# Lie algebraic Carroll/Galilei duality

José Figueroa-O’Farrill  Maxwell Institute and School of Mathematics, The
University of Edinburgh, James Clerk Maxwell Building, Peter Guthrie Tait
Road, Edinburgh EH9 3FD, Scotland, United Kingdom<EMAIL_ADDRESS>În
memoria Veronicăi Stanciu

###### Abstract.

We characterise Lie groups with bi-invariant bargmannian, galilean or
carrollian structures. Localising at the identity, we show that Lie algebras
with ad-invariant bargmannian, carrollian or galilean structures are actually
determined by the same data: a metric Lie algebra with a skew-symmetric
derivation. This is the same data defining a one-dimensional double extension
of the metric Lie algebra and, indeed, bargmannian Lie algebras coincide with
such double extensions, containing carrollian Lie algebras as an ideal and
projecting to galilean Lie algebras. This sets up a canonical correspondence
between carrollian and galilean Lie algebras mediated by bargmannian Lie
algebras. This reformulation allows us to use the structure theory of metric
Lie algebras to give a list of bargmannian, carrollian and galilean Lie
algebras in the positive-semidefinite case. We also characterise Lie groups
admitting a bi-invariant (ambient) leibnizian structure. Leibnizian Lie
algebras extend the class of bargmannian Lie algebras and also set up a non-
canonical correspondence between carrollian and galilean Lie algebras.

EMPG-22-20, ORCID: 0000-0002-9308-9360

###### Contents

  1. 1 Introduction
  2. 2 Bargmannian, carrollian and galilean structures
  3. 3 Bargmannian, carrollian and galilean Lie algebras
    1. 3.1 Metric Lie algebras and double extensions
    2. 3.2 Bargmannian Lie algebras
    3. 3.3 Carrollian Lie algebras
    4. 3.4 Galilean Lie algebras
    5. 3.5 Summary
  4. 4 Classification
  5. 5 Leibnizian Lie groups
    1. 5.1 Leibnizian structures
    2. 5.2 Leibnizian Lie algebras
    3. 5.3 A leibnizian Lie algebra which is not bargmannian
  6. 6 Conclusion
  7. A Skew-symmetric derivations of a reductive metric Lie algebra
  8. B Once more, with indices

##  1\. Introduction

The study of non-lorentzian spacetime geometries is coming of age (see, e.g.,
the recent reviews [1, 2]), yet there are still some simple questions which
have not been asked nor answered. A natural first step when studying an
unfamiliar geometric structure is to find examples of such structures with
lots of symmetries. A natural class of such examples are homogeneous spaces
and, in particular, Lie groups with bi-invariant structures. Bi-invariance is
typically quite strong and this often allows one to classify them or at least
to characterise them in linear algebraic terms. This is the case, for example,
with Lie groups admitting a bi-invariant metric (of any signature), which were
studied by Medina [3] and characterised in terms of their Lie algebras by
Medina and Revoy [4] (see also [5, 6, 7]).

The three protagonists of today’s tale are Bargmann, Carroll and Galilei. They
may be used to label Lie groups, Lie algebras, homogeneous spaces and also
Cartan geometries (here, equivalently, $G$-structures). It turns out that in
all of these settings, objects with these names sit in relation to each other
in a way which suggests a correspondence (loosely, a duality) between Carroll
and Galilei mediated by Bargmann. This correspondence was first pointed out in
a geometric context in [8], but before describing this result let us set the
stage by discussing the Lie algebras themselves.

The Carroll and Galilei Lie algebras are examples of kinematical Lie algebras
[9, 10]. In spatial dimension $n$, they are spanned by
$\left(L_{ab},B_{a},P_{a},H\right)$, where $L_{ab}=-L_{ba}$ span a Lie
subalgebra isomorphic to $\mathfrak{so}(n)$ under which $B_{a},P_{a}$ are
vectors and $H$ a scalar. These conditions translate into the following Lie
brackets which are common to all kinematical Lie algebras

$\begin{split}[L_{ab},L_{cd}]&=\delta_{bc}L_{ad}-\delta_{ac}L_{bd}-\delta_{bd}L_{ac}+\delta_{bd}L_{ac}\\\
[L_{ab},B_{b}]&=\delta_{bc}B_{a}-\delta_{ac}B_{b}\\\
[L_{ab},P_{b}]&=\delta_{bc}P_{a}-\delta_{ac}P_{b}\\\
[L_{ab},H]&=0.\end{split}$ (1.1)

The kinematical Lie algebra where all other brackets vanish is called the
static kinematical Lie algebra and denoted $\mathfrak{s}$. All kinematical Lie
algebras are deformations of $\mathfrak{s}$ [11, 12, 13]. The subalgebra
$\mathfrak{s}_{0}$ of $\mathfrak{s}$ spanned by
$\left(L_{ab},B_{a},P_{a}\right)$ will play a rôle in our discussion.

The Carroll Lie algebra $\mathfrak{c}$ is a central extension of
$\mathfrak{s}_{0}$, with $H$ the central element and additional nonzero
bracket

$[B_{a},P_{b}]=\delta_{ab}H.$ (1.2)

In contrast, the Galilei Lie algebra $\mathfrak{g}$ is an “extension-by-
derivation” of $\mathfrak{s}_{0}$. The derivation is
$\operatorname{ad}_{H}=[H,-]$ and is defined by
$\operatorname{ad}_{H}(L_{ab})=\operatorname{ad}_{H}(P_{a})=0$ and
$\operatorname{ad}_{H}(B_{a})=-P_{a}$, resulting in the additional nonzero
bracket

$[H,B_{a}]=-P_{a}.$ (1.3)

We may summarise these observations in the diagrammatical language of (short)
exact sequences of Lie algebras as follows:

${0}$${\mathbb{R}}$${\mathfrak{c}}$${0}$${\mathfrak{s}_{0}}$${\mathfrak{g}}$${\mathbb{R}}$${0}$${0}$
(1.4)

The exact row says that $\mathfrak{g}$ is an extension-by-derivation of
$\mathfrak{s}_{0}$, whereas the exact column says that $\mathfrak{c}$ is a
one-dimensional extension of $\mathfrak{s}_{0}$. The diagram does not fix the
brackets uniquely, since $\mathfrak{s}_{0}$ admits many derivations and also
other one-dimensional extensions, central or not.

The Bargmann Lie algebra $\mathfrak{b}$ is a central extension of the Galilei
Lie algebra $\mathfrak{g}$ with additional generator $M$ and additional
bracket

$[B_{a},P_{b}]=\delta_{ab}M,$ (1.5)

which is reminiscent of the bracket (1.2) with $M$ playing the rôle of $H$.
Under this relabelling of bases, we see that the Bargmann Lie algebra
$\mathfrak{b}$ is an extension-by-derivation of the Carroll Lie algebra
$\mathfrak{c}$. The derivation is $\operatorname{ad}_{H}=[H,-]$ where again
$\operatorname{ad}_{H}$ annihilates $L_{ab},P_{a},M$ and its action on $B_{a}$
is given by the bracket (1.3).

This allows us to complete the diagram (1.4) to the following commutative
diagram of Lie algebras:

${0}$${0}$${\mathbb{R}}$${\mathbb{R}}$${0}$${\mathfrak{c}}$${\mathfrak{b}}$${\mathbb{R}}$${0}$${0}$${\mathfrak{s}_{0}}$${\mathfrak{g}}$${\mathbb{R}}$${0}$${0}$${0}$
(1.6)

which will recur in other contexts in this work with other Lie algebras
playing the rôles of $\mathfrak{s}_{0}$, $\mathfrak{c}$, $\mathfrak{g}$ and
$\mathfrak{b}$. In fact, this very diagram has already appeared in [14, App.
B] in the context of the hamiltonian description of particle dynamics on the
(spatially isotropic) homogeneous galilean spacetimes classified in [15]. In
that context, the Lie algebras which play the rôles of $\mathfrak{s}_{0}$,
$\mathfrak{c}$, $\mathfrak{g}$ and $\mathfrak{b}$ are infinite-dimensional.
The two exact rows in the the diagram (1.6) say that $\mathfrak{b}$ (resp.
$\mathfrak{g}$) is an extension-by-derivation of $\mathfrak{c}$ (resp.
$\mathfrak{s}_{0}$), whereas the two exact columns say that $\mathfrak{b}$
(resp. $\mathfrak{c}$) is a one-dimensional extension of $\mathfrak{g}$ (resp.
$\mathfrak{s}_{0}$). We could say, borrowing the terminology from the theory
of metric Lie algebras, that the Bargmann Lie algebra $\mathfrak{b}$ is a
_double extension_ 111To be clear, this proposal expands the definition of a
double-extension, transcending its origin in the context of metric Lie
algebras, to a more general notion in which a double extension is the
composition of a one-dimensional extension with an extension-by-derivation; at
least in those cases where these two operations commute. of
$\mathfrak{s}_{0}$. This is more than a mere analogy and we will see that this
is exactly right for Lie algebras admitting ad-invariant bargmannian
structures (to be defined below).

The Bargmann Lie algebra $\mathfrak{b}$ can also be defined as the subalgebra
of the Poincaré Lie algebra $\mathfrak{iso}(n+1,1)$ in one dimension higher
which centralises a null translation. Since the Poincaré translations commute,
they are contained in $\mathfrak{b}$ and hence the (simply-connected) Bargmann
Lie group acts transitively on Minkowski spacetime $\mathbb{M}^{n+2}$. This
fact underlies the geometric Carroll/Galilei duality in [8].

