# Non-relativistic spin-$\mathbf{3}$ symmetries in $\mathbf{2+1}$ dimensions
from expanded/extended Nappi-Witten algebras

Ricardo Caroca  Diego M. Peñafiel  Patricio Salgado-Rebolledo

###### Abstract

We show that infinite families of Galilean spin-$3$ symmetries in $2+1$
dimensions, which include higher-spin extensions of the Bargmann, Newton-
Hooke, non-relativistic Maxwell, and non-relativistic AdS-Lorentz algebras,
can be obtained as Lie algebra expansions of two different spin-$3$ extensions
of the Nappi-Witten symmetry. These higher-spin Nappi-Witten algebras, in
turn, are obtained by means of Inönü-Wigner contractions applied to suitable
direct product extensions of $\mathfrak{sl}(3,\mathbb{R})$. Conversely, we
show that the same result can be obtained by considering contractions of
expanded $\mathfrak{sl}(3,\mathbb{R})$ algebras. The method can be used to
define non-relativistic higher-spin Chern-Simon gravity theories in $2+1$
dimensions in a systematic way.

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###### Contents

  1. 1 Introduction
  2. 2 Spin-$3$ extensions of the Nappi-Witten algebra
    1. 2.1 NR contraction of $\mathfrak{sl}(3,\mathbb{R})\oplus\mathfrak{so}(3)$
    2. 2.2 NR contraction of $\mathfrak{sl}(3,\mathbb{R})\oplus\mathfrak{sl}(2,\mathbb{R})\oplus\mathfrak{so}(2)$
  3. 3 Three-dimensional higher-spin gravity from spin-$3$ Nappi-Witten algebras
    1. 3.1 Expansions of higher-spin Nappi-Witten algebras
    2. 3.2 Invariant bilinear forms
    3. 3.3 Chern-Simons actions
    4. 3.4 Non-relativistic limit of expanded $\mathfrak{sl}(3,\mathbb{R})$ algebras
    5. 3.5 Flat limit
  4. 4 Examples
    1. 4.1 Spin-$3$ Bargmann algebras
    2. 4.2 Spin-$3$ non-relativistic Maxwell algebras
    3. 4.3 Spin-$3$ Newton-Hooke algebras
    4. 4.4 Spin-$3$ non-relativistic AdS-Lorentz algebras
    5. 4.5 Higher-spin gravity theories
  5. 5 Concluding Remarks
  6. A Non-relativistic spin-$3$ Chern-Simons theories
    1. A.1 Newton-Hooke and Bargmann cases
    2. A.2 Non-relativistic AdS-Lorentz and Maxwell cases

##  1 Introduction

The study of higher-spin fields in $2+1$ dimensions has received considerable
attention in recent years due to the possibility to overcome some of the
difficulties appearing in higher-dimensional cases, where in order to surpass
a series of old no-go theorems [1] and obtain a consistent theory, the
inclusion of a cosmological constant as well as an infinite tower of
interacting higher-spin fields is needed [2, 3]. In fact, in the massless
sector, a three-dimensional higher-spin gravity theory with negative
cosmological constant that includes a finite number of higher-spin fields can
be defined by means of a Chern-Simons (CS) action [4, 5, 6]. Even though CS
theories unavoidably lead to no local propagating degrees of freedom in the
bulk, higher-spin CS gravity has been shown to admit black hole solutions with
non-trivial properties (see the reviews [7, 8] and references therein).
Furthermore, considering suitable boundary conditions gives rise to rich
boundary dynamics and asymptotic symmetries given by $\mathcal{W}$-algebras
[9, 10]. Higher-spin CS theories were later generalized to the case of
vanishing cosmological constant [11, 12] and non-Lorentzian gravity [13, 14].

The recent interest in higher-spin fields in three space-time dimensions goes
beyond gravitational physics. A remarkable $(2+1)$-dimensional system where
higher-spin symmetries play an important role is the fractional quantum Hall
effect, where $\mathcal{W}$-algebras appear as the quantum realization of the
area-preserving diffeormorphism algebra [15, 16, 17]. Moreover, it has been
conjectured that, apart from the Magnetoroton [18], a spin-$2$ graviton-like
mode, additional collective excitations described by higher-spin fields should
be introduced in the known effective field theory description to reproduce the
full Girvin-Macdonald-Platzman algebra [19, 20]. Another exotic scenario where
higher-spin fields can be relevant is Fracton physics (see the reviews [21,
22] and references therein), which introduces quasiparticles with restricted
mobility whose local interactions are described by higher-rank gauge fields.
The higher-spin theories capable of describing these types of physical systems
are, however, non-relativistic (NR).

