb{F}^{n}$,
where $n=\dim_{\mathbb{F}}A=\dim_{\mathbb{F}}B$, and
$\operatorname{id}_{B}=\operatorname{id}_{\mathbb{F}^{n}}$.

From Definition 4.1, it turns out that a difference between isomorphisms and
isotopisms is that one can find a Lie algebra that is isotopic to a Leibniz
non-Lie algebra, as the following example shows.

######  Example 4.4.

Let $\mathfrak{h}_{3}$ be the three-dimensional Heisenberg Lie algebra, let
$A=(a_{ij})_{i,j}\in\operatorname{GL}_{2}(\mathbb{F})$,
$B=(b_{ij})_{i,j}=A\cdot\operatorname{diag}\\{\lambda,\mu\\}$, with
$\lambda,\mu\in\mathbb{F}^{*}$ and let $\mathfrak{g}_{\lambda,\mu}^{A}$ be the
nilpotent Leibniz algebra with one-dimensional commutator ideal with structure
matrix

$\begin{pmatrix}0&a_{11}b_{22}-a_{21}b_{12}&0\\\
a_{12}b_{21}-a_{22}b_{11}&0&0\\\ 0&0&0\\\ \end{pmatrix}.$

Then the triple of linear isomoprhisms
$(f,g,\operatorname{id}_{\mathbb{F}^{3}})\colon\mathfrak{g}_{\lambda,\mu}^{A}\leftrightarrows\mathfrak{h}_{3}$,
where $f$ and $g$ are defined respectively by the matrices

$\begin{pmatrix}a_{11}&a_{21}&0\\\ a_{12}&a_{22}&0\\\ 0&0&\alpha\\\
\end{pmatrix}\text{ and }\begin{pmatrix}b_{11}&b_{21}&0\\\ b_{12}&b_{22}&0\\\
0&0&\beta\\\ \end{pmatrix},\text{ with }\alpha,\beta\in\mathbb{F}^{\ast},$

is a principal isotopism. Indeed, for every
$x\equiv(x_{1},x_{2},x_{3}),y\equiv(y_{1},y_{2},y_{3})\in\mathbb{F}^{3}$, we
have

$\displaystyle\left[f(x),g(y)\right]_{\mathfrak{h}_{3}}=$
$\displaystyle(0,0,(a_{11}x_{1}+a_{12}x_{2})(b_{21}y_{1}+b_{22}y_{2})-(a_{21}x_{1}+a_{22}x_{2})(b_{11}y_{1}+b_{12}y_{2}))=$
$\displaystyle=$
$\displaystyle(0,0,(a_{11}b_{22}-a_{21}b_{12})x_{1}y_{2}-(a_{12}b_{21}-a_{22}b_{11})x_{2}y_{1})=\left[x,y\right]_{\mathfrak{g}_{\lambda,\mu}^{A}},$

since $a_{11}b_{21}-b_{11}a_{21}=a_{12}b_{22}-b_{12}a_{22}=0$.

One can check that, if $\lambda=\dfrac{2}{\det(A)}-\mu$, then
$\mathfrak{g}_{\lambda,\mu}^{A}=\mathfrak{l}_{3}^{\alpha}$, where
$\alpha=\mu\det(A)-1$. Indeed $\mathfrak{g}_{\lambda,\mu}^{A}$ and
$\mathfrak{l}_{3}^{\alpha}$ have the same structure matrix, that is

$\begin{pmatrix}0&\mu\det(A)&0\\\ -\lambda\det(A)&0&0\\\
0&0&0\end{pmatrix}=\begin{pmatrix}0&1+\alpha&0\\\ -1+\alpha&0&0\\\
0&0&0\end{pmatrix}.$

Conversely, for every $\alpha\in\mathbb{F}^{\ast}\setminus\\{\pm 1\\}$, if we
choose $A=\operatorname{I}_{2}$, $\lambda=1-\alpha$ and $\mu=1+\alpha$, then
$\mathfrak{g}_{\lambda,\mu}^{\operatorname{I}_{2}}=\mathfrak{l}_{3}^{\alpha}$.
Thus $\mathfrak{h}_{3}$ and $\mathfrak{l}_{3}^{\alpha}$ are isotopic for every
$\alpha\in\mathbb{F}$, $\alpha\neq\pm 1$. We note that this result is not true
if $\alpha=\pm 1$, in fact in the next section we will show that
$\mathfrak{l}_{3}^{\pm 1}$ and $\mathfrak{h}_{3}$ are not isotopic.

In a similar way, if $\mu=-\lambda=\dfrac{1}{\operatorname{det}(A)}$, then the
structure matrix of $\mathfrak{g}_{\lambda,\mu}^{A}$ is

$\begin{pmatrix}0&1&0\\\ 1&0&0\\\ 0&0&0\\\ \end{pmatrix},$

i.e. $\mathfrak{g}_{\lambda,\mu}^{A}=\mathfrak{k}_{1}$ is the Kronecker
Leibniz algebra. Thus $\mathfrak{l}_{3}^{\alpha}$ is isotopic to
$\mathfrak{k}_{1}$, for every $\alpha\neq\pm 1$.

##  5 Isotopisms of two-step nilpotent Leibniz algebras and Lie racks

The notion of isotopism was used in [15] and [16] for classifying some
families of Lie algebras, such as _filiform Lie algebras_ over finite fields.
The authors also determined some isotopism invariants for these algebras.

The main goal of this section is to investigate the isotopism classes of
indecomposable nilpotent Leibniz algebras with one-dimensional commutator
ideal over a field $\mathbb{F}$, with $\operatorname{char}(\mathbb{F})\neq 2$.
Moreover we will show that, when $\mathbb{F}=\mathbb{R}$, two Leibniz algebras
of this type are isotopic if and only if the Lie racks integrating them are
isotopic. For doing this, we need to find new isotopism invariants for Leibniz
algebras and Lie racks.

We start with the following definition, which was given by M. Elhamdadi and E.
M. Moutuou in [13].