Choosing a Witt frame
$(\boldsymbol{e}_{a},\boldsymbol{e}_{+},\boldsymbol{e}_{-})$ for Minkowski
spacetime $\mathbb{M}^{n+2}$ we may express the generators of the Poincaré
algebra as $\left(L_{ab},L_{+a},L_{-a},L_{+-},P_{a},P_{+},P_{-}\right)$. The
centraliser of $P_{+}$ is spanned by
$\left(L_{ab},L_{+a},P_{a},P_{+},P_{-}\right)$ and it is isomorphic to the
Bargmann Lie algebra $\mathfrak{b}$, with $P_{+}$ playing the rôle of $M$. If
$X$ is an element of the Poincaré Lie algebra, we let $\xi_{X}$ denote the
corresponding Killing vector field on Minkowski spacetime. The null vector
field $\xi_{P_{+}}$ on Minkowski spacetime is not just Killing but actually
parallel. It defines a distribution $\xi_{P_{+}}^{\perp}\subset T\mathbb{M}$
which is integrable. The leaves of the corresponding foliation are null
hypersurfaces and are copies of the Carroll spacetime $\mathsf{C}^{n+1}$.
Indeed, they are homogeneous spaces of the normal subgroup with Lie algebra
the ideal of $\mathfrak{b}$ spanned by
$\left(L_{ab},L_{+a},P_{a},P_{+}\right)$, which is isomorphic to the Carroll
Lie algebra $\mathfrak{c}$, with $P_{+}$ playing the rôle of $H$. On the other
hand, the null reduction [16, 17] of $\mathbb{M}$ by the one-parameter
subgroup generated by $\xi_{P_{+}}$ is isomorphic to Galilei spacetime
$\mathsf{G}^{n+1}$. Indeed, the quotient is homogeneous under the Lie group
whose Lie algebra is the quotient of $\mathfrak{b}$ by the line spanned by
$P_{+}$, which is isomorphic to the Galilei Lie algebra $\mathfrak{g}$. This
results in the following suggestive diagram

${\mathsf{C}^{n+1}}$${\mathbb{M}^{n+2}}$${\mathsf{G}^{n+1},}$ (1.7)

which is the fundamental example of the geometric Carroll/Galilei duality in
[8].

This duality seems to be broken when we take all (spatially isotropic)
homogeneous kinematical spacetimes into consideration. In the classification
of [15] there are three other homogeneous carrollian spacetimes besides the
Carroll spacetime: the carrollian limits $\mathsf{dSC}$ of de Sitter and
$\mathsf{AdSC}$ of anti de Sitter spacetimes, and the lightcone $\mathsf{LC}$,
which can be realised as null hypersurfaces in de Sitter, anti de Sitter and
Minkowski spacetimes, respectively [15, 18]. On the other hand, there are two
one-parameter families of homogeneous galilean spacetimes. The galilean limit
$\mathsf{dSG}$ of de Sitter spacetime is one point ($\gamma=-1$) in a
continuum $\mathsf{dSG}_{\gamma}$, for $\gamma\in[-1,1]$, of galilean
spacetimes. Similarly, the galilean limit $\mathsf{AdSG}$ of anti de Sitter
spacetime is one point $(\chi=0)$ in a continuum $\mathsf{AdSG}_{\chi}$, for
$\chi\geq 0$, of galilean spacetimes. These two continua have a point in
common, since $\lim_{\gamma\to
1}\mathsf{dSG}_{\gamma}=\lim_{\chi\to\infty}\mathsf{AdSG}_{\chi}$. It is often
convenient to describe homogeneous spaces infinitesimally via their Klein
pairs. The above homogeneous galilean spacetimes have Klein pairs
$(\mathfrak{g}_{\alpha,\beta},\mathfrak{h})$ where
$\mathfrak{g}_{\alpha,\beta}$ is the kinematical Lie algebra with additional
nonzero brackets

$[H,B_{a}]=-P_{a}\qquad\text{and}\qquad[H,P_{a}]=\alpha B_{a}+\beta P_{a},$
(1.8)

for some $(\alpha,\beta)\in\mathbb{R}^{2}$ and $\mathfrak{h}$ is the
subalgebra spanned by $\left(L_{ab},B_{a}\right)$. The parameters $\gamma$ and
$\chi$ labeling the continua of galilean spacetimes serve as convenient
parametrisations for the equivalence classes of pairs $(\alpha,\beta)$ under
the relation $(\alpha,\beta)\sim(s^{2}\alpha,s\beta)$ for any nonzero
$s\in\mathbb{R}$. Just like the Galilei Lie algebra
$\mathfrak{g}=\mathfrak{g}_{0,0}$, the Lie algebra
$\mathfrak{g}_{\alpha,\beta}$ also admits a one-dimensional extension
$\mathfrak{b}_{\alpha,\beta}$ with additional brackets

$[B_{a},P_{b}]=\delta_{ab}M\qquad\text{and}\qquad[H,M]=\beta M,$ (1.9)

which is central if and only if $\beta=0$. In that case, we may distinguish
three cases: $\alpha=0$, corresponding to the Galilei Lie algebra, $\alpha>0$
and $\alpha<0$, corresponding to the two Newton–Hooke Lie algebras.

The Lie algebra $\mathfrak{b}_{\alpha,\beta}$ can also be described as an
extension-by-derivation of the Carroll Lie algebra $\mathfrak{c}$. Again we
need to relabel the generators $M\leftrightarrow H$ and the derivation
$\operatorname{ad}_{H}$ is defined by $\operatorname{ad}_{H}(L_{ab})=0$,
$\operatorname{ad}_{H}(B_{a})=-P_{a}$, $\operatorname{ad}_{H}(P_{a})=\alpha
B_{a}+\beta P_{a}$ and $\operatorname{ad}_{H}(M)=\beta M$. In other words, we
once again arrive at a commutative diagram like (1.6)

${0}$${0}$${\mathbb{R}}$${\mathbb{R}}$${0}$${\mathfrak{c}}$${\mathfrak{b}_{\alpha,\beta}}$${\mathbb{R}}$${0}$${0}$${\mathfrak{s}_{0}}$${\mathfrak{g}_{\alpha,\beta}}$${\mathbb{R}}$${0}$${0}$${0}$
(1.10)

In this note we will exhibit yet other avatars of the above diagram. Indeed,
our main aim in this note is the classification of (connected, simply-
connected) Lie groups admitting bargmannian, carrollian or galilean
structures. The Lie correspondence allows us to work at the level of Lie
algebras with ad-invariant bargmannian, carrollian or galilean structures.
Without aiming to confuse the reader, we shall refer to such Lie algebras as
bargmannian, carrollian or galilean, for short. However please note that,
paradoxically perhaps, the Bargmann Lie algebra is not bargmannian, the
Carroll Lie algebra is not carrollian and the Galilei Lie algebra is not
galilean. Nevertheless bargmannian, carrollian and galilean Lie algebra sit in
relation to each other just like the Bargmann, Carroll and Galilei Lie
algebras; namely, in a commutative diagram such as (1.6). Neither should our
bargmannian, carrollian and galilean Lie algebras be confused with the
transitive Lie algebras of kinematical spacetimes with such structures. Metric
Lie algebras associated with kinematical Lie algebras were studied in [19] in
the context of Chern–Simons theories of gravity.

This note is organised as follows. In Section 2 we review the notions of
carrollian and galilean structures and their duality via bargmannian manifolds
and specialise to connected Lie groups admitting bi-invariant such structures.
This allows us to localise at the identity and discuss ad-invariant
bargmannian, carrollian and galilean structures on Lie algebras, which are
studied in Section 3. We will see that all three kinds of Lie algebras are
characterised in terms of the same data: namely, a metric Lie algebra with a
skew-symmetric derivation. This is also the data which defines a one-
dimensional double extension of a metric Lie algebra, which is a construction
due to Medina and Revoy [4] (see also [5, 6]) which results in a new metric
Lie algebra. This will allow us to identify bargmannian Lie algebras precisely
as one-dimensional double extensions of metric Lie algebras. In Section 4 we
specialise to strictly carrollian and galilean structures (as opposed to the
“pseudo” versions we treated before) and classify the relevant Lie algebras.
In Section 5 we discuss Lie groups with bi-invariant leibnizian structures in
the sense of [20] and show that they too appear in a commutative diagram such
as (1.6). Leibnizian groups (strictly) extend the class of bargmannian Lie
groups and they too mediate a duality between carrollian and galilean Lie
group, which is not canonical and hence in principle different from the one
discussed in Section 3.5. Finally in Section 6 we offer some concluding
remarks. There are two appendices: in Appendix A we give a proof of a result
needed in Section 4; and in Appendix B we write some of the Lie algebraic
structures in terms of a basis.

##  2\. Bargmannian, carrollian and galilean structures

Let $M$ be a finite-dimensional smooth manifold. Recall that a galilean
structure [21] on $M$ consists of a nowhere-vanishing one-form
$\tau\in\Omega^{1}(M)$ and a symmetric bivector field
$\gamma\in\Gamma(\odot^{2}TM)$, which is everywhere corank-$1$ as a field of
bilinear forms on one-forms and such that its radical is everywhere spanned by
$\tau$. Typically one demands that $\gamma$ is positive-semidefinite, but one
can also consider pseudo-galilean structures where $\gamma$ has any signature.
Galilean structures are particular examples of $G$-structures [21, 22].

Similarly, a carrollian structure [23, 8] on $M$ consists of a nowhere-
vanishing vector field $\kappa\in\mathcal{X}(M)$ and a symmetric
$(0,2)$-tensor field $h\in\Gamma(\odot^{2}T^{*}M)$ which is everywhere of
corank-$1$ and such that the characteristic distribution is generated by
$\kappa$. Again one typically demands that $h$ is positive-semidefinite, but
one can also consider pseudo-carrollian structures where $h$ has any
signature. For example, the blow-up of spatial infinity of Minkowski spacetime
is a pseudo-carrollian manifold [24], whereas the blow-up of either past or
future timelike infinity is carrollian [25]. Carrollian structures are also
examples of $G$-structures [22].

One may exhibit a correspondence between carrollian and galilean structures
[8] by passing to a higher-dimensional manifold $M$ with a bargmannian
structure, namely an indefinite metric $g$ together with a nowhere-vanishing
null vector field $\xi$. Strictly speaking, bargmannian structures require the
metric $g$ to be lorentzian, but one can also have pseudo-bargmannian
structures where $g$ is only assumed to be of indefinite signature. In any
case, the null vector field $\xi$ defines a distribution $\xi^{\perp}\subset
TM$ which, if integrable, foliates $M$ by null (since $\xi\in\xi^{\perp}$)
hypersurfaces $i:N\to M$ which inherit a carrollian structure $(\xi,i^{*}g)$.
If $\xi$ is Killing, then the null reduction [16, 17] defines (in the good
cases) a fibration $\pi:M\to N$ where $N$ inherits a galilean structure whose
clock one-form pulls back to the one-form $\xi^{\flat}$ dual to $\xi$ and
whose spatial cometric is induced by the inverse of $g$.