NR symmetries and NR geometry have attracted recent interest due to their
applicability in extensions of holography [23, 24, 25, 26] and condensed
matter physics [27, 28, 29, 30]. The extension of kinematical symmetries to
include higher-spin generators was introduced in [13], where in particular two
different spin-$3$ extensions of the Extended Bargmann and Newton-Hooke
algebras were given111Galilean and Carrollian conformal higher-spin symmetries
in arbitrary dimensions have been recently constructed in [31].. On the other
hand, Lie algebra expansions methods [32, 33, 34, 35, 36, 37] have been
extensively used in the construction of extended non-Lorentzian symmetries
[38, 39, 40, 41, 42, 43, 44, 45, 46, 47]. Particularly, in [41] it has been
shown that applying the expansion method to the Nappi-Witten algebra allows
one to define infinite families of NR symmetries in $2+1$ dimensions that
generalize the Bargmann and the Newton-Hooke symmetries. These are central
extensions of the Generalized Galilean symmetries presented in [48] for
$D=2+1$. This result can be contrasted with the families of relativistic
spin-$3$ symmetries in $2+1$ dimensions found in [49] by means of Lie algebra
expansions applied to the $\mathfrak{sl}(3,\mathbb{R})$ algebra. In
particular, two families were constructed explicitly, which define generalized
spin-$3$ symmetries of AdS type and Poincaré type. In this setup, the first
_level_ of the expansion yields the spin-$3$ extension of the AdS algebra,
$\mathfrak{sl}(3,\mathbb{R})\times\mathfrak{sl}(3,\mathbb{R})$, and the
spin-$3$ extension of the Poincaré algebra, respectively. At the second level,
spin-$3$ extensions of the $(2+1)$-dimensional version of the Maxwell algebra
and the AdS-Lorentz algebra arise, whereas higher levels correspond to
spin-$3$ extensions of more complicated examples of the so-called Generalized
AdS [50] and Generalized Poincaré [51] algebras in three space-time
dimensions. Therefore, if higher-spin extensions of the Nappi-Witten symmetry
can be defined, expansions of such symmetries can be used to define NR limits
of the higher-spin symmetries defined in [49]. In $2+1$ dimensions, NR limits
usually require to extend relativistic symmetries by introducing extra Abelian
generators as a way to obtain a non-degenerate invariant form in the resulting
NR algebra (see for example [52]). Similarly, to define proper NR contractions
of $\mathfrak{sl}(3,\mathbb{R})$, it will be necessary to consider suitable
extensions thereof. This time, however, the necessary extensions to be
introduced are non-Abelian. In particular, we find two higher-spin extensions
of the $\mathfrak{nw}$ symmetry, which we dub $\mathfrak{hs_{3}nw1}$ and
$\mathfrak{hs_{3}nw2}$, and follow from two inequivalent contractions applied
to $\mathfrak{sl}(3,\mathbb{R})\oplus\mathfrak{so}(3)$ and
$\mathfrak{sl}(3,\mathbb{R})\oplus\mathfrak{sl}(2,\mathbb{R})\oplus\mathfrak{so}(2)$,
respectively.

In this article, i) we define two different spin-$3$ extensions of the Nappi-
Witten symmetry, which we denominate $\mathfrak{hs_{3}nw1}$ and
$\mathfrak{hs_{3}nw2}$, by means of Inönü-Wigner contractions [53] of suitable
direct product extensions of the $\mathfrak{sl}(3,\mathbb{R})$ algebra, ii) we
show that expansions of these spin-$3$ Nappi-Witten algebras define NR higher-
spin symmetries in $2+1$ dimensions, iii) we show that the same result can be
achieved by first expanding the $\mathfrak{sl}(3,\mathbb{R})$ algebra and
subsequently defining a NR contraction of (an extension of) the resulting
expanded symmetry, iv) we construct non-degenerate invariant bilinear forms on
the resulting NR higher-spin algebras and define the corresponding higher-spin
CS gravity theories.

The paper is organized as follows: in Section 2 we show how the
$\mathfrak{nw}$ algebra can be defined as a NR contraction of the
$\mathfrak{sl}(2,\mathbb{R})\oplus\mathfrak{so}(2)$ algebra and extend this
result to define a NR contraction of the
$\mathfrak{sl}(3,\mathbb{R})\oplus\mathfrak{so}(3)$ algebra and the
$\mathfrak{sl}(3,\mathbb{R})\oplus\mathfrak{sl}(2,\mathbb{R})\oplus\mathfrak{so}(2)$
algebra, which lead to two different higher-spin extensions of the
$\mathfrak{nw}$ symmetry. In Section 3 we construct expanded Nappi-Witten
algebras by means of Lie algebra expansions based on Abelian semigroups. The
method is also used to define invariant bilinear forms on the expanded
algebras and CS actions. In Section 4, we show that the spin-$3$ extensions of
the Extended Bargmann algebra and the NR Maxwell algebra in $2+1$ dimensions
correspond to particular examples of expanded higher-spin Nappi-Witten
algebras. By considering a different semigroup, a cosmological constant can be
introduced, leading to a spin-$3$ extension of the Newton-Hooke and the NR
AdS-Lorentz algebras. The algebraic structure of these symmetries is discussed
and the explicit form of the corresponding CS higher-spin gravity theories is
given. Finally, we end with a general discussion of our results and elaborate
on possible future directions.

Note added: during the preparation of the manuscript, it came to our knowledge
the development of Ref. [54], where a spin-$3$ generalization of the Lie
algebra expansions considered [38, 42] for $D=2+1$ are worked out, and
presents some overlap with the results included in Section 4.3 of our article.

##  2 Spin-$3$ extensions of the Nappi-Witten algebra

In this section, we construct two different spin-$3$ extensions of the Nappi-
Witten algebra $\mathfrak{nw}$ by considering suitable Inöü-Wigner
contractions of extensions of the $\mathfrak{sl}(3,\mathbb{R})$ algebra. To
make our construction more transparent, we start by reviewing how the
$\mathfrak{nw}$ algebra itself can be obtained as a contraction of an Abelian
extension of $\mathfrak{sl}(2,\mathbb{R})$222This type of contraction was
introduced by Saletan [55] and later generalized to higher dimensions in [56,
57].. The starting point is the $\mathfrak{sl}(2,\mathbb{R})$ algebra

$\left[\boldsymbol{L}_{A},\boldsymbol{L}_{B}\right]=\epsilon_{AB}^{\quad
C}\boldsymbol{L}_{C},\qquad A=0,1,2,$ (2.1)

which is isomorphic to the Lorentz algebra in $2+1$ dimensions,
$\mathfrak{so}(2,1)$. Here $\epsilon_{012}=1$ and the relativistic indices are
lowered and raised with the Minkowski metric $\eta_{AB}={\rm diag}(-1,1,1)$
and its inverse. A NR limit can be defined by introducing a $\mathfrak{so}(2)$
generator $\boldsymbol{X}$ and redefining the Lie algebra generators in the
following way