######  Definition 5.1.

Let $X$ be a left rack. The _center_ of $X$ is

$\operatorname{Z}(X)=\\{x\in X\;|\;x\rhd y=y,\;\forall y\in X\\}.$

We observe that $\operatorname{Z}(X)$ is a _subrack_ of $X$ since $1\rhd y=y$
and

$(x\rhd x^{\prime})\rhd y=(x\rhd x^{\prime})\rhd(x\rhd y)=x\rhd(x^{\prime}\rhd
y)=x\rhd y=y,$

for every $x,x^{\prime}\in\operatorname{Z}(X)$ and for every $y\in X$. In [13]
the center of $X$ is called the _stabilizer_ of $X$ and it is denoted by
$\operatorname{Stab}(X)$. We refer to it as the center for the following
reasons:

  * •

If $G$ is a group, then
$\operatorname{Z}(\operatorname{Conj}(G))=\operatorname{Z}(G)$;

  * •

If $X$ is a Lie rack and $\mathfrak{g}=\operatorname{T}_{1}X$, then
$\operatorname{T}_{1}\operatorname{Z}(X)=\operatorname{Z}_{l}(\mathfrak{g})$.

######  Proposition 5.2.

  * (i)

Let $\mathfrak{g},\mathfrak{h}$ be Leibniz algebras and let
$(f,g,h)\colon\mathfrak{g}\leftrightarrows\mathfrak{h}$ be an isotopism. Then

$f(\operatorname{Z}_{l}(\mathfrak{g}))=\operatorname{Z}_{l}(\mathfrak{h}),\;\;g(\operatorname{Z}_{r}(\mathfrak{g}))=\operatorname{Z}_{r}(\mathfrak{h}),\;\;h([\mathfrak{g},\mathfrak{g}])=[\mathfrak{h},\mathfrak{h}]$

and the dimensions of the left center, of the right center and of the
commutator ideal are _isotopism invariants_.

  * (ii)

Let $X,Y$ be left racks and let $(f,g,g)\colon X\leftrightarrows Y$ be a _left
isotopism_. Then

$f(\operatorname{Z}(X))=\operatorname{Z}(Y).$

In the case of Lie racks, the dimension of the center is a _left isotopism
inviariant_.

###### Proof.

  * (i)

Let $x,y\in\mathfrak{h}$ such that $x=f(\bar{x})$, $y=g(\bar{y})$, with
$\bar{x},\bar{y}\in\mathfrak{g}$. If
$\bar{x}\in\operatorname{Z}_{l}(\mathfrak{g})$, then

$[x,y]_{\mathfrak{h}}=[f(\bar{x}),g(\bar{y})]_{\mathfrak{h}}=h([\bar{x},\bar{y}])=h(0)=0$

and $x\in\operatorname{Z}_{l}(\mathfrak{h})$. Conversely, if
$x\in\operatorname{Z}_{l}(\mathfrak{h}$), then

$0=[x,y]_{\mathfrak{h}}=[f(\bar{x}),g(\bar{y})]_{\mathfrak{h}}=h([x,y]).$

Thus $[\bar{x},\bar{y}]\in\operatorname{Ker}(h)=0$, i.e.
$\bar{x}\in\operatorname{Z}_{l}(\mathfrak{g})$ and
$f(\operatorname{Z}_{l}(\mathfrak{g}))=\operatorname{Z}_{l}(\mathfrak{h})$. In
a similar way, one can check that
$g(\operatorname{Z}_{r}(\mathfrak{g}))=\operatorname{Z}_{r}(\mathfrak{h})$.
Finally, for every $\bar{x},\bar{y}\in\mathfrak{g}$

$h([\bar{x},\bar{y}])=[f(\bar{x}),g(\bar{y})]_{\mathfrak{h}}\in[\mathfrak{h},\mathfrak{h}].$

Coversely, for any $x,y\in\mathfrak{h}$, with $x=f(\bar{x})$ and
$y=g(\bar{y})$, we have

$[x,y]_{\mathfrak{h}}=h([\bar{x},\bar{y}])\in h([\mathfrak{g},\mathfrak{g}])$

and $h([\mathfrak{g},\mathfrak{g}])=[\mathfrak{h},\mathfrak{h}]$.

  * (ii)

Let $x,y\in Y$ such that $x=f(\bar{x})$, $y=g(\bar{y})$, with
$\bar{x},\bar{y}\in X$. If $\bar{x}\in\operatorname{Z}(X)$, then

$x\rhd y=f(\bar{x})\rhd_{Y}g(\bar{y})=g(\bar{x}\rhd\bar{y})=g(\bar{y})=y$

and $x\in\operatorname{Z}(Y)$. Conversely, if $x\in\operatorname{Z}(Y)$, then

$g(\bar{y})=y=x\rhd_{Y}y=f(\bar{x})\rhd_{Y}g(\bar{y})=g(\bar{x}\rhd\bar{y})$

and the statement is proved, since $g$ is injective.

∎

The last proposition allows us to conclude that the algebras
$\mathfrak{l}_{3}^{\pm 1}$ and $\mathfrak{h}_{3}$ are not isotopic, since
$\dim_{\mathbb{F}}\operatorname{Z}_{l}(\mathfrak{l}_{3}^{1})=2$ (see Section
2.1) and $\dim_{\mathbb{F}}\operatorname{Z}(\mathfrak{h}_{3})=1$. Moreover we
recall that $\mathfrak{l}_{3}^{1}$ and $\mathfrak{l}_{3}^{-1}$ are isomorphic,
and $\mathfrak{h}_{3}$ is isotopic to $\mathfrak{l}_{3}^{\alpha}$, for every
$\alpha\neq\pm 1$, thus $\mathfrak{l}_{3}^{\alpha}$ and $\mathfrak{l}_{3}^{\pm
1}$ are not isotopic. This remark, combined with the one we illustrated at the
end of the previous section, leeds to the following.

######  Proposition 5.3.

Let $\mathfrak{g}$ be a three-dimensional indecomposable nilpotent Leibniz
algebra with one-dimensional commutator ideal over a field $\mathbb{F}$, with
$\operatorname{char}(\mathbb{F})\neq 2$. Then either $\mathfrak{g}$ is
isomorphic to the Heisenberg Leibniz algebra $\mathfrak{l}_{3}^{1}$, or
$\mathfrak{g}$ is isotopic to the Heisenberg Lie algebra $\mathfrak{h}_{3}$.

We want now to extend this result to the case that
$\dim_{\mathbb{F}}\mathfrak{g}>3$.