In this note we are interested in Lie groups admitting bi-invariant
carrollian, galilean or bargmannian structures; that is, triples
$(G,\kappa,h)$, $(G,\tau,\gamma)$ or $(G,g,\xi)$ where $G$ is a connected Lie
group and $\kappa$ and $h$, $\tau$ and $\gamma$ or $g$ and $\xi$ are invariant
under Lie derivatives by both left- and right-invariant vector fields. Such
tensor fields are determined uniquely by their value at the identity, where
they define tensors of the Lie algebra $\mathfrak{g}$ of $G$ which are
invariant under the adjoint representation of $G$ on $\mathfrak{g}$. Since we
assume assume that $G$ is connected, it will be enough to demand that the
tensors are ad-invariant; that is, invariant under the adjoint action of
$\mathfrak{g}$ on itself.

##  3\. Bargmannian, carrollian and galilean Lie algebras

In this section we characterise Lie algebras admitting ad-invariant
bargmannian, carrollian or galilean structures. They turn out to be intimately
linked to metric Lie algebras, which we review first.

###  3.1. Metric Lie algebras and double extensions

Let $\mathfrak{g}$ be a finite-dimensional real Lie algebra. We say that
$\mathfrak{g}$ is a metric Lie algebra if it admits an ad-invariant inner
product $\left<-,-\right>$; that is, one for which

$\left<[W,X],Y\right>=-\left<X,[W,Y]\right>$ (3.1)

for all $W,X,Y\in\mathfrak{g}$. Medina and Revoy [4] introduced the notion of
a double extension, as a method of constructing new metric Lie algebras from
old. In this note we will only need a special case of this construction:
namely, a one-dimensional double extension.

The ingredients for a one-dimensional double extension are a metric Lie
algebra $\mathfrak{g}_{0}$ with ad-invariant scalar product
$\left<-,-\right>_{0}$ and a skew-symmetric derivation $D_{0}$ of
$\mathfrak{g}_{0}$; that is, for all $X,Y\in\mathfrak{g}_{0}$,

$D_{0}[X,Y]_{0}=[D_{0}X,Y]_{0}+[X,D_{0}Y]_{0}\qquad\text{and}\qquad\left<D_{0}X,Y\right>_{0}=-\left<X,D_{0}Y\right>_{0},$
(3.2)

where $[-,-]_{0}$ is the Lie bracket in $\mathfrak{g}_{0}$. On the vector
space $\mathfrak{g}=\mathfrak{g}_{0}\oplus\mathbb{R}D\oplus\mathbb{R}Z$, where
$D$ and $Z$ are two additional generators, we define the brackets

$\begin{split}[X,Y]&=[X,Y]_{0}+\left<D_{0}X,Y\right>_{0}Z\\\ [D,X]&=D_{0}X\\\
[Z,-]&=0,\end{split}$ (3.3)

for all $X,Y\in\mathfrak{g}_{0}$. We can define a scalar product on
$\mathfrak{g}$ by

$\left<X,Y\right>=\left<X,Y\right>_{0},\qquad\left<X,D\right>=\left<X,Z\right>=0,\qquad\left<D,Z\right>=1\qquad\text{and}\qquad\left<D,D\right>\in\mathbb{R},$
(3.4)

for all $X,Y\in\mathfrak{g}_{0}$, which can be seen to be ad-invariant. We may
(and will) perform a Lie algebra automorphism $D\mapsto
D-\tfrac{1}{2}\left<D,D\right>Z$ which allows us to set $\left<D,D\right>=0$.
The Lie algebra $\mathfrak{g}$ is said to be the double extension of
$\mathfrak{g}_{0}$ relative to the skew-symmetric derivation $D_{0}$. This
construction is due to Medina and Revoy [4], refined and applied in the
context of two-dimensional conformal field theory in [5, 6], and further
refined by Kath and Olbrich [7].

###  3.2. Bargmannian Lie algebras

A metric Lie algebra $(\mathfrak{g},\left<-,-\right>)$ is said to be
bargmannian if there is a nonzero null $Z\in\mathfrak{g}$, which is central.
This of course requires $\left<-,-\right>$ to be indefinite, yet still
nondegenerate. We will show that bargmannian Lie algebras are one-dimensional
double-extensions of metric Lie algebras.

######  Proposition 3.1.

Every bargmannian Lie algebra $\mathfrak{g}$ is isomorphic to a double
extension of a metric Lie algebra $\mathfrak{g}_{0}$ by a skew-symmetric
derivation.

###### Proof.

Let $\mathfrak{g}$ be a bargmannian Lie algebra with scalar product
$\left<-,-\right>$ and central null element $Z\in\mathfrak{g}$. The one-
dimensional isotropic ideal spanned by $Z$ is clearly minimal and hence by the
structure theorem of Medina and Revoy [4] (see also [6, §3]) the Lie algebra
$\mathfrak{g}$ is a double extension of the metric Lie algebra
$\mathfrak{g}_{0}:=Z^{\perp}/\mathbb{R}Z$, which inherits an inner product
from that of $\mathfrak{g}$, by a skew-symmetric derivation defined as
follows. Let $D\in\mathfrak{g}$ be such that $\left<D,Z\right>=1$. Then
$\operatorname{ad}_{D}$ preserves $Z^{\perp}$ since $Z^{\perp}$ is an ideal of
$\mathfrak{g}$ and $[D,Z]=0$ since $Z$ in central, hence
$\operatorname{ad}_{D}$ induces a derivation $D_{0}$ of $\mathfrak{g}_{0}$ as
follows: if $X\in Z^{\perp}$ and letting $\overline{X}\in\mathfrak{g}_{0}$
denote its projection, we define the derivation $D_{0}$ by
$D_{0}\overline{X}=\overline{[D,X]}$, which is well defined. It is clearly a
derivation and also skew-symmetric under the inner product
$\left<-,-\right>_{0}$ on $\mathfrak{g}_{0}$ defined by

$\left<\overline{X},\overline{Y}\right>_{0}=\left<X,Y\right>.$ (3.5)

The fact that $D_{0}$ is skew-symmetric follows from the fact that so is
$\operatorname{ad}_{D}$:

$\displaystyle\left<D_{0}\overline{X},\overline{Y}\right>_{0}$
$\displaystyle=\left<\overline{[D,X]},\overline{Y}\right>_{0}$
$\displaystyle=\left<[D,X],Y\right>$ $\displaystyle=-\left<X,[D,Y]\right>$
$\displaystyle=-\left<\overline{X},\overline{[D,Y]}\right>_{0}$
$\displaystyle=-\left<\overline{X},D_{0}\overline{Y}\right>_{0}.$

As vector spaces, $\mathfrak{g}=Z^{\perp}\oplus\mathbb{R}D$, where
$Z^{\perp}=\mathfrak{g}_{0}\oplus\mathbb{R}Z$ and the Lie brackets are given
by

$\left[X,Y\right]=\left[X,Y\right]_{0}+\left<D_{0}X,Y\right>_{0}Z,\qquad\left[D,X\right]=D_{0}X\qquad\text{and}\qquad\left[Z,-\right]=0,$
(3.6)

for all $X,Y\in\mathfrak{g}_{0}$, where we have adorned the bracket and the
inner product of $\mathfrak{g}_{0}$ with a subscript 0. The inner product can
be brought to the form

$\left<X,Y\right>=\left<X,Y\right>_{0},\qquad\left<X,D\right>=0,\qquad\left<X,Z\right>=0,\qquad\left<D,Z\right>=1,\qquad\text{and}\qquad\left<D,D\right>=0,$
(3.7)

for all $X,Y\in\mathfrak{g}_{0}$. In other words, comparing with Section 3.1,
we see that $\mathfrak{g}$ is the double extension of
$\left(\mathfrak{g}_{0},\left<-,-\right>_{0}\right)$ by the skew-symmetric
derivation $D_{0}$. ∎

Notice that if the inner product on $\mathfrak{g}_{0}$ has signature $(p,q)$,
the one on $\mathfrak{g}$ has signature $(p+1,q+1)$. Hence if $\mathfrak{g}$
is lorentzian, then $\mathfrak{g}_{0}$ must have a positive-definite inner
product. In that case, it follows from the structure theorem of Medina and
Revoy [4] (see also [6]) that $\mathfrak{g}_{0}$ is isomorphic to an
orthogonal direct sum of compact simple Lie algebras $\mathfrak{s}_{i}$ (each
with a negative multiple of the Killing form) and an abelian Lie algebra
$\mathfrak{a}$ with a euclidean inner product. We will use this in Section 4
in order to classify (strictly) carrollian and galilean Lie algebras.

###  3.3. Carrollian Lie algebras

Let $\mathfrak{g}$ be a finite-dimensional real Lie algebra. By an ad-
invariant carrollian structure on $\mathfrak{g}$ we mean a pair $(Z,h)$
consisting of a (nonzero) central element $Z\in\mathfrak{g}$ and an ad-
invariant $h\in\odot^{2}\mathfrak{g}^{*}$ whose radical is one-dimensional and
spanned by $Z$.

######  Proposition 3.2.

Every carrollian Lie algebra $\mathfrak{g}$ is isomorphic to a one-dimensional
central extension of a metric Lie algebra $\mathfrak{g}_{0}$.

###### Proof.

Let $(\mathfrak{g},Z,h)$ be a carrollian Lie algebra and let
$\mathfrak{g}_{0}=\mathfrak{g}/\mathbb{R}Z$. Then $\mathfrak{g}$ is a central
extension of $\mathfrak{g}_{0}$, which we may summarise by the following short
exact sequence of Lie algebras:

${0}$${\mathbb{R}Z}$${\mathfrak{g}}$${\mathfrak{g}_{0}}$${0.}$ (3.8)

Moreover, $h$ induces an inner product $\left<-,-\right>_{0}$ on
$\mathfrak{g}_{0}$ by

$\left<\overline{X},\overline{Y}\right>_{0}=h(X,Y),$ (3.9)

where $\overline{X}\in\mathfrak{g}_{0}$ is the image of $X\in\mathfrak{g}$
under the canonical map $\mathfrak{g}\to\mathfrak{g}/\mathbb{R}Z$. The inner
product is well-defined since $h(Z,-)=0$. Since $Z$ spans the radical,
$\left<-,-\right>_{0}$ is nondegenerate and since $h$ is
$\mathfrak{g}$-invariant, $\left<-,-\right>_{0}$ is
$\mathfrak{g}_{0}$-invariant. In other words,
$\left(\mathfrak{g}_{0},\left<-,-\right>_{0}\right)$ is a metric Lie algebra.
∎

As vector spaces, $\mathfrak{g}=\mathfrak{g}_{0}\oplus\mathbb{R}Z$ and the Lie
bracket is given by