$\displaystyle\boldsymbol{L}_{0}$
$\displaystyle=\dfrac{1}{2}\boldsymbol{k}+c^{2}\boldsymbol{\sigma},\qquad\boldsymbol{L}_{a}$
$\displaystyle=c\,\boldsymbol{k}_{a},\qquad\boldsymbol{X}$
$\displaystyle=\frac{1}{2}\boldsymbol{k}-c^{2}\boldsymbol{\sigma},$ (2.2)

where have introduced the parameter $c$, which stands for the speed of light,
and decomposed the relativistic indices as $A=(0,a=1,2)$. The new generators
$\left\\{\boldsymbol{k},\boldsymbol{k}_{a},\boldsymbol{\sigma}\right\\}$ are
then expressed in terms of $\boldsymbol{L}_{a}$ and $\boldsymbol{X}$ through
the relations

$\displaystyle\boldsymbol{k}$
$\displaystyle=\boldsymbol{L}_{0}+\boldsymbol{X},\qquad\boldsymbol{k}_{a}$
$\displaystyle=\frac{1}{c}\boldsymbol{L}_{a},\qquad\boldsymbol{\sigma}$
$\displaystyle=\frac{1}{2c^{2}}\left(\boldsymbol{L}_{0}-\boldsymbol{X}\right),$
(2.3)

and, in the limit $c\rightarrow\infty$, their commutators reproduce the Nappi-
Witten algebra $\mathfrak{nw}$

$\left[\boldsymbol{k},\boldsymbol{k}_{a}\right]=\epsilon_{ab}\,\boldsymbol{k}_{b},\qquad\left[\boldsymbol{k}_{a},\boldsymbol{k}_{b}\right]=-\epsilon_{ab}\,\boldsymbol{\sigma}\qquad(\epsilon_{ab}\equiv\epsilon_{0ab}).$
(2.4)

Due to the central term $\boldsymbol{\sigma}$, the $\mathfrak{nw}$ algebra
admits a non-degenerate invariant bilinear form (see Section 3.2). The Nappi-
Witten symmetry is the central extension of the Euclidean symmetry in two
dimensions333This corresponds to the Euclidean version of the Nappi-Witten
algebra presented in [58]. [59]. As we have just shown, it can be interpreted
as a centrally extended NR limit of the $\mathfrak{sl}(2,\mathbb{R})$ algebra,
pretty much in the same way as the Bargmann algebra is a centrally extended NR
limit of the Galilean one. It is thus evident that, in order to generalize the
previous construction and define a symmetry that could be understood as a NR
contraction of $\mathfrak{sl}(3,\mathbb{R})$ in the sense of (2.2), it will be
necessary to properly extend the $\mathfrak{sl}(3,\mathbb{R})$ algebra. Its
commutation relations read444The convention for the symmetrization of indices
is $x_{(A}y_{B)}=x_{A}y_{B}+x_{B}y_{A}$, without normalization factor. [10]

$\left[\boldsymbol{L}_{A},\boldsymbol{L}_{B}\right]=\epsilon_{AB}^{\quad
C}\boldsymbol{L}_{C}\,\quad\left[\boldsymbol{L}_{A},\boldsymbol{\mathcal{L}}_{BC}\right]=\epsilon_{A(B}^{\quad\;D}\boldsymbol{\mathcal{L}}_{C)D}\,\quad\left[\boldsymbol{\mathcal{L}}_{AB},\boldsymbol{\mathcal{L}}_{CD}\right]=-\eta_{(A(C}\epsilon_{D)B)}^{\quad\;E}\boldsymbol{L}_{E},$
(2.5)

where $\boldsymbol{L}_{A}$ defines an $\mathfrak{sl}(2,\mathbb{R})$
sublagelbra and $\boldsymbol{\mathcal{L}}_{AB}$ is traceless symmetric
generator, i.e.

$\boldsymbol{\mathcal{L}}_{AB}=\boldsymbol{\mathcal{L}}_{BA},\quad\eta^{AB}\boldsymbol{\mathcal{L}}_{AB}=0\quad\Rightarrow\quad\boldsymbol{\mathcal{L}}_{00}=\delta^{ab}\boldsymbol{\mathcal{L}}_{ab}\equiv\boldsymbol{\mathcal{L}}_{aa}.$
(2.6)

Splitting the relativistic indices into temporal and spatial components yields

$\displaystyle\left[\boldsymbol{L}_{0},\boldsymbol{L}_{a}\right]=\epsilon_{ab}\boldsymbol{L}_{b},$
$\displaystyle\left[\boldsymbol{L}_{a},\boldsymbol{L}_{b}\right]=-\epsilon_{ab}\boldsymbol{L}_{0},$
$\displaystyle\left[\boldsymbol{L}_{0},\boldsymbol{\mathcal{L}}_{a0}\right]=\epsilon_{ab}\boldsymbol{\mathcal{L}}_{b0},$
(2.7)
$\displaystyle\left[\boldsymbol{L}_{0},\boldsymbol{\mathcal{L}}_{ab}\right]=-\epsilon_{c(a}\boldsymbol{\mathcal{L}}_{b)c},$
$\displaystyle\left[\boldsymbol{L}_{a},\boldsymbol{\mathcal{L}}_{bc}\right]=-\epsilon_{a(b}\boldsymbol{\mathcal{L}}_{c)0},$
$\displaystyle\left[\boldsymbol{L}_{a},\boldsymbol{\mathcal{L}}_{b0}\right]=-\epsilon_{a(c}\boldsymbol{\mathcal{L}}_{b)c},$
$\displaystyle\left[\boldsymbol{\mathcal{L}}_{a0},\boldsymbol{\mathcal{L}}_{b0}\right]=\epsilon_{ab}\boldsymbol{L}_{0},$
$\displaystyle\left[\boldsymbol{\mathcal{L}}_{a0},\boldsymbol{\mathcal{L}}_{bc}\right]=\delta_{a(b}\epsilon_{c)d}\boldsymbol{L}_{d},$
$\displaystyle\left[\boldsymbol{\mathcal{L}}_{ab},\boldsymbol{\mathcal{L}}_{cd}\right]=\delta_{(a(c}\epsilon_{d)b)}\boldsymbol{L}_{0}.$