######  Theorem 5.4.

Let $\mathbb{F}$ be a field with $\operatorname{char}(\mathbb{F})\neq 2$ and
let $\mathfrak{g}$ be a nilpotent Leibniz algebra with
$\dim_{\mathbb{F}}\mathfrak{g}=t\geq 3$ and
$\dim_{\mathbb{F}}[\mathfrak{g},\mathfrak{g}]=1$.

  * (i)

If $t=2n+2$, then $\mathfrak{g}$ is isomorphic to the _Dieudonné_ Leibniz
algebra $\mathfrak{d}_{n}$;

  * (ii)

If $t=2n+1$, then either $\mathfrak{g}$ is isomorphic to the Heisenberg
Leibniz algebra $\mathfrak{l}_{2n+1}^{J_{1}}$, where $J_{1}$ is the $n\times
n$ Jordan block of eigenvalue $1$, or $\mathfrak{g}$ is isotopic to the
Heisenberg Lie algebra $\mathfrak{h}_{2n+1}$;

  * (iii)

The Heisenberg Leibniz algebras $\mathfrak{l}_{2n+1}^{J_{1}}$ and
$\mathfrak{l}_{2n+1}^{J_{-1}}$ are not isotopic to any Lie algebra.

###### Proof.

  * (i)

If $\dim_{\mathbb{F}}\mathfrak{g}$ is even, then $\mathfrak{g}$ must be
isomorphic to the Dieudonné algebra $\mathfrak{d}_{n}$, where $t=2n+2$;

  * (ii)

If $\dim_{\mathbb{F}}\mathfrak{g}=2n+1$ and $\mathfrak{g}$ is not isomorphic
to $\mathfrak{l}_{2n+1}^{J_{1}}$, then $\mathfrak{g}\cong\mathfrak{k}_{n}$ or
$\mathfrak{g}\cong\mathfrak{l}_{2n+1}^{A}$, where $A$ is the companion matrix
of the power of an irreducible monic polynomial $p(x)^{k}\in\mathbb{F}[x]$ and
it is not similar to the Jordan blocks $J_{1}$ adn $J_{-1}$. We want to show
that both these Leibniz algebras are isotopic to the Heisenberg Lie algebra
$\mathfrak{h}_{2n+1}$. The Leibniz algebra $\mathfrak{k}_{n}$ is isotopic to
the $(2n+1)$-dimensional Heisenberg Lie algebra via the left principal
isotopism
$(f,\operatorname{id}_{\mathbb{F}^{t}},\operatorname{id}_{\mathbb{F}^{t}})$,
where

$\displaystyle f(x_{1},\ldots,x_{n},y_{1},\ldots,y_{n},z)=$
$\displaystyle=(x_{1}+x_{2},x_{2}+x_{3},\ldots,x_{n-1}+x_{n},x_{n},-y_{1},y_{1}-y_{2},y_{2}-y_{3},\ldots,y_{n-1}-y_{n},z),$

for every $(x_{1},\ldots,x_{n},y_{1},\ldots,y_{n},z)\in\mathbb{F}^{t}$. The
Heisenberg Leibniz algebra $\mathfrak{l}_{2n+1}^{A}$ is isotopic to
$\mathfrak{h}_{2n+1}$ via the triple
$(f,\operatorname{id}_{\mathbb{F}^{t}},\operatorname{id}_{\mathbb{F}^{t}})$,
where $f$ is the linear isomorphism defined by the matrix

$\begin{pmatrix}&&&\vline&&&&\vline&0\\\ &I_{n}+A&&\vline&&\mbox{\Large
0}&&\vline&\vdots\\\ &&&\vline&&&&\vline&0\\\
\hline\cr&&&\vline&&&&\vline&0\\\ &\mbox{\Large
0}&&\vline&&I_{n}-A^{T}&&\vline&\vdots\\\ &&&\vline&&&&\vline&0\\\ \hline\cr
0&\cdots\cdots&0&\vline&0&\cdots\cdots&0&\vline&1\end{pmatrix}$

We observe that $f$ is a bijection since the matrices $I_{n}+A$ and
$I_{n}-A^{T}$ are invertible. Indeed one has

$\operatorname{det}(xI_{n}+A)=(-1)^{n}f(x),$
$\operatorname{det}(xI_{n}-A^{T})=\operatorname{det}(xI_{n}-A)=f(x)$

and for $x=1$

$\operatorname{det}(I_{n}+A)=(-1)^{n}f(-1),$
$\operatorname{det}(I_{n}-A^{T})=f(1).$

Thus $f$ is not invertible if and only if $p(1)=0$ or $p(-1)=0$. Since $p(x)$
is irreducible over $\mathbb{F}$, this would imply that $p(x)=x-1$ or
$p(x)=x+1$ and $A$ would be similar to the $n\times n$ Jordan block $J_{1}$ or
$J_{-1}$. We have a contradiction since we supposed that $\mathfrak{g}$ is not
isomorphic to the Heisenberg Leibniz algebra $\mathfrak{l}_{2n+1}^{J_{1}}$.
Finally, by Proposition 5.2, we have that $\mathfrak{l}_{2n+1}^{J_{1}}$ and
$\mathfrak{l}_{2n+1}^{J_{-1}}$ are not isotopic to $\mathfrak{h}_{2n+1}$ since

$\dim_{\mathbb{F}}\operatorname{Z}_{l}(\mathfrak{l}_{2n+1}^{J_{\pm 1}})=2,$
$\dim_{\mathbb{F}}\operatorname{Z}(\mathfrak{h}_{2n+1})=1$

and the dimension of the left center is an isotopism invariant.