$\begin{split}[\overline{X},\overline{Y}]&=[\overline{X},\overline{Y}]_{0}+\alpha(\overline{X},\overline{Y})Z\\\
[Z,-]&=0\end{split}$ (3.10)

for all $\overline{X},\overline{Y}\in\mathfrak{g}_{0}$ and where $[-,-]_{0}$
is the Lie bracket on $\mathfrak{g}_{0}$. It follows that
$\alpha\in\wedge^{2}\mathfrak{g}_{0}^{*}$ is a $2$-cocycle. Corresponding to a
$2$-cocycle on any metric Lie algebra, there is always a skew-symmetric
derivation $D_{0}$, defined by

$\left<D_{0}\overline{X},\overline{Y}\right>_{0}=\alpha(\overline{X},\overline{Y})$
(3.11)

for all $\overline{X},\overline{Y}\in\mathfrak{g}_{0}$. Since
$\left<-,-\right>_{0}$ is nondegenerate, $D_{0}$ is uniquely defined. Since
$\alpha$ is alternating, $D_{0}$ is skew-symmetric:

$\left<D_{0}\overline{X},\overline{Y}\right>_{0}=\alpha(\overline{X},\overline{Y})=-\alpha(\overline{Y},\overline{X})=-\left<\overline{X},D_{0}\overline{Y}\right>_{0}.$
(3.12)

Finally, the cocycle condition for $\alpha$, says that $D_{0}$ is a
derivation: for every
$\overline{X},\overline{Y},\overline{U}\in\mathfrak{g}_{0}$,

$\displaystyle\left<D_{0}[\overline{X},\overline{Y}]_{0},\overline{U}\right>_{0}$
$\displaystyle=\alpha([\overline{X},\overline{Y}]_{0},\overline{U})$
$\displaystyle=\alpha(\overline{X},[\overline{Y},\overline{U}]_{0})+\alpha(\overline{Y},[\overline{U},\overline{X}]_{0})$
$\displaystyle=\left<D_{0}\overline{X},[\overline{Y},\overline{U}]_{0}\right>_{0}+\left<D_{0}\overline{Y},[\overline{U},\overline{X}]_{0}\right>_{0}$
$\displaystyle=\left<[D_{0}\overline{X},\overline{Y}]_{0},\overline{U}\right>_{0}-\left<[D_{0}\overline{Y},\overline{X}]_{0},\overline{U}\right>_{0}$
$\displaystyle=\left<[D_{0}\overline{X},\overline{Y}]_{0},\overline{U}\right>_{0}+\left<[\overline{X},D_{0}\overline{Y}]_{0},\overline{U}\right>_{0},$

so that
$D_{0}[\overline{X},\overline{Y}]_{0}=[D_{0}\overline{X},\overline{Y}]_{0}+[\overline{X},D_{0}\overline{Y}]_{0}$.

Let $\widehat{\mathfrak{g}}$ denote the one-dimensional double-extension of
$\mathfrak{g}_{0}$ relative to the skew-symmetric derivation $D_{0}$, as
discussed in Section 3.1. Then the carrollian Lie algebra $\mathfrak{g}$
embeds into $\widehat{\mathfrak{g}}$ as the ideal $Z^{\perp}$, resulting in
the following short exact sequence of Lie algebras:

${0}$${\mathfrak{g}}$${\widehat{\mathfrak{g}}}$${\mathbb{R}D}$${0.}$ (3.13)

###  3.4. Galilean Lie algebras

Let $\mathfrak{g}$ be again a finite-dimensional real Lie algebra. By an ad-
invariant galilean structure on $\mathfrak{g}$ we mean a nonzero ad-invariant
$\tau\in\mathfrak{g}^{*}$ and an ad-invariant
$\gamma\in\odot^{2}\mathfrak{g}$, defining a symmetric bilinear form on
$\mathfrak{g}^{*}$ which has a one-dimensional radical spanned by $\tau$.

######  Proposition 3.3.

Every galilean Lie algebra is an “extension by skew-symmetric derivation” of a
metric Lie algebra $\mathfrak{g}_{0}$.

###### Proof.

Let $(\mathfrak{g},\tau,\gamma)$ be a galilean Lie algebra. Since $\tau$ is
ad-invariant, it annihilates brackets ($\tau([X,Y])=0$) and hence
$\mathfrak{g}_{0}:=\ker\tau$ is an ideal of $\mathfrak{g}$. We thus get a
short exact sequence of Lie algebras

${0}$${\mathfrak{g}_{0}}$${\mathfrak{g}}$${\mathbb{R}}$${0.}$$\scriptstyle{\tau}$
(3.14)

This sequence always splits: let $D\in\mathfrak{g}$ be any element with
$\tau(D)=1$. Then since $\tau$ annihilates brackets, $\operatorname{ad}_{D}$
is a derivation of $\mathfrak{g}_{0}$. Dualising the above sequence we see
that $\mathfrak{g}_{0}^{*}\cong\mathfrak{g}^{*}/\mathbb{R}\tau$. The tensor
$\gamma$ defines an inner product $\gamma_{0}$ on $\mathfrak{g}_{0}^{*}$.
Since $\gamma$ is $\mathfrak{g}$-invariant, so is $\gamma_{0}$. The inverse of
$\gamma_{0}$ is an invariant inner product $\left<-,-\right>_{0}$ on
$\mathfrak{g}_{0}$ relative to which $\operatorname{ad}_{D}$ is skewsymmetric.
In summary, a galilean Lie algebra is an “extension by skew-symmetric
derivation” of a metric Lie algebra. ∎

So just as in the case of carrollian Lie algebras, the underlying data is
again a metric Lie algebra and a skew-symmetric derivation. If we again let
$\widehat{\mathfrak{g}}$ denote the (one-dimensional) double extension of
$\mathfrak{g}_{0}$ by the skew-symmetric derivation $\operatorname{ad}_{D}$,
we have that the galilean Lie algebra $\mathfrak{g}$ is a quotient of
$\widehat{\mathfrak{g}}$ by the central line $\mathbb{R}Z$, resulting the
short exact sequence

${0}$${\mathbb{R}Z}$${\widehat{\mathfrak{g}}}$${\mathfrak{g}}$${0.}$ (3.15)

###  3.5. Summary

Let $\left(\mathfrak{g}_{0},\left<-,-\right>_{0}\right)$ be a metric Lie
algebra and $D_{0}$ a skew-symmetric derivation. This data allows us to define
three other Lie algebras:

  1. (1)

a bargmannian Lie algebra $\widehat{g}$, the one-dimensional double extension
of $\mathfrak{g}_{0}$ by $D_{0}$;

  2. (2)

a carrollian Lie algebra $\mathfrak{g}_{\text{car}}$ which is a central
extension of $\mathfrak{g}_{0}$ with cocycle given by $D_{0}$ via (3.11) or,
equivalently, an ideal of $\widehat{\mathfrak{g}}$; and

  3. (3)

a galilean Lie algebra $\mathfrak{g}_{\text{gal}}$ which is an extension by
the skew-symmetric derivation $D_{0}$ of $\mathfrak{g}_{0}$ or, equivalently,
a quotient of $\widehat{\mathfrak{g}}$.

These Lie algebras fit into the following commutative diagram:

${0}$${0}$${\mathbb{R}}$${\mathbb{R}}$${0}$${\mathfrak{g}_{\text{car}}}$${\widehat{\mathfrak{g}}}$${\mathbb{R}}$${0}$${0}$${\mathfrak{g}_{0}}$${\mathfrak{g}_{\text{gal}}}$${\mathbb{R}}$${0}$${0}$${0}$
(3.16)

where the two short exact rows are extensions-by-derivations and the two short
exact columns are central extensions. This defines a canonical correspondence
between carrollian and galilean Lie algebras. For example, we start with a
carrollian Lie algebra $(\mathfrak{g}_{\text{car}},Z,h)$ given by the data
$(\mathfrak{g}_{0},D_{0})$. This Lie algebra is an ideal of the double
extension $\widehat{\mathfrak{g}}$ of $\mathfrak{g}_{0}$ by $D_{0}$ and then
we define the galilean dual $\mathfrak{g}_{\text{gal}}$ of
$\mathfrak{g}_{\text{car}}$ to be the quotient of $\widehat{\mathfrak{g}}$ by
the ideal generated by $Z$. Conversely, let
$(\mathfrak{g}_{\text{gal}},\tau,\gamma)$ be a galilean Lie algebra given by
the data $(\mathfrak{g}_{0},D_{0})$ and let $\widehat{\mathfrak{g}}$ again
denote the double extension of $\mathfrak{g}_{0}$ by $D_{0}$. Then
$\mathfrak{g}_{\text{gal}}$ is a quotient of $\widehat{\mathfrak{g}}$ by a
one-dimensional ideal spanned by $Z$ and we define the carrollian dual
$\mathfrak{g}_{\text{car}}$ of $\mathfrak{g}_{\text{gal}}$ as the ideal
$Z^{\perp}$ of $\widehat{\mathfrak{g}}$.

In Section 5 we will discuss a non-canonical correspondence between carrollian
and galilean algebras which is mediated by a leibnizian Lie algebra (to be
defined below), but first we will use the canonical correspondence just
established to classify strictly carrollian, galilean and bargmannian Lie
algebras.

##  4\. Classification

In this section we specialise to strict carrollian and galilean structures,
where $h$ and $\gamma$ are positive-semidefinite. This means that the metric
Lie algebra $\mathfrak{g}_{0}$ has a positive-definite inner product or,
equivalently, that the bargmannian Lie algebra $\widehat{\mathfrak{g}}$ is
lorentzian. Lorentzian Lie algebras were classified by Medina [3], but we are
interested only in those which are isomorphic to one-dimensional double
extensions of a positive signature metric Lie algebra. Since double extensions
always make the inner product indefinite, the structure theorem says that
metric Lie algebras with positive-definite inner products are necessarily
orthogonal direct sums of (compact) simple Lie algebras (with a negative
multiple of the Killing form) and an abelian Lie algebra with a (trivially
invariant) euclidean inner product. Therefore we will let
$\mathfrak{g}_{0}=\mathfrak{s}_{1}\oplus\cdots\oplus\mathfrak{s}_{k}\oplus\mathfrak{a}$,
where the $\mathfrak{s}_{i}$ are simple and $\mathfrak{a}$ is abelian. The
direct sums are orthogonal and the inner product on $\mathfrak{s}_{i}$ is
given by $-\lambda_{i}\kappa_{i}$, where $\lambda_{i}>0$ and $\kappa_{i}$ is
the Killing form of $\mathfrak{s}_{i}$, which for compact simple Lie algebras
is negative-definite, hence the sign. The inner product on $\mathfrak{a}$ is
any desired euclidean inner product. Up to an isomorphism, we can think of
$\mathfrak{a}=\mathbb{R}^{m}$ for some $m$ and the inner product being the
standard euclidean inner product on $\mathbb{R}^{m}$.