Thus, one would like to keep the definitions (2.2) for the
$\mathfrak{sl}(2,\mathbb{R})$ generators and find some analog relations for
the spin-$3$ generators $\boldsymbol{\mathcal{L}}_{ab}$ and
$\boldsymbol{\mathcal{L}}_{a0}$ in such a way that the limit
$c\rightarrow\infty$ produces a spin-$3$ extension of $\mathfrak{nw}$ with a
non-degenerate invariant form. This requires supplementing the Abelian
generator $\boldsymbol{X}$ with extra generators that, as we will see, do not
commute with $\boldsymbol{X}$. In the following, we consider two extensions of
$\mathfrak{sl}(3,\mathbb{R})$ that admit a well-defined NR limit. These are
$\mathfrak{sl}(3,\mathbb{R})\oplus\mathfrak{so}(3)$ and
$\mathfrak{sl}(3,\mathbb{R})\oplus\mathfrak{sl}(2,\mathbb{R})\oplus\mathfrak{so}(2)$.
In each case, NR contractions lead to novel extensions of the $\mathfrak{nw}$
algebra that include higher-rank generators, which can be interpreted as NR
higher-spin symmetries.

###  2.1 NR contraction of $\mathfrak{sl}(3,\mathbb{R})\oplus\mathfrak{so}(3)$

In order to extend the $\mathfrak{sl}(3,\mathbb{R})$ commutation relations
(2.7), we introduce a $\mathfrak{so}(3)$ algebra spanned by generators
$\\{\boldsymbol{X},\boldsymbol{X}_{a}\\}$ satisfying

$\displaystyle\left[\boldsymbol{X},\boldsymbol{X}_{a}\right]=$
$\displaystyle\epsilon_{ab}\boldsymbol{X}_{b},\qquad\left[\boldsymbol{X}_{a},\boldsymbol{X}_{b}\right]=$
$\displaystyle\epsilon_{ab}\boldsymbol{X},$ (2.8)

and consider the direct product
$\mathfrak{sl}(3,\mathbb{R})\times\mathfrak{so}(3)$ generated by
$\boldsymbol{L}_{A}=\\{\boldsymbol{L}_{0},\boldsymbol{L}_{a}\\}$,
$\boldsymbol{\mathcal{L}}_{AB}=\\{\boldsymbol{\mathcal{L}}_{a0},\boldsymbol{\mathcal{L}}_{ab}\\}$,
and $\\{\boldsymbol{X},\boldsymbol{X}_{a}\\}$. The NR limit can be defined by
means of the change of basis

$\displaystyle\boldsymbol{L}_{0}$
$\displaystyle=\dfrac{1}{2}\boldsymbol{k}+c^{2}\boldsymbol{\sigma},$
$\displaystyle\boldsymbol{L}_{a}$ $\displaystyle=c\,\boldsymbol{k}_{a},$
$\displaystyle\boldsymbol{X}$
$\displaystyle=\dfrac{1}{2}\boldsymbol{k}-c^{2}\boldsymbol{\sigma},$ (2.9)
$\displaystyle\boldsymbol{\mathcal{L}}_{a0}$
$\displaystyle=\dfrac{1}{2}\boldsymbol{\mathcal{K}}_{a}+c^{2}\boldsymbol{\Sigma}_{a},$
$\displaystyle\boldsymbol{\mathcal{L}}_{ab}$
$\displaystyle=c\,\boldsymbol{\mathcal{K}}_{ab},$
$\displaystyle\boldsymbol{X}_{a}$
$\displaystyle=\dfrac{1}{2}\boldsymbol{\mathcal{K}}_{a}-c^{2}\boldsymbol{\Sigma}_{a},$

with inverse relations given by

$\displaystyle\boldsymbol{k}$
$\displaystyle=\boldsymbol{L}_{0}+\boldsymbol{X},$
$\displaystyle\boldsymbol{k}_{a}$
$\displaystyle=\frac{1}{c}\boldsymbol{L}_{a},$
$\displaystyle\boldsymbol{\sigma}$
$\displaystyle=\frac{1}{2c^{2}}\left(\boldsymbol{L}_{0}-\boldsymbol{X}\right),$
(2.10) $\displaystyle\boldsymbol{\mathcal{K}}_{a}$
$\displaystyle=\boldsymbol{\mathcal{L}}_{a0}+\boldsymbol{X}_{a},$
$\displaystyle\boldsymbol{\mathcal{K}}_{ab}$
$\displaystyle=\frac{1}{c}\boldsymbol{\mathcal{L}}_{ab},$
$\displaystyle\boldsymbol{\Sigma}_{a}$
$\displaystyle=\frac{1}{2c^{2}}\left(\boldsymbol{\mathcal{L}}_{a0}-\boldsymbol{X}_{a}\right).$

In the limit $c\rightarrow\infty$ we find the following non-vanishing
commutation relations for the generators (2.10)