  * (iii)

If we suppose that there exists a Lie algebra $\mathfrak{p}$ and an isotopism

$(f,g,h)\colon\mathfrak{l}_{2n+1}^{J_{1}}\leftrightarrows\mathfrak{p},$

then, by Proposition 5.2, $\dim_{\mathbb{F}}[\mathfrak{p},\mathfrak{p}]=1$ and
$\dim_{\mathbb{F}}\operatorname{Z}(\mathfrak{p})=2$. For the classification of
Lie algebras with one-dimensional commutator ideal (see Section 3 of [14]),
$\mathfrak{p}=\mathfrak{p}_{1}\oplus\mathfrak{p}_{2}$, where
$\mathfrak{p}_{1}$ is an abelian Lie algebra and $\mathfrak{p}_{2}$ is either
the non-abelian Lie algebra of dimension $2$ (i.e. $\mathfrak{p}_{2}$ has
basis $\\{e_{1},e_{2}\\}$ and the the Lie bracket is defined by
$[e_{1},e_{2}]=e_{1}$), or it is an Heisenberg algebra $\mathfrak{h}_{2k+1}$.
In the first case, $\dim_{\mathbb{F}}\mathfrak{p}_{1}$ must be odd. Thus
$\dim_{\mathbb{F}}\operatorname{Z}(\mathfrak{p})$ is also odd, since
$\operatorname{Z}(\mathfrak{p})=\mathfrak{p}_{1}$. In the second one,
$\dim_{\mathbb{F}}\mathfrak{p}_{1}$ must be even and

$\dim_{\mathbb{F}}\operatorname{Z}(\mathfrak{p})=\dim_{\mathbb{F}}\mathfrak{p}_{1}+\dim_{\mathbb{F}}\operatorname{Z}(\mathfrak{h}_{2k+1})=\dim_{\mathbb{F}}\mathfrak{p}_{1}+1$

is odd. Hence, in both cases we have a contradiction and
$\mathfrak{l}_{2n+1}^{J_{1}}$ cannot be isotopic to a Lie algebra. The
statement is also valid for the Leibniz algebra
$\mathfrak{l}_{2n+1}^{J_{-1}}$, since it is isomorpich to
$\mathfrak{l}_{2n+1}^{J_{1}}$.

∎

In the last theorem we showed that an isotopism between two nilpotent Leibniz
algebras with one-dimensional commutator ideal $\mathfrak{g},\mathfrak{h}$ can
be always chosen of the form
$(f,\operatorname{id}_{\mathbb{F}^{t}},\operatorname{id}_{\mathbb{F}^{t}})$.
In the next lemma we show that, if $\mathbb{F}$ is the field of real numbers,
such an isotopism turns out to be also an isotopism between the Lie racks
integrating $\mathfrak{g}$ and $\mathfrak{h}$. We recall that, by Theorem 3.9,
we can identify the underlying vector space of a Lie rack $X$ integrating a
two step nilpotent Leibniz algebra, with its tangent space
$\operatorname{T}_{1}X$.

######  Lemma 5.5.

Let $\mathbb{F}=\mathbb{R}$ and let $\mathfrak{g},\mathfrak{h}$ be two-step
nilpotent Leibniz algebras with
$\dim_{\mathbb{R}}\mathfrak{g}=\dim_{\mathbb{R}}\mathfrak{h}=t$. Let $X,Y$ be
the Lie racks integrating $\mathfrak{g}$ and $\mathfrak{h}$ respectively.

  * (i)

If $(f,g,g)\colon\mathfrak{g}\leftrightarrows\mathfrak{h}$ is a left Leibniz
algebra isotopism, then it is also a Lie rack isotopism between $X$ and $Y$;

  * (ii)

If $(f,g,h)\colon X\leftrightarrows Y$ is a Lie rack isotopism, then $g=h$ and
it turns out to be a Leibniz algebra isotopism between $\mathfrak{g}$ and
$\mathfrak{h}$;

  * (iii)

Let
$\dim_{\mathbb{R}}[\mathfrak{g},\mathfrak{g}]=\dim_{\mathbb{R}}[\mathfrak{h},\mathfrak{h}]=1$.
Then $\mathfrak{g}$ is isotopic to $\mathfrak{h}$ if and only if $X$ is
isotopic to $Y$.

###### Proof.

The multiplications of the Lie racks $X$ and $Y$ can be written as

$x\rhd
y=y+[x,y]_{\mathfrak{g}},\;\;x^{\prime}\rhd_{Y}y^{\prime}=y^{\prime}+[x^{\prime},y^{\prime}]_{\mathfrak{h}},\;\;\forall
x,y\in X,\;\forall x^{\prime},y^{\prime}\in Y.$

  * (i)

If $(f,g,g)\colon\mathfrak{g}\leftrightarrows\mathfrak{h}$ is an isotopism of
Leibniz algebra, then

$f(x)\rhd_{Y}g(y)=g(y)+[f(x),g(y)]_{\mathfrak{h}}=g(y)+g([x,y])=g(x\rhd y),$

for every $x,y\in X$;

  * (ii)

If $(f,g,h)\colon X\leftrightarrows Y$ is an isotopism of Lie racks, then

$g(y)=1_{Y}\rhd_{Y}g(y)=f(1_{X})\rhd_{Y}g(y)=h(1_{X}\rhd y)=h(y),\;\;\forall
y\in X,$

thus $g=h$ and the triple $(f,g,g)$ becomes an isotopism between
$\mathfrak{g}$ and $\mathfrak{h}$, since

$g(y)+[f(x),g(y)]_{\mathfrak{h}}=f(x)\rhd_{Y}g(y)=g(x\rhd y)=g(y)+g([x,y]),$

for every $x,y\in X$, and then

$[f(x),g(y)]_{\mathfrak{h}}=g([x,y]).$

  * (iii)

If $X$ and $Y$ are isotopic Lie racks, then from (ii) $\mathfrak{g}$ and
$\mathfrak{h}$ are isotopic Leibniz algebra. Conversely, if $\mathfrak{g}$ and
$\mathfrak{h}$ are isotopic nilpotent Leibniz algebras with one-dimensional
commutator ideal, then by Theorem 5.4, an isotopism between them can be chosen
of the form
$(f,\operatorname{id}_{\mathbb{F}^{t}},\operatorname{id}_{\mathbb{F}^{t}})$
and, from (ii), it becomes an isotopism between $X$ and $Y$.

∎

The last result allows us to describe the isotopism classes of Lie global
racks integrating the indecomposable nilpotent Leibniz algebras with one-
dimensional commutator ideal.

######  Theorem 5.6.

Let $\mathbb{F}=\mathbb{R}$ and let $X$ be a Lie rack integrating a nilpotent
Leibniz algebra $\mathfrak{g}$ with $\dim_{\mathbb{F}}\mathfrak{g}=t$ and
$\dim_{\mathbb{F}}[\mathfrak{g},\mathfrak{g}]=1$.