Next we determine the skew-symmetric derivations of $\mathfrak{g}_{0}$. The
following proposition, proved in Appendix A, states that any skew-symmetric
derivation of $\mathfrak{g}_{0}$ is the sum of an inner derivation of the
semisimple part
$\mathfrak{ss}=\mathfrak{s}_{1}\oplus\cdots\oplus\mathfrak{s}_{k}$ and a skew-
symmetric endomorphism of the abelian part $\mathfrak{a}$.

######  Proposition 4.1.

Let $D_{0}$ be a skew-symmetric derivation of $\mathfrak{g}_{0}$. Then

$D_{0}=T+\sum_{i}\operatorname{ad}_{X_{i}},$

where $T\in\mathfrak{so}(\mathfrak{a})$ and $X_{i}\in\mathfrak{s}_{i}$.

The double extension $\widehat{\mathfrak{g}}$ of $\mathfrak{g}_{0}$ by the
skew-symmetric derivation $D_{0}$ has underlying vector space
$\mathfrak{g}_{0}\oplus\mathbb{R}D\oplus\mathbb{R}Z$ and the brackets are
given by (3.3). We may apply the following general linear transformation of
the vector space: it is the identity on $\mathfrak{g}_{0}\oplus\mathbb{R}Z$
and maps $D\mapsto D-\sum_{i}X_{i}$. In that basis,
$[D,X]=T(X_{\mathfrak{a}})$, where $X=X_{\mathfrak{ss}}+X_{\mathfrak{a}}$ with
$X_{\mathfrak{ss}}\in\mathfrak{ss}$ and $X_{\mathfrak{a}}\in\mathfrak{a}$. In
other words, we may change basis so that effectively
$D_{0}\in\mathfrak{so}(\mathfrak{a})$. Using the euclidean inner product on
$\mathfrak{a}$, we may identify $D_{0}\in\mathfrak{so}(\mathfrak{a})$ with
$\omega\in\wedge^{2}\mathfrak{a}^{*}$ where
$\omega(A,B)=\left<D_{0}A,B\right>_{0}$ for all $A,B\in\mathfrak{a}$. Let
$\omega$ have rank $2\ell$ for some
$\ell=0,1,\dots,\lfloor\frac{\dim\mathfrak{a}}{2}\rfloor$.

If $\ell=0$, then $D_{0}=0$ and

$\widehat{\mathfrak{g}}=\mathfrak{g}_{0}\oplus\underbrace{\mathbb{R}D\oplus\mathbb{R}Z}_{\mathfrak{a}_{2}},$
(4.1)

where $\mathfrak{a}_{2}$ is a two-dimensional abelian Lie algebra with a
lorentzian inner product.

If $\ell>0$, we may split
$\mathfrak{a}=\mathfrak{a}_{0}\oplus\mathfrak{a}_{1}$, where
$\mathfrak{a}_{0}=\ker D_{0}$ and $\mathfrak{a}_{1}=\mathfrak{a}_{0}^{\perp}$,
and $\omega$ defines a symplectic structure on $\mathfrak{a}_{1}$. Then the
double extension is

$\widehat{\mathfrak{g}}=\mathfrak{ss}\oplus\mathfrak{a}_{0}\oplus\mathfrak{nw}_{2\ell+2},$
(4.2)

where $\mathfrak{nw}_{2\ell+2}$ is a lorentzian Nappi–Witten algebra [26, 27,
28, 5] of dimension $2\ell+2$ with underlying vector space
$\mathfrak{a}_{1}\oplus\mathbb{R}D\oplus\mathbb{R}Z$ and brackets

$[A,B]=\omega(A,B)Z\qquad\text{and}\qquad[D,A]=D_{0}A$ (4.3)

for all $A,B\in\mathfrak{a}_{1}$ and with lorentzian inner product

$\left<A,B\right>=\left<A,B\right>_{0},\qquad\left<D,Z\right>=1\qquad\text{and}\qquad\left<D,D\right>=0,$
(4.4)

where the choice $\left<D,D\right>=0$ can be arrived at via a Lie algebra
automorphism $D\mapsto D-\tfrac{1}{2}\left<D,D\right>Z$.

The corresponding carrollian Lie algebra is the ideal $Z^{\perp}$ of
$\widehat{\mathfrak{g}}$. As a Lie algebra,

$Z^{\perp}=\mathfrak{ss}\oplus\mathfrak{a}_{0}\oplus\mathfrak{heis}_{2\ell+1}$
(4.5)

where the Heisenberg Lie algebra $\mathfrak{heis}_{2\ell+1}$ has underlying
vector space $\mathfrak{a}_{1}\oplus\mathbb{R}Z$ and brackets

$[A,B]=\omega(A,B)Z$ (4.6)

for all $A,B\in\mathfrak{a}_{1}$ with $Z$ central.

Finally, the corresponding galilean Lie algebra is the quotient
$\widehat{\mathfrak{g}}/\mathbb{R}Z$ with underlying vector space
$\mathfrak{ss}\oplus\mathfrak{a}_{0}\oplus\mathfrak{a}_{1}\oplus\mathbb{R}D$,
where the bracket on $\mathfrak{a}_{1}\oplus\mathbb{R}D$ is given by

$[A,B]=0\qquad\text{and}\qquad[D,A]=D_{0}A,$ (4.7)

for all $A,B\in\mathfrak{a}_{1}$.

In summary, the ingredients to construct any bargmannian, carrollian and
galilean Lie algebras in diagram (3.16) are as follows:

  * •

a compact semisimple Lie algebra $\mathfrak{ss}$ with a choice of positive-
definite ad-invariant inner product: a negative multiple of the Killing form
on each of its simple factors;

  * •

a euclidean vector space $\mathfrak{a}_{0}$, whose only invariant is the
dimension; and

  * •

a euclidean vector space $\mathfrak{a}_{1}$ with a symplectic structure
$\omega$.

Any Lie algebra with an ad-invariant bargmannian, carrollian or galilean
structure can be constructed out of these ingredients. The metric Lie algebra
$\mathfrak{ss}\oplus\mathfrak{a}_{0}$ is always an orthogonal summand of these
Lie algebras, so we may concentrate on the euclidean/symplectic vector space
$(\mathfrak{a}_{1},\left<-,-\right>,\omega)$.

The bargmannian Lie algebra is of Nappi–Witten type and has underlying vector
space $\mathfrak{a}_{1}\oplus\mathbb{R}Z\oplus\mathbb{R}D$ and brackets, for
all $A,B\in\mathfrak{a}_{1}$, given by

$[A,B]=\omega(A,B)Z\qquad\text{and}\qquad[D,A]=D_{0}A,$ (4.8)

with $Z$ central and where $D_{0}$ is defined by
$\left<D_{0}A,B\right>=\omega(A,B)$. The underlying inner product extends the
one on $\mathfrak{a}_{1}$ by declaring $Z$ to be null and
$\left<D,Z\right>=1$. We may always redefine $D\mapsto
D-\tfrac{1}{2}\left<D,D\right>Z$, which is a Lie algebra automorphism, to
ensure that $\left<D,D\right>=0$.

The carrollian Lie algebra is a Heisenberg Lie algebra with underlying vector
space $\mathfrak{a}_{1}\oplus\mathbb{R}Z$ and brackets

$[A,B]=\omega(A,B)Z,$ (4.9)

with $Z$ central.

The galilean Lie algebra is an extension of the abelian Lie algebra
$\mathfrak{a}_{1}$ by the derivation $D_{0}$. It has underlying vector space
$\mathfrak{a}_{1}\oplus\mathbb{R}D$ and the only nonzero bracket is the action
$[D,A]=D_{0}A$, for all $A\in\mathfrak{a}_{1}$, of the derivation $D_{0}$.

In Appendix B we describe these Lie algebras in a basis.

Finally, we tackle the isomorphism problem. Which triples
$(\mathfrak{a}_{1},\left<-,-\right>,\omega)$ correspond to non-isomorphic Lie
algebras. Clearly if two such triples are in the same orbit of
$\operatorname{GL}(\mathfrak{a}_{1})$, then the resulting Lie algebras
(bargmannian, carrollian and galilean) are isomorphic. Hence we need to
classify triples $(\mathfrak{a}_{1},\left<-,-\right>,\omega)$ up to the action
of $\operatorname{GL}(\mathfrak{a}_{1})$. We may use
$\operatorname{GL}(\mathfrak{a}_{1})$ to relate any two scalar products
$\left<-,-\right>$, leaving the freedom to act with the stabiliser
$\operatorname{O}(\mathfrak{a}_{1})$ of the scalar product on the symplectic
form. In other words, which are the possible symplectic forms $\omega$ on
$\mathfrak{a}_{1}$ up to orthogonal transformations? This is the same problem
as classifying $D_{0}\in\mathfrak{so}(\mathfrak{a}_{1})$ up to the adjoint
action of $\operatorname{O}(\mathfrak{a}_{1})$. Using
$\operatorname{SO}(\mathfrak{a}_{1})$ we may always conjugate to a Cartan
subalgebra, or said differently, we can skew-diagonalise $D_{0}$ to a matrix
of the form

$\begin{pmatrix}{\color[rgb]{0.5,0.5,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.5,0.5}0}&\mu_{1}&&&\\\
-\mu_{1}&{\color[rgb]{0.5,0.5,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.5,0.5}0}&&&\\\
&&\ddots&&\\\
&&&{\color[rgb]{0.5,0.5,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.5,0.5}0}&\mu_{n}\\\
&&&-\mu_{n}&{\color[rgb]{0.5,0.5,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.5,0.5}0}\end{pmatrix}$
(4.10)

where $\mu_{i}$ are nonzero and ordered. Conjugating by the full orthogonal
group we may assume that $\mu_{1}\geq\mu_{2}\geq\dots\geq\mu_{n}>0$, where
$\dim\mathfrak{a}_{1}=2n$.

We may summarise this discussion as follows.

######  Theorem 4.2.

The isomorphism classes of bargmannian (and hence also carrollian and
galilean) Lie algebras are parametrised by the following data:

  * •

a compact Lie algebra with a choice of positive-definite ad-invariant scalar
product (i.e., the orthogonal direct sum of a semisimple and an abelian Lie
algebras); and

  * •

$(\mu_{1},\dots,\mu_{n})\in\mathbb{R}^{n}$ with
$\mu_{1}\geq\mu_{2}\geq\dots\geq\mu_{n}>0$, which are the skew-eigenvalues of
a symplectic form on the euclidean space $\mathbb{E}^{2n}$.