$\displaystyle\left[\boldsymbol{k},\boldsymbol{k}_{a}\right]=\epsilon_{ab}\boldsymbol{k}_{b},$
$\displaystyle\left[\boldsymbol{k}_{a},\boldsymbol{k}_{b}\right]=-\epsilon_{ab}\boldsymbol{\sigma},$
$\displaystyle\left[\boldsymbol{k},\boldsymbol{\mathcal{K}}_{a}\right]=\epsilon_{ab}\boldsymbol{\mathcal{K}}_{b},$
(2.11)
$\displaystyle\left[\boldsymbol{k},\boldsymbol{\mathcal{K}}_{ab}\right]=-\epsilon_{c(a}\boldsymbol{\mathcal{K}}_{b)c},$
$\displaystyle\left[\boldsymbol{k},\boldsymbol{\Sigma}_{a}\right]=\epsilon_{ab}\boldsymbol{\Sigma}_{b},$
$\displaystyle\left[\boldsymbol{k}_{a},\boldsymbol{\mathcal{K}}_{b}\right]=-\epsilon_{a(c}\boldsymbol{\mathcal{K}}_{b)c},$
$\displaystyle\left[\boldsymbol{k}_{a},\boldsymbol{\mathcal{K}}_{bc}\right]=-\epsilon_{a(b}\boldsymbol{\Sigma}_{c)},$
$\displaystyle\left[\boldsymbol{\sigma},\boldsymbol{\mathcal{K}}_{a}\right]=\epsilon_{ab}\boldsymbol{\Sigma}_{b},$
$\displaystyle\left[\boldsymbol{\mathcal{K}}_{a},\boldsymbol{\mathcal{K}}_{b}\right]=\epsilon_{ab}\boldsymbol{k},$
$\displaystyle\left[\boldsymbol{\mathcal{K}}_{a},\boldsymbol{\Sigma}_{b}\right]=\epsilon_{ab}\boldsymbol{\sigma},$
$\displaystyle\left[\boldsymbol{\mathcal{K}}_{a},\boldsymbol{\mathcal{K}}_{bc}\right]=\delta_{a(b}\epsilon_{c)d}\boldsymbol{k}_{d},$
$\displaystyle\left[\boldsymbol{\mathcal{K}}_{ab},\boldsymbol{\mathcal{K}}_{cd}\right]=\delta_{(a(c}\epsilon_{d)b)}\boldsymbol{\sigma}.$

We will refer to this algebra as $\mathfrak{hs_{3}nw1}$ since it defines a
higher-spin extension of the Nappi-Witten algebra555Note that
$\mathfrak{hs_{3}nw1}$ is isomorphic to the homogeneous part of spin-$3$
Bargmann algebra in $2+1$ dimensions $\mathfrak{hs_{3}barg1_{2+1}}$ found in
[13]. .

###  2.2 NR contraction of
$\mathfrak{sl}(3,\mathbb{R})\oplus\mathfrak{sl}(2,\mathbb{R})\oplus\mathfrak{so}(2)$

As a second possibility to extend $\mathfrak{sl}(3,\mathbb{R})$, we introduce
the $\mathfrak{sl}(2,\mathbb{R})\oplus\mathfrak{so}(2)$ algebra

$[\boldsymbol{\mathcal{A}},\boldsymbol{\mathcal{B}}]=\boldsymbol{\mathcal{B}},\quad[\boldsymbol{\mathcal{A}},\boldsymbol{\mathcal{C}}]=-\boldsymbol{\mathcal{C}},\quad[\boldsymbol{\mathcal{B}},\boldsymbol{\mathcal{C}}]=2\boldsymbol{\mathcal{A}},\quad[\boldsymbol{\mathcal{D}},{\rm
all}]=0,$ (2.12)

where the set
$\\{\boldsymbol{\mathcal{A}},\boldsymbol{\mathcal{B}},\boldsymbol{\mathcal{C}}\\}$
spans $\mathfrak{sl}(2,\mathbb{R})$, whereas $\boldsymbol{\mathcal{D}}$ is the
Abelian $\mathfrak{so}(2)$ generator. By redefining the generators as

$\displaystyle\boldsymbol{\mathcal{A}}$
$\displaystyle=\frac{1}{4}\left(\boldsymbol{X}_{22}-\boldsymbol{X}_{11}\right),$
$\displaystyle\boldsymbol{\mathcal{B}}$
$\displaystyle=\frac{1}{2}\left(\boldsymbol{X}_{12}+\boldsymbol{X}\right),$
(2.13) $\displaystyle\boldsymbol{\mathcal{C}}$
$\displaystyle=\frac{1}{2}\left(\boldsymbol{X}_{12}-\boldsymbol{X}\right),$
$\displaystyle\boldsymbol{\mathcal{D}}$
$\displaystyle=\frac{1}{4}\left(\boldsymbol{X}_{22}+\boldsymbol{X}_{11}\right),$

the $\mathfrak{sl}(2,\mathbb{R})\oplus\mathfrak{so}(2)$ algebra (2.12) can be
written in a form that is more convenient for our purposes, namely

$\displaystyle\left[\boldsymbol{X},\boldsymbol{X}_{ab}\right]=$
$\displaystyle-\epsilon_{c(a}\boldsymbol{X}_{b)c},\qquad\left[\boldsymbol{X}_{ab},\boldsymbol{X}_{cd}\right]=$
$\displaystyle\delta_{(a(c}\epsilon_{d)b)}\boldsymbol{X},$ (2.14)

with $\boldsymbol{X}_{ab}=\boldsymbol{X}_{ba}$. Thus, we consider the
$\mathfrak{sl}(3,\mathbb{R})\oplus\mathfrak{sl}(2,\mathbb{R})\oplus\mathfrak{so}(2)$
algebra generated by
$\boldsymbol{L}_{A}=\\{\boldsymbol{L}_{0},\boldsymbol{L}_{a}\\}$,
$\boldsymbol{\mathcal{L}}_{AB}=\\{\boldsymbol{\mathcal{L}}_{a0},\boldsymbol{\mathcal{L}}_{ab}\\}$,
which satisfy the commutation relations (2.7), and
$\\{\boldsymbol{X},\boldsymbol{X}_{ab}\\}$. The NR contraction follows from
the generator redefinition