  * (i)

If $t=2n+2$, then $X$ is isomorphic to the _Dieudonné_ rack $D_{n}$;

  * (ii)

If $t=2n+1$, then either $X$ is isomorphic to the Heisenberg rack
$R_{2n+1}^{J_{1}}$, where $J_{1}$ is the $n\times n$ Jordan block of
eigenvalue $1$, or $X$ is isotopic to the conjugation of the Heisenberg Lie
group $\operatorname{Conj}(H_{2n+1})$;

  * (iii)

The Heisenberg racks $R_{2n+1}^{J_{1}}$ and $R_{2n+1}^{J_{-1}}$ are not
isotopic to any Lie quandle.

###### Proof.

  * (i)

If $t$ is even, then $\mathfrak{g}\cong\mathfrak{d}_{n}$ and $X$ is isomorphic
to the Dieudonné rack $D_{n}$;

  * (ii)

If $t=2n+1$ and $\mathfrak{g}\cong\mathfrak{l}_{2n+1}^{J_{1}}$, then $X$ is
isomorphic to the Heisenberg rack $R_{2n+1}^{J_{1}}$. If this is not the case,
then $\mathfrak{g}$ is isotopic to $\mathfrak{h}_{2n+1}$ and $X$ is isomorphic
either to the Kronecker rack $K_{n}$, or to $R_{2n+1}^{A}$, where
$A=J_{\alpha}$ with $\alpha\in\mathbb{R}\setminus\\{\pm 1\\}$, or $A=J_{R}$
and $R=R_{\alpha,\beta}$ as in Section 2.2. Note that the last case is
admissible if and only if $n$ in even. Since both $\mathfrak{k}_{n}$ and
$\mathfrak{l}_{2n+1}^{A}$ are isotopic to the Heisenberg Lie algebra
$\mathfrak{h}_{2n+1}$ with a left principal isotopism
$(f,\operatorname{id}_{\mathbb{F}^{t}},\operatorname{id}_{\mathbb{F}^{t}})$,
by Lemma 5.5 we can conclude that both $K_{n}$ and $R_{2n+1}^{A}$ are isotopic
to $\operatorname{Conj}(H_{2n+1})$. Thus $X$ is isotopic to the conjugation of
the Heisenberg Lie group. Finally $R_{2n+1}^{J_{1}}$ and
$\operatorname{Conj}(H_{2n+1})$ are not isotopic. Indeed, if we suppose that
there exists an isotopism

$(f,g,h)\colon R_{2n+1}^{J_{1}}\leftrightarrows\operatorname{Conj}(H_{2n+1}),$

then, by Lemma 5.5, $g=h$ and we obtain a contradiction, since $(f,g,g)$ would
become an isotopism between the Leibniz algebras
$\mathfrak{l}_{2n+1}^{J_{\\`{\i}1}}$ and $\mathfrak{h}_{2n+1}$. Another way to
show that $R_{2n+1}^{J_{1}}$ and $\operatorname{Conj}(H_{2n+1})$ are not
isotopic racks is to see that

$\operatorname{Z}(R_{2n+1}^{J_{1}})=\\{(0,y,z)\;|\;y,z\in\mathbb{R}\\},$
$\operatorname{Z}(\operatorname{Conj}(H_{2n+1}))=\\{(0,0,z)\;|\;z\in\mathbb{R}\\},$

where we use the same notation of Example 3.10, and we proved in Proposition
5.2 that the dimension of the center of a Lie rack is a left isotopism
invariant.

  * (iii)

If $R_{2n+1}^{J_{1}}$ were isotopic to a Lie quandle $X$, then
$\mathfrak{l}_{2n+1}^{J_{1}}$ would be isotopic to the Leibniz algebra
$\mathfrak{g}=\operatorname{T}_{1}X$. By Theorem 3.7, $\mathfrak{g}$ is a Lie
algebra and $X=\operatorname{Conj}(G)$, where
$\operatorname{Lie}(G)=\mathfrak{g}$. Thus $\mathfrak{l}_{2n+1}^{J_{1}}$ would
be isotopic to a Lie algebra and, by (iii) of Theorem 5.4, this is a
contradiction. The same argument can be used for the Lie rack
$R_{2n+1}^{J_{-1}}$.

∎

In the last section we saw that, though the problem of finding isomorphism
classes of the complex Heisenberg Leibniz algebras
$\mathfrak{l}_{2n+1}^{J_{a}}$ is still open, the notion of isotopism allow us
to classify all the possible non-isotopic nilpotent Leibniz algebras with one-
dimensional commutator ideal. Moreover, this induces also a classification of
the non-isotopic Lie racks integrating such class of nilpotent Leibniz
algebras.

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# Neural Laplace for learning Stochastic Differential Equations

Adrien Carrel

###### Abstract

Neural Laplace is a unified framework for learning diverse classes of
differential equations (DE). For different classes of DE, this framework
outperforms other approaches relying on neural networks that aim to learn
classes of ordinary differential equations (ODE). However, many systems can’t
be modelled using ODEs. Stochastic differential equations (SDE) are the
mathematical tool of choice when modelling spatiotemporal DE-dynamics under
the influence of randomness. In this work, we review the potential
applications of Neural Laplace to learn diverse classes of SDE, both from a
theoretical and a practical point of view.

Machine Learning, ICML

##  1 Introduction

Differential equations (DE) are important in many domains as they can model a
wide variety of real-world situations. However, given a set of points sampled
from a DE, it is difficult to predict the future trajectory without knowing
the DE and its parameters. Many methods relying on neural networks have been
developed to learn different classes of DE. Most of them have only been
designed to predict Ordinary differential equations (ODE).

More recently, (Holt et al., 2022) developed a powerful framework based on the
Laplace Transform for learning many classes of DE. However, this work has only
been applied to certain types of DE. In this paper, we expand the scope of
this method to another major class of DE, stochastic differential equations
(SDE), to provide a more complete picture of this field. SDEs are used mostly
in finance but may be applied in the foreseeable future in many other domains
where randomness is part of the system, such as healthcare (Breda & Canci,
2023). In this paper, we investigate, in particular, the Geometric Brownian
Motion.

Some applications of neural networks to learn SDE have been developed, like
Neural SDE (Yang et al., 2022; Salvi et al., 2022). However, they have been
created to only solve this particular class of DE and may not be able to learn
diverse classes of DE like Neural Laplace. Thus, in this paper, we focus on
the potential applications of Neural Laplace to SDE to further confirm its
unified framework status.