The Lie brackets and the ad-invariant bargmannian, carrollian and galilean
structures associated to that data can be constructed as explained above.

In other words, the list of bargmannian Lie algebras, from which we can then
obtain all carrollian and galilean dual pairs is parametrised by compact
semisimple Lie algebras with a choice of positive-definite invariant scalar
product, a euclidean space of dimension $m\geq 0$ and $n$ positive real
numbers $\mu_{1}\geq\dots\geq\mu_{n}>0$.

##  5\. Leibnizian Lie groups

In this section we characterise Lie groups with bi-invariant leibnizian
structures. Such Lie groups extend the class of bargmannian Lie groups and we
will see that they also contain carrollian Lie group as a normal subgroup and
they quotient to a galilean Lie group. At the Lie algebra level, they too sit
in relation to each other in a way described by a diagram isomorphic to (1.6).
Hence they define a non-canonical correspondence between carrollian and
galilean Lie algebras.

###  5.1. Leibnizian structures

In [20] Bekaert and Morand introduced the notion of an (ambient) leibnizian
structure on a manifold $M$. This is a triplet $(\xi,\psi,h)$ where
$\xi\in\mathcal{X}(M)$ is a nowhere-vanishing vector field,
$\psi\in\Omega^{1}(M)$ is a nowhere-vanishing one-form with $\psi(\xi)=0$, and
$h\in\Gamma(\odot^{2}(\ker\psi)^{*})$ is a corank-$1$, positive-semidefinite,
bilinear form on $\ker\psi\subset TM$ whose radical is spanned by $\xi$. Every
bargmannian manifold $(M,g,\xi)$ has a natural leibnizian structure given by
$\psi=\xi^{\flat}$ and $h=\left.g\right|_{\xi^{\perp}=\ker\xi^{\flat}}$.

A natural question is: _which Lie groups admit a bi-invariant leibnizian
structure?_ For the same price, we may (and will) actually consider pseudo-
leibnizian structures, where $h$ is not necessarily positive-semidefinite, but
still corank-$1$ on $\ker\psi$. We shall refer to such Lie groups and their
Lie algebras as leibnizian.

###  5.2. Leibnizian Lie algebras

######  Definition 5.1.

A Lie algebra $\mathfrak{g}$ is said to be leibnizian if it admits a nonzero
central element $Z\in\mathfrak{g}$, a nonzero ad-invariant
$\psi\in\mathfrak{g}^{*}$ (i.e.,
$\left.\psi\right|_{[\mathfrak{g},\mathfrak{g}]}=0$) with $\psi(Z)=0$ and an
ad-invariant $h\in\odot^{2}(\ker\psi)^{*}$ of corank $1$ whose radical is
spanned by $Z$.

It is immediate that if $(\mathfrak{g},Z,\psi,h)$ is leibnizian, then
$(\ker\psi,Z,h)$ is carrollian. As in Section 3.3,
$\mathfrak{g}_{0}=\ker\psi/\mathbb{R}Z$ is a metric Lie algebra. Let
$\pi:\ker\psi\to\mathfrak{g}_{0}$ send $X\mapsto\overline{X}$. Then the Lie
bracket and inner product on $\mathfrak{g}_{0}$ are defined by

$[\overline{X},\overline{Y}]_{0}=\overline{[X,Y]}\qquad\text{and}\qquad\left<\overline{X},\overline{Y}\right>_{0}=h(X,Y).$

As before, $\ker\psi$ is a central extension of the metric Lie algebra
$\mathfrak{g}_{0}$

${0}$${\mathbb{R}Z}$${\ker\psi}$${\mathfrak{g}_{0}}$${0,}$$\scriptstyle{\pi}$
(5.1)

and the central extension defines a skew-symmetric derivation $D_{0}$ of
$\mathfrak{g}_{0}$ as in equation (3.11).

Similarly, if $(\mathfrak{g},Z,\psi,h)$ is leibnizian, let
$\overline{\mathfrak{g}}:=\mathfrak{g}/\mathbb{R}Z$, so that $\mathfrak{g}$ is
a central extension of $\overline{\mathfrak{g}}$:

${0}$${\mathbb{R}Z}$${\mathfrak{g}}$${\overline{\mathfrak{g}}}$${0,}$$\scriptstyle{\overline{\pi}}$
(5.2)

where $\overline{\pi}:\mathfrak{g}\to\overline{\mathfrak{g}}$ sends
$X\mapsto\overline{X}$. The map $\pi:\ker\psi\to\mathfrak{g}_{0}$ above is the
restriction of $\overline{\pi}$ to the ideal $\ker\psi$, which explains why we
use the same notation. Since $\ker\psi$ is a codimension-$1$ ideal of
$\mathfrak{g}$, $\mathfrak{g}$ is an extension-by-derivation of $\ker\psi$:

${0}$${\ker\psi}$${\mathfrak{g}}$${\mathbb{R}}$${0}$$\scriptstyle{\psi}$ (5.3)

and hence it follows that $\mathfrak{g}_{0}$ is a codimension-$1$ ideal of
$\overline{\mathfrak{g}}$, so that $\overline{\mathfrak{g}}$ is an extension
by derivation of $\mathfrak{g}_{0}$:

${0}$${\mathfrak{g}_{0}}$${\overline{\mathfrak{g}}}$${\mathbb{R}}$${0,}$$\scriptstyle{\overline{\psi}}$
(5.4)

where $\overline{\psi}(\overline{X})=\psi(X)$.

In summary, we have again an instance of our favourite diagram:

${0}$${0}$${\mathbb{R}Z}$${\mathbb{R}Z}$${0}$${\ker\psi}$${\mathfrak{g}}$${\mathbb{R}}$${0}$${0}$${\mathfrak{g}_{0}}$${\overline{\mathfrak{g}}}$${\mathbb{R}}$${0}$${0}$${0}$$\scriptstyle{\pi}$$\scriptstyle{\psi}$$\scriptstyle{\overline{\pi}}$$\scriptstyle{\overline{\psi}}$
(5.5)

where $\ker\psi$ is carrollian and $\mathfrak{g}_{0}$ metric. We will now show
that $\overline{\mathfrak{g}}$ is galilean.

Let us split the exact sequences (5.3) and (5.4) by choosing
$D\in\mathfrak{g}$ with $\psi(D)=1$. Let $\overline{D}=\overline{\pi}(D)$ so
that $\overline{\psi}(\overline{D})=1$. As vector spaces,
$\mathfrak{g}=\ker\psi\oplus\mathbb{R}D$ and hence
$\overline{\mathfrak{g}}=\mathfrak{g}_{0}\oplus\mathbb{R}\overline{D}$.

######  Lemma 5.2.

$\operatorname{ad}_{D}$ induces
$\operatorname{ad}_{\overline{\mathfrak{g}}}\overline{D}$ which in turn
induces a skew-symmetric derivation $\overline{D}_{0}$ of
$(\mathfrak{g}_{0},\left<-,-\right>_{0})$ defined by

$\overline{D}_{0}\overline{X}:=[\overline{D},\overline{X}]=\overline{[D,X]},$

where the first bracket is in $\overline{\mathfrak{g}}$.

###### Proof.

The endomorphism $\overline{D}_{0}$ is well-defined because
$[D,X]\in[\mathfrak{g},\mathfrak{g}]\subset\ker\psi$ and $Z$ is central. We
show that it is a derivation. Let $X,Y\in\ker\psi$ and calculate

$\displaystyle\overline{D}_{0}[\overline{X},\overline{Y}]_{0}$
$\displaystyle=\overline{D}_{0}\overline{[X,Y]}$
$\displaystyle=\overline{[D,[X,Y]]}$
$\displaystyle=\overline{[[D,X],Y]+[X,[D,Y]]}$ (by Jacobi)
$\displaystyle=\overline{[[D,X],Y]}+\overline{[X,[D,Y]]}$
$\displaystyle=[\overline{[D,X]},\overline{Y}]_{0}+[\overline{X},\overline{[D,Y]}]_{0}$
$\displaystyle=[\overline{D}_{0}\overline{X},\overline{Y}]_{0}+[\overline{X},\overline{D}_{0}\overline{Y}]_{0}.$

To show that it is skew-symmetric we let $X,Y\in\ker\psi$ and calculate

$\displaystyle\left<\overline{D}_{0}\overline{X},\overline{Y}\right>_{0}$
$\displaystyle=\left<\overline{[D,X]},\overline{Y}\right>_{0}$
$\displaystyle=h([D,X],Y)$ $\displaystyle=-h(X,[D,Y])$ (since $h$ is
invariant) $\displaystyle=-\left<\overline{X},\overline{[D,Y]}\right>_{0}$
$\displaystyle=-\left<\overline{X},\overline{D}_{0}\overline{Y}\right>_{0}.$

∎

The covector $\overline{\psi}\in\overline{\mathfrak{g}}^{*}$, which is induced
from $\psi$, is $\overline{\mathfrak{g}}$-invariant since $\psi$ is
$\mathfrak{g}$-invariant. The inverse of the $\mathfrak{g}_{0}$-invariant
scalar product on $\mathfrak{g}_{0}=\ker\overline{\psi}$ pushes forward to a
symmetric tensor $\gamma\in\odot^{2}\overline{\mathfrak{g}}$. Since the
derivation $\overline{D}_{0}$ of $\mathfrak{g}_{0}$ induced by
$\operatorname{ad}_{\overline{\mathfrak{g}}}\overline{D}$ is skew-symmetric,
it follows that $\gamma$ is actually $\overline{\mathfrak{g}}$-invariant. In
other words, $(\overline{\mathfrak{g}},\overline{\psi},\gamma)$ is a galilean
Lie algebra.

###  5.3. A leibnizian Lie algebra which is not bargmannian

Comparing the commutative diagrams (5.5) and (3.16), we see that both
$\ker\psi$ and $\mathfrak{g}_{\text{car}}$ are carrollian, $\mathfrak{g}_{0}$
is metric in both diagrams, $\overline{\mathfrak{g}}$ and
$\mathfrak{g}_{\text{gal}}$ are both galilean, but whereas
$\widehat{\mathfrak{g}}$ is bargmannian, $\mathfrak{g}$ is leibnizian. It
would be tempting to conjecture that leibnizian Lie algebras are actually
bargmannian, but this turns out to be false.