$\displaystyle\boldsymbol{L}_{0}$
$\displaystyle=\dfrac{1}{2}\boldsymbol{k}+c^{2}\boldsymbol{\sigma},$
$\displaystyle\boldsymbol{L}_{a}$ $\displaystyle=c\,\boldsymbol{k}_{a},$
$\displaystyle\boldsymbol{X}$
$\displaystyle=\dfrac{1}{2}\boldsymbol{k}-c^{2}\boldsymbol{\sigma},$ (2.15)
$\displaystyle\boldsymbol{\mathcal{L}}_{ab}$
$\displaystyle=\dfrac{1}{2}\tilde{\boldsymbol{\mathcal{K}}}_{ab}+c^{2}\tilde{\boldsymbol{\Sigma}}_{ab},$
$\displaystyle\boldsymbol{\mathcal{L}}_{a0}$
$\displaystyle=c\,\tilde{\boldsymbol{\mathcal{K}}}_{a},$
$\displaystyle\boldsymbol{X}_{ab}$
$\displaystyle=\dfrac{1}{2}\tilde{\boldsymbol{\mathcal{K}}}_{ab}-c^{2}\tilde{\boldsymbol{\Sigma}}_{ab}.$

analog to (2.9). Inverting the transformation yields the relations

$\displaystyle\boldsymbol{k}$
$\displaystyle=\boldsymbol{L}_{0}+\boldsymbol{X},$
$\displaystyle\boldsymbol{k}_{a}$
$\displaystyle=\frac{1}{c}\boldsymbol{L}_{a},$
$\displaystyle\boldsymbol{\sigma}$
$\displaystyle=\frac{1}{2c^{2}}\left(\boldsymbol{L}_{0}-\boldsymbol{X}\right),$
(2.16) $\displaystyle\tilde{\boldsymbol{\mathcal{K}}}_{ab}$
$\displaystyle=\boldsymbol{\mathcal{L}}_{ab}+\boldsymbol{X}_{ab},$
$\displaystyle\tilde{\boldsymbol{\mathcal{K}}}_{a}$
$\displaystyle=\frac{1}{c}\boldsymbol{\mathcal{L}}_{a0},$
$\displaystyle\tilde{\boldsymbol{\Sigma}}_{ab}$
$\displaystyle=\frac{1}{2c^{2}}\left(\boldsymbol{\mathcal{L}}_{ab}-\boldsymbol{X}_{ab}\right).$

which can be used to find the commutation relations in the new basis. In the
limit $c\rightarrow\infty$ the non-vanishing commutators read

$\displaystyle[\boldsymbol{k},\boldsymbol{k}_{a}]=\epsilon_{ab}\boldsymbol{k}_{b},$
$\displaystyle[\boldsymbol{k}_{a},\boldsymbol{k}_{b}]=-\epsilon_{ab}\boldsymbol{\sigma},$
$\displaystyle[\boldsymbol{k},\tilde{\boldsymbol{\mathcal{K}}}_{a}]=\epsilon_{ab}\tilde{\boldsymbol{\mathcal{K}}}_{b},$
(2.17)
$\displaystyle[\boldsymbol{k},\tilde{\boldsymbol{\mathcal{K}}}_{ab}]=-\epsilon_{c(a}\tilde{\boldsymbol{\mathcal{K}}}_{b)c},$
$\displaystyle[\boldsymbol{k},\tilde{\boldsymbol{\Sigma}}_{ab}]=-\epsilon_{c(a}\tilde{\boldsymbol{\Sigma}}_{b)c},$
$\displaystyle[\boldsymbol{k}_{a},\tilde{\boldsymbol{\mathcal{K}}}_{b}]=-\epsilon_{a(c}\tilde{\boldsymbol{\Sigma}}_{b)c},$
$\displaystyle[\boldsymbol{k}_{a},\tilde{\boldsymbol{\mathcal{K}}}_{bc}]=-\epsilon_{a(b}\tilde{\boldsymbol{\mathcal{K}}}_{c)},$
$\displaystyle[\boldsymbol{\sigma},\tilde{\boldsymbol{\mathcal{K}}}_{ab}]=-\epsilon_{c(a}\tilde{\boldsymbol{\Sigma}}_{b)c},$
$\displaystyle[\tilde{\boldsymbol{\mathcal{K}}}_{ab},\tilde{\boldsymbol{\mathcal{K}}}_{cd}]=\delta_{(a(c}\epsilon_{d)b)}\boldsymbol{k}.$
$\displaystyle[\tilde{\boldsymbol{\mathcal{K}}}_{a},\tilde{\boldsymbol{\mathcal{K}}}_{b}]=\epsilon_{ab}\boldsymbol{\sigma},$
$\displaystyle[\tilde{\boldsymbol{\mathcal{K}}}_{a},\tilde{\boldsymbol{\mathcal{K}}}_{bc}]=\delta_{a(b}\epsilon_{c)d}\boldsymbol{k}_{d},$
$\displaystyle[\tilde{\boldsymbol{\mathcal{K}}}_{ab},\tilde{\boldsymbol{\Sigma}}_{cd}]=\delta_{(a(c}\epsilon_{d)b)}\boldsymbol{\sigma},$

and define a second higher-spin extension of the Nappi-Witten symmetry666As in
the previous case, the algebra $\mathfrak{hs_{3}nw2}$ is isomorphic to the
homogenous part of the second spin-$3$ extension of the Extended Bargmann
algebra in $2+1$ dimensions $\mathfrak{hs_{3}barg2_{2+1}}$ found in [13],
which can be shown using the redefinition given in Eq. (2.18)., which we dub
as $\mathfrak{hs_{3}nw2}$. Note that it is is possible to define a new
generator $\tilde{\boldsymbol{\Sigma}}_{ab}$