##  2 Problem formalism

###  2.1 Goal

After introducing some definitions from the Neural Laplace paper, we extend
the Laplace Transform to stochastic processes. Then, we take a look at the
well-known Geometric Brownian Motion and we show that the expectancy of its
Laplace Transform can be calculated and that the variance of the norm of the
Laplace Transform can be bounded. Therefore, under certain conditions, the
norm of the Laplace Transform can be approximated with low uncertainty. As the
imaginary part is driven by the imaginary part of the input (which is known),
we can consider that the Laplace Transform is almost known and, therefore,
that Neural Laplace can provide correct estimations after learning this class
of DE.

###  2.2 General definitions

Notation. For a system with $D\in\mathbb{N}^{*}$ dimensions, we note for all
$d\in\llbracket 1,D\rrbracket$ and for all $t\in\mathbb{R}$, the state of
dimension $d$ at time $t$: $x_{d}(t)$. We define the function of time
$x_{d}:\mathbb{R}\longrightarrow\mathbb{R}$, which is trajectory along the
dimension $d$. $x_{d}(t)$ is the point on this trajectory at time $t$. Let
$\textbf{x}(t)=\left[x_{1}(t),\ldots,x_{D}(t)\right]^{T}\in\mathbb{R}^{D}$ be
the state vector at time $t$ and $\textbf{x}=\left[x_{1},\ldots,x_{D}\right]$
be the vector-valued trajectory. When conducting experiments later, we will
consider a discrete set of observations $\mathcal{T}=\\{t_{1},\ldots,T\\}$ to
evaluate the state observations where $T=t_{N}$ and $N\in\mathbb{N}^{*}$ is
the number of observations.

Laplace Transform. The Laplace transform of trajectory x is defined as:

$\textbf{F}(\textbf{s})=\mathcal{L}\\{\textbf{x}\\}(s)=\int_{0}^{\infty}e^{-\textbf{s}t}\textbf{x}(t)dt$
(1)

where $\textbf{s},\textbf{F}(\textbf{s})\in\mathbb{C}^{D}$.
$\textbf{F}(\textbf{s})$, also called the Laplace representation, may have
singularities ($\|\textbf{F}(\textbf{s})\|\rightarrow+\infty$ for some
$\textbf{s}\in\mathbb{C}^{D}$).

Stochastic Differential Equations. A SDE is a differential equation in which
one or more of the terms is a stochastic process. The solution to this type of
equation is also a stochastic process.

Stochastic process. Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability
space and $(E,\mathcal{E})$ be a measurable space. A stochastic process is a
collection of random variables $(X_{t})_{t\in\mathcal{T}}$ indexed by a set
$\mathcal{T}$ where for all $t\in\mathcal{T}$,
$X_{t}:(\Omega,\mathcal{F},\mathbb{P})\longrightarrow(E,\mathcal{E})$.

Brownian motion. Let
$(\Omega,\mathcal{F},\mathbb{P},(\mathcal{F}_{t})_{t\in\mathbb{R}_{+}})$ be a
probability space with a filtration. A standard Brownian motion (or standard
Wiener process) is a continuous centered gaussian process
$\\{B_{t},t\in\mathbb{R}_{+}\\}$ such that $\forall s,t\in\mathbb{R}_{+}$,
$Cov(B_{t},B_{s})=min(t,s)$, and for all $t\geq 0$, $B_{t}$ is
$\mathcal{F}_{t}$-measurable.

In particular, a Brownian motion verifies the following properties:

  * •

$B_{0}=0$ (standard)

  * •

$t\mapsto B_{t}(w)$ is continuous for almost all $w\in\Omega$

  * •

For all $s,t\in\mathbb{R}_{+}$ such that $s<t$,
$B_{t}-B_{s}\sim\mathcal{N}(0,t-s)$ (gaussian distribution with mean 0 and
variance $t-s$)

  * •

Independent increments: for all $n\in\mathbb{N}^{*}$, for all
$t_{1},\ldots,t_{n}\in\mathbb{R}_{+}$ such that $0=t_{0}<t_{1}<\ldots<t_{n}$,
the random variables $(B_{t_{k}}-B_{t_{k-1}})_{k\in\llbracket 1,n\rrbracket}$
are independent.

This stochastic process is often used to model randomness in SDEs, including
the Geometric Brownian Motion.

Geometric Brownian Motion. Let
$(\Omega,\mathcal{F},\mathbb{P},(\mathcal{F}_{t})_{t\in\mathbb{R}_{+}})$ be a
probability space with a filtration and $X=(X_{t})_{t\in\mathbb{R}_{+}}$ be a
stochastic process that takes value in $(\mathbb{R},\mathcal{B}(\mathbb{R}))$.

$X$ is a Geometric Brownian Motion (GBM) if it is a solution of the SDE:

$dX_{t}=\mu X_{t}dt+\sigma X_{t}B_{t}$ (2)

where $\mu\in\mathbb{R}$ and $\sigma\geq 0$ are two constants and
$B=(B_{t})_{t\in\mathbb{R}_{+}}$ is a Brownian motion.

Using Itô’s calculus, one can calculate an explicit solution for GBMs:

$X_{t}=X_{0}e^{\left(\mu-\frac{\sigma^{2}}{2}\right)t+\sigma B_{t}}$ (3)

###  2.3 Properties of Geometric Brownian Motions

######  Lemma 2.1.

If $Z$ is a standard normal distribution ($Z\sim\mathcal{N}(0,1)$), then
$\forall\lambda\in\mathbb{R},\mathbb{E}\left[e^{\lambda
Z}\right]=e^{\frac{\lambda^{2}}{2}}$

###### Proof.

Let $Z$ be a random variable such that $Z\sim\mathcal{N}(0,1)$. For all
$\lambda\in\mathbb{R}$,

$\displaystyle\mathbb{E}\left[e^{\lambda Z}\right]$
$\displaystyle=\int_{-\infty}^{+\infty}e^{\lambda
x}\frac{1}{\sqrt{2\pi}}e^{-\frac{x^{2}}{2}}dx$
$\displaystyle=\int_{-\infty}^{+\infty}\frac{1}{\sqrt{2\pi}}e^{-\frac{(x-\lambda)^{2}}{2}}dx$
$\displaystyle=e^{\frac{\lambda^{2}}{2}}\int_{-\infty}^{+\infty}\frac{1}{\sqrt{2\pi}}e^{-\frac{x^{2}}{2}}dx$
$\displaystyle=e^{\frac{\lambda^{2}}{2}}$

∎

######  Proposition 2.2.

If $X=(X_{t})_{t\in\mathbb{R}_{+}}$ is a GBM, then for all $t\geq 0$:

  * •

$\mathbb{E}[X_{t}]=X_{0}e^{\mu t}$

  * •

$\mathbb{V}[X_{t}]=X^{2}_{0}e^{2\mu t}\left(e^{\sigma^{2}t}-1\right)$

###### Proof.

Let $X$ be a GBM and $t\in\mathbb{R}_{+}$.