In diagram (5.5) there are two skew-symmetric derivations of the metric Lie
algebra $\mathfrak{g}_{0}$ at play: the derivation $D_{0}$ associated to the
central extension defining $\ker\psi$ and the derivation $\overline{D}_{0}$ by
which we extend $\mathfrak{g}_{0}$ in order to construct
$\overline{\mathfrak{g}}$. If $\mathfrak{g}$ were bargmannian, and hence the
double extension of $\mathfrak{g}_{0}$ by a skewsymmetric derivation, both of
these derivations would coincide. In the general leibnizian case, however,
they need not coincide and this gives us a hint how to prove that the class of
bargmannian Lie algebras is a proper subclass of the leibnizian Lie algebras.

We will do this by constructing a leibnizian Lie algebra which is not
bargmannian. Let us take $\mathfrak{g}_{0}$ to be abelian with a euclidean
inner product $\left<-,-\right>_{0}$. Let
$D_{0}\in\mathfrak{so}(\mathfrak{g}_{0})$ and define a Lie algebra
$\mathfrak{g}_{\text{car}}$ with underlying vector space
$\mathfrak{g}_{0}\oplus\mathbb{R}Z$ and Lie brackets

$[X,Y]=\left<D_{0}X,Y\right>_{0}Z\qquad\text{and $Z$ central,}$ (5.6)

for all $X,Y\in\mathfrak{g}_{0}$. Pick $D\in\mathfrak{so}(\mathfrak{g}_{0})$
in a different $\operatorname{SO}(\mathfrak{g}_{0})$-orbit than $D_{0}$. Let
$\mathfrak{g}$ be the Lie algebra with underlying vector space
$\mathfrak{g}_{0}\oplus\mathbb{R}Z\oplus\mathbb{R}W$ with Lie brackets

$\begin{split}[X,Y]&=\left<D_{0}X,Y\right>_{0}Z\\\ [W,X]&=DX\\\
[Z,-]&=0,\end{split}$ (5.7)

for all $X,Y\in\mathfrak{g}_{0}$. We will show one example of this
construction for which $\mathfrak{g}$ is not bargmannian. Let
$\mathfrak{g}_{0}$ be four-dimensional, with orthonormal basis $e_{i}$,
$i=1,\dots,4$ and let $D_{0},D\in\mathfrak{so}(\mathfrak{g}_{0})$ be given
relative to this basis by the matrices

$D_{0}=\begin{pmatrix}{\color[rgb]{0.5,0.5,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.5,0.5}0}&1&{\color[rgb]{0.5,0.5,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.5,0.5}0}&{\color[rgb]{0.5,0.5,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.5,0.5}0}\\\
-1&{\color[rgb]{0.5,0.5,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.5,0.5}0}&{\color[rgb]{0.5,0.5,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.5,0.5}0}&{\color[rgb]{0.5,0.5,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.5,0.5}0}\\\
{\color[rgb]{0.5,0.5,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.5,0.5}0}&{\color[rgb]{0.5,0.5,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.5,0.5}0}&{\color[rgb]{0.5,0.5,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.5,0.5}0}&\alpha\\\
{\color[rgb]{0.5,0.5,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.5,0.5}0}&{\color[rgb]{0.5,0.5,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.5,0.5}0}&-\alpha&{\color[rgb]{0.5,0.5,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.5,0.5}0}\end{pmatrix}\qquad\text{and}\qquad
D=\begin{pmatrix}{\color[rgb]{0.5,0.5,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.5,0.5}0}&\beta&{\color[rgb]{0.5,0.5,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.5,0.5}0}&{\color[rgb]{0.5,0.5,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.5,0.5}0}\\\
-\beta&{\color[rgb]{0.5,0.5,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.5,0.5}0}&{\color[rgb]{0.5,0.5,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.5,0.5}0}&{\color[rgb]{0.5,0.5,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.5,0.5}0}\\\
{\color[rgb]{0.5,0.5,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.5,0.5}0}&{\color[rgb]{0.5,0.5,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.5,0.5}0}&{\color[rgb]{0.5,0.5,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.5,0.5}0}&\gamma\\\
{\color[rgb]{0.5,0.5,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.5,0.5}0}&{\color[rgb]{0.5,0.5,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.5,0.5}0}&-\gamma&{\color[rgb]{0.5,0.5,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.5,0.5}0}\end{pmatrix}.$
(5.8)

Let $\mathfrak{g}$ be the six-dimensional Lie algebra spanned by
$e_{1},\dots,e_{4},e_{+},e_{-}$ with nonzero brackets

$[e_{i},e_{j}]=(D_{0})_{ij}e_{+}\qquad\text{and}\qquad[e_{i},e_{-}]=D_{ij}e_{j},$
(5.9)

or explicitly,

$\begin{aligned} [e_{1},e_{2}]&=e_{+}\\\ [e_{1},e_{-}]&=\beta e_{2}\\\
[e_{2},e_{-}]&=-\beta e_{1}\\\
\end{aligned}\qquad\text{and}\qquad\begin{aligned} [e_{3},e_{4}]&=\alpha
e_{+}\\\ [e_{3},e_{-}]&=\gamma e_{4}\\\ [e_{4},e_{-}]&=-\gamma e_{3}\\\
\end{aligned}$ (5.10)

Let $\theta^{1},\dots,\theta^{4},\theta^{+},\theta^{-}$ denote the canonical
dual basis for $\mathfrak{g}^{*}$. Then the Lie algebra $\mathfrak{g}$ is
leibnizian with $Z=e_{+}$, $\psi=\theta^{-}$ and
$h=\sum_{i=1}^{4}(\theta^{i})^{2}$. A short calculation shows that the most
general ad-invariant symmetric bilinear form on $\mathfrak{g}$ is given up to
scale by

$\begin{pmatrix}\gamma&{\color[rgb]{0.5,0.5,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.5,0.5}0}&{\color[rgb]{0.5,0.5,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.5,0.5}0}&{\color[rgb]{0.5,0.5,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.5,0.5}0}&{\color[rgb]{0.5,0.5,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.5,0.5}0}&{\color[rgb]{0.5,0.5,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.5,0.5}0}\\\
{\color[rgb]{0.5,0.5,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.5,0.5}0}&\gamma&{\color[rgb]{0.5,0.5,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.5,0.5}0}&{\color[rgb]{0.5,0.5,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.5,0.5}0}&{\color[rgb]{0.5,0.5,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.5,0.5}0}&{\color[rgb]{0.5,0.5,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.5,0.5}0}\\\
{\color[rgb]{0.5,0.5,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.5,0.5}0}&{\color[rgb]{0.5,0.5,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.5,0.5}0}&\alpha\beta&{\color[rgb]{0.5,0.5,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.5,0.5}0}&{\color[rgb]{0.5,0.5,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.5,0.5}0}&{\color[rgb]{0.5,0.5,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.5,0.5}0}\\\
{\color[rgb]{0.5,0.5,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.5,0.5}0}&{\color[rgb]{0.5,0.5,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.5,0.5}0}&{\color[rgb]{0.5,0.5,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.5,0.5}0}&\alpha\beta&{\color[rgb]{0.5,0.5,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.5,0.5}0}&{\color[rgb]{0.5,0.5,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.5,0.5}0}\\\
{\color[rgb]{0.5,0.5,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.5,0.5}0}&{\color[rgb]{0.5,0.5,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.5,0.5}0}&{\color[rgb]{0.5,0.5,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.5,0.5}0}&{\color[rgb]{0.5,0.5,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.5,0.5}0}&\lambda&-\beta\gamma\\\
{\color[rgb]{0.5,0.5,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.5,0.5}0}&{\color[rgb]{0.5,0.5,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.5,0.5}0}&{\color[rgb]{0.5,0.5,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.5,0.5}0}&{\color[rgb]{0.5,0.5,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.5,0.5}0}&-\beta\gamma&{\color[rgb]{0.5,0.5,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.5,0.5}0}\\\
\end{pmatrix}$ (5.11)

for any $\lambda\in\mathbb{R}$. This matrix is non-degenerate if and only if
$\alpha\beta\gamma\neq 0$. So we can take either of $\alpha$, $\beta$ or
$\gamma$ to be zero (with, say, the other two nonzero) and arrive at a
leibnizian Lie algebra which is not bargmannian.

##  6\. Conclusion

In this paper we have characterised Lie algebras admitting ad-invariant
bargmannian, carrollian, and galilean structures. We have shown that they are
all given by the same data: namely a metric Lie algebra and a skew-symmetric
derivation. This sets up a canonical correspondence between carrollian and
galilean Lie algebras.

We have shown that these Lie algebras, together with the underlying metric Lie
algebra, fit into a commutative diagram (3.16) of Lie algebras which is
isomorphic to a similar diagram (1.6) involving the Bargmann, Carroll and
Galilei Lie algebras (despite them not being bargmannian, carrollian nor
galilean) and also a similar diagram [14, App. B] involving some infinite-
dimensional Lie algebras playing a rôle in the hamiltonian description of
particle dynamics on the homogeneous galilean spacetimes, whose transitive Lie
algebras together with their Bargmann extension and the Carroll Lie algebra
itself also fit in yet another commutative diagram (1.10) isomorphic to the
other commutative diagrams. This diagrammatic coincidence, if indeed it is a
coincidence, deserves to be further studied.

For the strict carrollian and galilean cases, the underlying metric Lie
algebra is positive-definite and this allows a classification up to isometric
isomorphism and the Lie algebras are orthogonal direct sums of a compact Lie
algebra with a choice of positive-definite inner product and either a
Nappi–Witten Lie algebra (in the bargmannian case), a Heisenberg Lie algebra
(in the carrollian case) or an extension by skew-symmetric derivation of an
abelian metric Lie algebra (in the galilean case).