$\tilde{\boldsymbol{\Sigma}}_{ab}\;\longrightarrow\;\tilde{\boldsymbol{\Sigma}}^{\prime}_{ab}=\tilde{\boldsymbol{\Sigma}}_{ab}+\delta_{ab}\tilde{\boldsymbol{\Sigma}}_{cc},$
(2.18)

which has the effect of modifying one of the commutation relations in (2.17)
as

$[\boldsymbol{k}_{a},\tilde{\boldsymbol{\mathcal{K}}}_{b}]=-\epsilon_{a(c}\tilde{\boldsymbol{\Sigma}}_{b)c}\quad\longrightarrow\quad[\boldsymbol{k}_{a},\tilde{\boldsymbol{\mathcal{K}}}_{b}]=-\epsilon_{ac}\tilde{\boldsymbol{\Sigma}}^{\prime}_{bc},$
(2.19)

while leaving the other commutation relations involving
$\tilde{\boldsymbol{\Sigma}}_{ab}$ invariant. This relation will be useful in
Section 4, when connecting some of our results with the ones found in [13].

##  3 Three-dimensional higher-spin gravity from spin-$3$ Nappi-Witten
algebras

In the previous section we obtained two different higher-spin extensions of
the $\mathfrak{nw}$ algebra, namely $\mathfrak{hs_{3}nw1}$ and
$\mathfrak{hs_{3}nw2}$, by defining NR contractions of suitable extensions of
$\mathfrak{sl}(3,\mathbb{R})$. This can be summarized as follows:

$\displaystyle\mathfrak{sl}(3,\mathbb{R})\oplus\mathfrak{so}(3)\;$
$\displaystyle\underset{c\rightarrow\infty}{\longrightarrow}$
$\displaystyle\qquad\mathfrak{hs_{3}nw1}$ (3.1)
$\displaystyle\mathfrak{sl}(3,\mathbb{R})\oplus\mathfrak{sl}(2,\mathbb{R})\oplus\mathfrak{so}(2)\;$
$\displaystyle\underset{c\rightarrow\infty}{\longrightarrow}$
$\displaystyle\qquad\mathfrak{hs_{3}nw2}.$

The next step in our construction is to apply the Lie algebra expansion method
based on Abelian semigroups [35] to these NR symmetries and construct CS
actions invariant under them. In this way, we will obtain two different
families of NR spin-$3$ algebras.

###  3.1 Expansions of higher-spin Nappi-Witten algebras

In order to define expansions of the spin-$3$ Nappi-Witten algebras (2.11) and
(2.17), we introduce an Abelian semigroup
$S=\left\\{\lambda_{0},\lambda_{1},\dots,\lambda_{N}\right\\}$, endowed with
the associative product

$\lambda_{i}\cdot\lambda_{j}=\lambda_{j}\cdot\lambda_{i}=\lambda_{i\diamond
j}\,,$ (3.2)

where $\diamond$ defines the semigroup product law. Since the direct product
of $S$ with a Lie algebra is again a Lie algebra [35], one can define the
expanded Nappi-Witten algebra [41]

$S\times\mathfrak{nw}=\mathrm{Span}\left\\{S\otimes\\{\boldsymbol{k},\boldsymbol{k}_{a},\boldsymbol{\sigma}\\}\right\\}.$
(3.3)

whose generators are obtained by considering the tensor product of the
semigroup elements and the generators of the $\mathfrak{nw}$ algebra, i.e.

$\boldsymbol{k}^{(i)}_{a}=\lambda_{i}\otimes\boldsymbol{k}_{a}\,,\quad\boldsymbol{k}^{(i)}=\lambda_{i}\otimes\boldsymbol{k}\,,\quad\boldsymbol{\sigma}^{(i)}=\lambda_{i}\otimes\boldsymbol{\sigma}\,,$
(3.4)

The commutation relations of the expanded Nappi-Witten algebra are

$[\boldsymbol{k}^{(i)},\boldsymbol{k}^{(j)}_{a}]=\epsilon_{a}^{\;\;b}\boldsymbol{k}^{(i\diamond
j)}_{b},\quad[\boldsymbol{k}^{(i)}_{a},\boldsymbol{k}^{(j)}_{b}]=-\epsilon_{ab}\boldsymbol{\sigma}^{(i\diamond
j)}.$ (3.5)

Similarly, one can apply the expansion procedure to the symmetries
$\mathfrak{hs_{3}nw1}$ and $\mathfrak{hs_{3}nw2}$, which were defined in Eqs.
(2.11) and (2.17), as follows:

  * •

Expansion of $\mathfrak{hs_{3}nw1}$: we consider the direct product of $S$ and
the generators of (2.11), which has the form

$S\times\mathfrak{hs_{3}nw1}=\mathrm{Span}\left\\{S\otimes\\{\boldsymbol{k},\boldsymbol{k}_{a},\boldsymbol{\sigma},\boldsymbol{\mathcal{K}}_{a},\boldsymbol{\mathcal{K}}_{ab},\boldsymbol{\Sigma}_{a}\\}\right\\},$
(3.6)

with the expanded generators defined in (3.4) plus

$\boldsymbol{\mathcal{K}}^{(i)}_{ab}=\lambda_{i}\otimes\boldsymbol{\mathcal{K}}_{ab}\,,\quad\boldsymbol{\mathcal{K}}_{a}^{(i)}=\lambda_{i}\otimes\boldsymbol{\mathcal{K}}_{a}\,,\quad\boldsymbol{\Sigma}_{a}^{(i)}=\lambda_{i}\otimes\boldsymbol{\Sigma}_{a}\,.$
(3.7)