1)

$\displaystyle\mathbb{E}[X_{t}]$
$\displaystyle=\mathbb{E}\left[X_{0}e^{\left(\mu-\frac{\sigma^{2}}{2}\right)t+\sigma
B_{t}}\right]$
$\displaystyle=X_{0}e^{\left(\mu-\frac{\sigma^{2}}{2}\right)t}\mathbb{E}\left[e^{\sigma
B_{t}}\right]$
$\displaystyle=X_{0}e^{\left(\mu-\frac{\sigma^{2}}{2}\right)t}\mathbb{E}\left[e^{\sigma\sqrt{t}B_{1}}\right]$

$B_{1}\sim\mathcal{N}(0,1)$ so using Lemma 2.1, we obtain that
$\mathbb{E}\left[e^{\sigma\sqrt{t}B_{1}}\right]=e^{\frac{\sigma^{2}t}{2}}$,
hence the result.

2)
$\mathbb{V}[X_{t}]=\mathbb{E}\left[X^{2}_{t}\right]-\mathbb{E}\left[X_{t}\right]^{2}$,
and

$\displaystyle\mathbb{E}[X^{2}_{t}]$
$\displaystyle=\mathbb{E}\left[X^{2}_{0}e^{\left(2\mu-\sigma^{2}\right)t+2\sigma
B_{t}}\right]$
$\displaystyle=X^{2}_{0}e^{\left(2\mu-\sigma^{2}\right)t}\mathbb{E}\left[e^{2\sigma
B_{t}}\right]$

Applying Lemma 2.1 again, we obtain $\mathbb{E}\left[e^{2\sigma
B_{t}}\right]=e^{2\sigma^{2}t}$. Thus,

$\displaystyle\mathbb{V}[X_{t}]$
$\displaystyle=X^{2}_{0}e^{\left(2\mu+\sigma^{2}\right)t}-\left(X^{2}_{0}e^{\mu
t}\right)^{2}$ $\displaystyle=X^{2}_{0}e^{2\mu
t}\left(e^{\sigma^{2}t}-1\right)$

∎

Laplace Transform of a real continuous stochastic process. To apply Neural
Laplace to a SDE, we first need to define the Laplace Transform of a
stochastic process. Let $X=(X_{t})_{t\in\mathbb{R}_{+}}$ be a stochastic
process defined on a probability space
$(\Omega,\mathcal{F},\mathbb{P},(\mathcal{F}_{t})_{t\in\mathbb{R}_{+}})$, that
takes values in $(\mathbb{R}^{D},\mathcal{B}(\mathbb{R}^{D}))$ where $D>0$ is
the dimension, and such that $t\mapsto X_{t}$ is almost surely continuous.

We can extend the Laplace Transform to a stochastic process by defining a
function F such that, for all $\textbf{s}\in\mathbb{C}^{D}$,
$\textbf{F}(\textbf{s})$ is a stochastic process defined on the same
probability space, that takes values in
$(\mathbb{C}^{D},\mathcal{B}(\mathbb{C}^{D}))$, and that verifies:

$\forall\omega\in\Omega,\textbf{F}(\textbf{s})(\omega)=\mathcal{L}\\{X(\omega)\\}(s)=\int_{0}^{\infty}e^{-\textbf{s}t}X_{t}(\omega)dt$

We will note
$\textbf{F}(\textbf{s})=\int_{0}^{\infty}e^{-\textbf{s}t}X_{t}dt$.

###  2.4 Laplace Transform of a Geometric Brownian Motion

######  Theorem 2.3.

Let $X$ be a GBM with parameters $\mu\in\mathbb{R},\sigma\geq 0$. For all
$s\in\mathbb{C}$ verifying $\Re(s)>\mu$, we have

$\mathbb{E}\left[F(s)\right]=\frac{X_{0}}{s-\mu}$

Moreover, if $s$ also verifies $\Re(s)>0$ and if $\mu-\frac{\sigma^{2}}{2}>0$,
then

$\mathbb{V}\left[\|F(s)\|\right]\leq\frac{\|X_{0}\|^{2}}{4\Re(s)(\mu-\frac{\sigma^{2}}{2})}$

###### Proof.

Let $X=(X_{t})_{t\in\mathbb{R}_{+}}$ be a Geometric Brownian Motion with
parameters $\mu\in\mathbb{R},\sigma\geq 0$, and $s\in\mathbb{C}$.

1)

$\displaystyle\mathbb{E}[F(s)]$
$\displaystyle=\mathbb{E}\left[\int_{0}^{+\infty}X_{t}e^{-st}dt\right]$
$\displaystyle=\int_{0}^{+\infty}\mathbb{E}\left[X_{t}e^{-st}\right]dt\text{
(*)}\textsuperscript{1}$ $\displaystyle=\int_{0}^{+\infty}X_{0}e^{(\mu-s)t}dt$
$\displaystyle=\frac{X_{0}}{s-\mu}$

*: We can apply Fubini’s theorem because $\int_{X\times T}|f(x,y)|d(x,y)<+\infty$ and $f$ is measurable, where $X=[-\infty,+\infty[$, $T=[0,+\infty[$, and $f:(x,t)\mapsto X_{0}e^{\left(\mu-s-\frac{\sigma^{2}}{2}\right)t+\sigma\sqrt{t}x}\frac{1}{\sqrt{2\pi}}e^{-\frac{x^{2}}{2}}$

2) We now that
$\mathbb{V}\left[\|F(s)\|\right]\leq\mathbb{E}\left[\|F(s)\|^{2}\right]$, Now
let’s find an upper bound of $\mathbb{E}\left[\|F(s)\|^{2}\right]$.