We also characterised Lie algebras admitting an ad-invariant leibnizian
structure and we saw that they too mediate a Lie algebraic Carroll/Galilei
correspondence which is however not canonical. Indeed there is some choice in
defining the galilean dual of a carrollian Lie algebra. This is the phenomenon
already observed at the level of the spatially isotropic homogeneous galilean
spacetimes. Indeed, as we could see already in diagram (1.10), the Carroll
spacetime (as the unique kinematical spacetime associated to the Carroll
algebra $\mathfrak{c}$) is apparently dual to any one of the galilean
spacetimes simply because $\mathfrak{c}$ is an ideal of any of the
$\mathfrak{b}_{\alpha,\beta}$ which are extensions by (different) derivations
of $\mathfrak{c}$. Here too, given a carrollian Lie algebra
$\mathfrak{g}_{\text{car}}$ we arrive at a metric Lie algebra
$\mathfrak{g}_{0}$ with a skew-symmetric derivation $D_{0}$. We may choose to
extend $\mathfrak{g}_{0}$ by this very derivation in order to obtain the
galilean dual Lie algebra $\mathfrak{g}_{\text{gal}}$ in diagram (3.16) (with
$\widehat{\mathfrak{g}}$ bargmannian) or we could choose any other skew-
symmetric derivation and arrive in this way at a leibnizian Lie algebra
$\mathfrak{g}$. In a way, the bargmannian case is canonical, since it is
uniquely determined by the carrollian Lie algebra, but the leibnizian case
provides added flexibility in the construction.

## Acknowledgments

I first reported some of these results at the 2nd Carroll Workshop in Mons on
12 September 2022. It is a pleasure to thank Nicolas Boulanger, Andrea
Campoleoni and Yannick Herfray for the invitation to participate in the
meeting. It is also a pleasure to thank Kevin Morand for a useful conversation
during the conference and for some very insightful correspondence afterwards;
Quim Gomis for interesting conversations about Carroll/Galilei duality; Stefan
Prohazka for comments and Dieter Van den Bleeken for a careful reading of a
previous version of this note; and Luminita Maxim for the Romanian lesson.
Last, but by far not least, I would like to dedicate this paper to the memory
of Veronica Stanciu (1925-2022). May she finally rest in peace.

## Data availability

Data sharing not applicable to this article as no datasets were generated or
analysed during the current study.

##  Appendix A Skew-symmetric derivations of a reductive metric Lie algebra

In this appendix we give a proof of Proposition 4.1, which we rephrase
slightly.

Let $\mathfrak{g}_{0}=\mathfrak{ss}\oplus\mathfrak{a}$ be an orthogonal direct
sum of a compact semisimple Lie algebra $\mathfrak{ss}$ with a choice of a
positive-definite inner product (i.e., a negative multiple of the Killing form
on each simple summand) and an abelian Lie algebra $\mathfrak{a}$ with a
choice of euclidean inner product.

######  Proposition A.1.

Let $D$ be a skew-symmetric derivation of $\mathfrak{g}_{0}$. Then
$D=D_{1}+D_{2}$ where $D_{1}=\operatorname{ad}_{X}$ for some
$X\in\mathfrak{ss}$ and $D_{2}\in\mathfrak{so}(\mathfrak{a})$.

###### Proof.

Let $X_{1}\in\mathfrak{ss}$ and $X_{2}\in\mathfrak{a}$. Then since they
commute and $D$ is a derivation

$0=D[X_{1},X_{2}]=[DX_{1},X_{2}]+[X_{1},DX_{2}].$

Let $Y_{1}\in\mathfrak{ss}$ and calculate the inner product

$\displaystyle 0$ $\displaystyle=\left<D[X_{1},X_{2}],Y_{1}\right>$
$\displaystyle=\left<[DX_{1},X_{2}]+[X_{1},DX_{2}],Y_{1}\right>$
$\displaystyle=\left<DX_{1},[X_{2},Y_{1}]\right>-\left<DX_{2},[X_{1},Y_{1}]\right>.$

The first term in the RHS is zero because $[X_{2},Y_{1}]=0$ and hence the
second term says that $DX_{2}$ is perpendicular to
$[\mathfrak{ss},\mathfrak{ss}]=\mathfrak{ss}$. In other words,
$DX_{2}\in\mathfrak{a}$ and hence $D$ preserves $\mathfrak{a}$. But then,
since $\mathfrak{ss}$ and $\mathfrak{a}$ are orthogonal and $D$ is skew-
symmetric,

$0=\left<DX_{1},X_{2}\right>+\left<X_{1},DX_{2}\right>.$

where the second term vanishes since
$DX_{2}\in\mathfrak{a}=\mathfrak{ss}^{\perp}$. Therefore $DX_{1}$ is
perpendicular to $\mathfrak{a}$ and hence $D$ also preserves $\mathfrak{ss}$.
In other words, $D=D_{1}+D_{2}$, where $D_{1}$ is a skew-symmetric derivation
of $\mathfrak{ss}$ and $D_{2}$ is a skew-symmetric derivation of
$\mathfrak{a}$. Since $\mathfrak{ss}$ is semisimple, all derivations are inner
and since the inner product is built out of the Killing form, all inner
derivations are also skew-symmetric, so that $D_{1}=\operatorname{ad}_{X}$ for
some $X\in\mathfrak{ss}$. Finally, since $\mathfrak{a}$ is abelian, any
endomorphism is a derivation and hence $D_{2}\in\mathfrak{so}(\mathfrak{a})$
is any skew-symmetric endomorphism. ∎

Now Proposition 4.1 follows because if
$\mathfrak{ss}=\mathfrak{s}_{1}\oplus\cdots\oplus\mathfrak{s}_{k}$, then
$X\in\mathfrak{ss}$ can be written as $X=\sum_{i}X_{i}$.

##  Appendix B Once more, with indices

In this appendix we will write down the bargmannian, carrollian and galilean
Lie algebras in terms of a basis.

Let $\mathfrak{g}_{0}$ be a metric Lie algebra with basis $(X_{a})$, Lie
brackets $[X_{a},X_{b}]=f_{ab}{}^{c}X_{c}$ and scalar product
$\left<X_{a},X_{b}\right>=\eta_{ab}$. Invariance of the inner product simply
says that $f_{abc}:=f_{ab}{}^{d}\eta_{dc}$ is totally skew-symmetric.

Let $D_{0}:\mathfrak{g}_{0}\to\mathfrak{g}_{0}$ be a skew-symmetric
derivation. Relative to the above basis for $\mathfrak{g}_{0}$, we write
$D_{0}X_{a}=X_{b}\omega^{b}{}_{a}$, where the skew-symmetry condition says
that $\omega_{ab}=\eta_{ac}\omega^{c}{}_{b}=-\omega_{ba}$ and the fact that
$D_{0}$ is a derivation says that

$\omega^{d}{}_{c}f_{ab}{}^{c}=f_{cb}{}^{d}\omega^{c}{}_{a}+f_{ac}{}^{d}\omega^{c}{}_{b}.$
(B.1)

The bargmannian Lie algebra $\widehat{\mathfrak{g}}$ associated to the data
$(\mathfrak{g}_{0},\left<-,-\right>,D_{0})$ is the double-extension, which has
basis $(X_{a},X_{+},X_{-})$ and Lie brackets

$[X_{a},X_{b}]=f_{ab}{}^{c}X_{c}+\omega_{ab}X_{+},\qquad[X_{-},X_{a}]=X_{b}\omega^{b}{}_{a}\qquad\text{and}\qquad[X_{+},-]=0,$
(B.2)

and the invariant scalar product can always be brought to a form where the
only nonzero entries are:

$\left<X_{a},X_{b}\right>=\eta_{ab}\qquad\text{and}\qquad\left<X_{+},X_{-}\right>=1.$
(B.3)

The associated carrollian Lie algebra $\mathfrak{g}_{\text{car}}$ is the ideal
of $\widehat{\mathfrak{g}}$ spanned by $(X_{a},X_{+})$ with induced brackets

$[X_{a},X_{b}]=f_{ab}{}^{c}X_{c}+\omega_{ab}X_{+}\qquad\text{and}\qquad[X_{+},-]=0.$
(B.4)

The invariant carrollian structure consists of $X_{+}$ and the invariant
symmetric bilinear form $\left<-,-\right>$ which is degenerate since
$\left<X_{+},-\right>=0$.

The associated galilean Lie algebra $\mathfrak{g}_{\text{gal}}$ is the
quotient of $\widehat{\mathfrak{g}}$ by the ideal spanned by $X_{+}$. It is
spanned by $(X_{a},X_{-})$ and the brackets are

$[X_{a},X_{b}]=f_{ab}{}^{c}X_{c}\qquad\text{and}\qquad[X_{-},X_{a}]=X_{b}\omega^{b}{}_{a}.$
(B.5)

Let $(\theta^{a},\theta^{-})$ be a canonical dual basis for
$\mathfrak{g}_{\text{gal}}^{*}$. Then the invariant galilean structure is
given by $\theta^{-}$ and $\gamma=\eta^{ab}X_{a}X_{b}$, where $\eta^{ab}$ are
the entries of the inverse of the restriction of $\eta$ to $\mathfrak{g}_{0}$.

In the special case where $\eta_{ab}$ is positive-definite we discussed in
Section 4, we showed that these Lie algebras are the orthogonal direct sums of
certain “ primitive” Lie algebras with a compact Lie algebra with a positive-
definite invariant inner product (i.e., the direct sum of a compact semisimple
Lie algebra with a choice of positive-definite invariant inner product and an
abelian Lie algebra with a euclidean inner product). These primitive Lie
algebras can be read off from the above expressions by setting
$f_{ab}{}^{c}=0$, $\eta_{ab}$ positive-definite and $\omega_{ab}$ symplectic.

Explicitly, the bargmannian Lie algebra is a Nappi–Witten Lie algebra with
brackets

$[X_{a},X_{b}]=\omega_{ab}X_{+},\qquad[X_{-},X_{a}]=X_{b}\omega^{b}{}_{a}\qquad\text{and}\qquad[X_{+},-]=0,$
(B.6)

with inner product $\left<X_{a},X_{b}\right>=\eta_{ab}$ and
$\left<X_{+},X_{-}\right>=1$. Of course, we can always choose an orthonormal
basis where $\eta_{ab}=\delta_{ab}$ and then use orthogonal transformations to
bring $\omega_{ab}$ to a normal form

$\omega=\lambda_{1}\theta^{1}\wedge\theta^{2}+\lambda_{2}\theta^{3}\wedge\theta^{4}+\cdots+\lambda_{\ell}\theta^{2\ell+1}\wedge\theta^{2\ell}$
(B.7)

where $\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{\ell}\geq 0$, where
the Nappi–Witten Lie algebra has dimension $2\ell+2$.

The carrollian Lie algebra is a Heisenberg Lie algebra with brackets

$[X_{a},X_{b}]=\omega_{ab}X_{+}\qquad\text{and}\qquad[X_{+},-]=0,$ (B.8)

and the only nonzero brackets of the galilean Lie algebra are

$[X_{-},X_{a}]=X_{b}\omega^{b}{}_{a}.$ (B.9)

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