$\displaystyle\mathbb{E}\left[\|F(s)\|^{2}\right]$
$\displaystyle=\mathbb{E}\left[\left\|\int_{0}^{+\infty}X_{t}e^{-st}dt\right\|^{2}\right]$
$\displaystyle\leq\mathbb{E}\left[\left(\int_{0}^{+\infty}\|X_{t}e^{-st}\|dt\right)^{2}\right]$
$\displaystyle\leq\mathbb{E}\bigg{[}\left(\int_{0}^{+\infty}\|X_{0}\|^{2}e^{-2\Re(s)t}dt\right)\times$
$\displaystyle\left(\int_{0}^{+\infty}e^{-\sigma^{2}t-2\mu
t+2\sigma\sqrt{t}B_{1}}dt\right)\bigg{]}\text{ (**)}\textsuperscript{2}$
$\displaystyle\leq\frac{\|X_{0}\|^{2}}{2\Re(s)}\int_{0}^{+\infty}e^{-\sigma^{2}t-2\mu
t}\mathbb{E}\left[e^{2\sigma\sqrt{t}B_{1}}\right]dt\text{
(*)}\textsuperscript{1}$
$\displaystyle\leq\frac{\|X_{0}\|^{2}}{2\Re(s)}\int_{0}^{+\infty}e^{(\sigma^{2}-2\mu)t}dt$
$\displaystyle\leq\frac{\|X_{0}\|^{2}}{4\Re(s)(\mu-\frac{\sigma^{2}}{2})}$

∎

00footnotetext: 1*: Fubini’s Theorem00footnotetext: 2**: Cauchy-Schwarz
inequality

##  3 Proposed method

According to Theorem 2.3, for a stochastic process $X$ solution of the GBM SDE
with a low initial value $|X_{0}|$, a low volatility $\sigma\geq 0$, and a
high drift $\mu$, we can expect that for most $s$ in the complex domain
$\mathbb{C}$, $F(s)$ is well-defined and
$F(s)\approx\mathbb{E}\left[F(s)\right]$. This is coherent with the fact that
$X$ is essentially driven by the drift and not really affected by the Brownian
motion, thus limiting the randomness.

Experiment. We generate $N_{trajectories}=200$ trajectories with
$N_{samples}=200$ points each sampled uniformly in $[\frac{T}{N_{samples}},T]$
where $T=1$. For all $i\in\llbracket 1,N_{trajectories}\rrbracket$, a
trajectory $\tau_{i}$ is characterized by its three parameters
$X_{0,i},\mu_{i},\sigma_{i}$ where $X_{0,i}\sim\mathcal{U}(0.1,1)$,
$\mu_{i}\sim\mathcal{U}(4,8)$ and $\sigma_{i}\sim\mathcal{U}(0.1,1)$.

We run 100 epochs to train the Neural Laplace and all the other baseline
models. The Root Mean Squared Error (RMSE) will be evaluated on a test set for
3 different seeds and the mean and the standard deviation (std) across the
different training are reported in the table Table 1 in the Results section
below.

######  Remark 3.1.

Due to the choice of parameters to generate the trajectories, especially the
drift $\mu$, $X_{t}\approx e^{\mu t}$, so we can expect that the other
baseline models will also perform well as $X$ is approximately the solution of
the ODE: $\frac{dX}{dt}(t)-\mu X=0$.

##  4 Results

Table 1: Mean and Std of the test RMSE across 3 different seeds for the
different methods.
Method | Mean | Std
---|---|---
ANODE (euler) | 0.387012 | 0.025194
Latent ODE (ODE enc.) | 0.529489 | 0.074544
NODE (euler) | 0.383214 | 0.033135
Neural Flow Coupling | 0.577819 | 0.075091
Neural Flow ResNet | 0.605377 | 0.037672
Neural Laplace | 0.597754 | 0.095084

Some of the baseline methods seem to perform better than Neural Laplace to
predict future values of a Geometric Brownian Motion (see Table 1). However,
this can be explained with the remark Remark 3.1.

Nonetheless, the mean of the test RMSE for the Neural Laplace method is still
low, which highlights the fact the method works.

##  5 Conclusion

This short piece of work extended the definition of the Laplace Transform to
stochastic processes in order to apply the Neural Laplace method to solve
stochastic differential equations. From a theoretical point of view, we
established conditions that a Geometric Brownian Motion must satisfy to
potentially lead to better predictions when training a Neural Laplace method.
Then, we conducted an experiment to show that Neural Laplace can learn from
this class of differential equation in this particular case, and we compared
the results with baseline models that work for ODE.

##  6 Future work

Among all the stochastic differential equations, we only investigated the
Geometric Brownian Motion. Looking at other SDEs to assess the feasibility of
using Neural Laplace in such settings both theoretically and empirically would
be the first things to add to this work. Let alone the fact that GBM is not
the best example, as highlighted by Remark 3.1. Moreover, for GBM, testing
other sets of values for $\mu$ and $\sigma$ may lead to some other results
that could be investigated.

When comparing Neural Laplace with baseline methods, we didn’t add Neural
Stochastic Differential Equations methods.

Finally, as the Laplacian Transform of a stochastic process is a random
variable, we could research how to provide confidence intervals to the
predictions made by the Neural Laplace methods based on the uncertainty that
we have on $F(s)$.

## References

  * Breda & Canci (2023) Breda, D. and Canci, Jung Kyuand D’Ambrosio, R.  _An Invitation to Stochastic Differential Equations in Healthcare_ , pp. 97–110.  Springer International Publishing, Cham, 2023.  ISBN 978-3-031-11814-2.  doi: 10.1007/978-3-031-11814-2˙6.  URL https://doi.org/10.1007/978-3-031-11814-2_6.
  * Holt et al. (2022) Holt, S., Qian, Z., and van der Schaar, M.  Neural laplace: Learning diverse classes of differential equations in the laplace domain, 2022.
  * Salvi et al. (2022) Salvi, C., Lemercier, M., and Gerasimovics, A.  Neural stochastic pdes: Resolution-invariant learning of continuous spatiotemporal dynamics, 2022.
  * Yang et al. (2022) Yang, L., Gao, T., Lu, Y., Duan, J., and Liu, T.  Neural network stochastic differential equation models with applications to financial data forecasting, 2022.