The commutation relations of the $S\times\mathfrak{hs_{3}nw1}$ algebra are
directly obtained from (2.11) and the semigroup multiplication law (3.2). They
are given by

$\displaystyle[\boldsymbol{k}^{(i)},\boldsymbol{k}_{a}^{(j)}]=\epsilon_{ab}\boldsymbol{k}_{b}^{(i\diamond
j)},$
$\displaystyle[\boldsymbol{k}_{a}^{(i)},\boldsymbol{k}_{b}^{(j)}]=-\epsilon_{ab}\boldsymbol{\sigma}^{(i\diamond
j)},$
$\displaystyle[\boldsymbol{k}^{(i)},\boldsymbol{\mathcal{K}}_{a}^{(j)}]=\epsilon_{ab}\boldsymbol{\mathcal{K}}_{b}^{(i\diamond
j)},$ (3.8)
$\displaystyle[\boldsymbol{k}^{(i)},\boldsymbol{\mathcal{K}}_{ab}^{(j)}]=-\epsilon_{c(a}\boldsymbol{\mathcal{K}}_{b)c}^{(i\diamond
j)},$
$\displaystyle[\boldsymbol{k}^{(i)},\boldsymbol{\Sigma}_{a}^{(j)}]=\epsilon_{ab}\boldsymbol{\Sigma}_{b}^{(i\diamond
j)},$
$\displaystyle[\boldsymbol{k}_{a}^{(i)},\boldsymbol{\mathcal{K}}_{b}^{(j)}]=-\epsilon_{a(c}\boldsymbol{\mathcal{K}}_{b)c}^{(i\diamond
j)},$
$\displaystyle[\boldsymbol{k}_{a}^{(i)},\boldsymbol{\mathcal{K}}_{bc}^{(j)}]=-\epsilon_{a(b}\boldsymbol{\Sigma}_{c)}^{(i\diamond
j)},$
$\displaystyle[\boldsymbol{\sigma}^{(i)},\boldsymbol{\mathcal{K}}_{a}^{(j)}]=\epsilon_{ab}\boldsymbol{\Sigma}_{b}^{(i\diamond
j)},$
$\displaystyle[\boldsymbol{\mathcal{K}}_{a}^{(i)},\boldsymbol{\mathcal{K}}_{b}^{(j)}]=\epsilon_{ab}\boldsymbol{k}^{(i\diamond
j)},$
$\displaystyle[\boldsymbol{\mathcal{K}}_{a}^{(i)},\boldsymbol{\Sigma}_{b}^{(j)}]=\epsilon_{ab}\boldsymbol{\sigma}^{(i\diamond
j)},$
$\displaystyle[\boldsymbol{\mathcal{K}}_{a}^{(i)},\boldsymbol{\mathcal{K}}_{bc}^{(j)}]=\delta_{a(b}\epsilon_{c)d}\boldsymbol{k}_{d}^{(i\diamond
j)},$
$\displaystyle[\boldsymbol{\mathcal{K}}_{ab}^{(i)},\boldsymbol{\mathcal{K}}_{cd}^{(j)}]=\delta_{(a(c}\epsilon_{d)b)}\boldsymbol{\sigma}^{(i\diamond
j)}.$

  * •

Expansion of $\mathfrak{hs_{3}nw2}$: applying the same method to (2.17) leads
to

$S\times\mathfrak{hs_{3}nw2}=\mathrm{Span}\left\\{S\otimes\\{\boldsymbol{k},\boldsymbol{k}_{a},\boldsymbol{\sigma},\tilde{\boldsymbol{\mathcal{K}}}_{a},\tilde{\boldsymbol{\mathcal{K}}}_{ab},\tilde{\boldsymbol{\Sigma}}_{ab}\\}\right\\}.$
(3.9)

The expanded generators in this case are given by (3.4) together with

$\tilde{\boldsymbol{\mathcal{K}}}^{(i)}_{ab}=\lambda_{i}\otimes\tilde{\boldsymbol{\mathcal{K}}}_{ab}\,,\quad\tilde{\boldsymbol{\mathcal{K}}}_{a}^{(i)}=\lambda_{i}\otimes\tilde{\boldsymbol{\mathcal{K}}}_{a}\,,\quad\tilde{\boldsymbol{\Sigma}}_{ab}^{(i)}=\lambda_{i}\otimes\tilde{\boldsymbol{\Sigma}}_{ab}\,,$
(3.10)

and the commutation relations of $S\times\mathfrak{hs_{3}nw2}$ read

$\displaystyle[\boldsymbol{k}^{(i)},\boldsymbol{k}_{a}^{(j)}]=\epsilon_{ab}\boldsymbol{k}_{b}^{(i\diamond
j)},$
$\displaystyle[\boldsymbol{k}_{a}^{(i)},\boldsymbol{k}_{b}^{(j)}]=-\epsilon_{ab}\boldsymbol{\sigma}^{(i\diamond
j)},$
$\displaystyle[\boldsymbol{k}^{(i)},\tilde{\boldsymbol{\mathcal{K}}}_{a}^{(j)}]=\epsilon_{ab}\tilde{\boldsymbol{\mathcal{K}}}_{b}^{(i\diamond
j)},$ (3.11)
$\displaystyle[\boldsymbol{k}^{(i)},\tilde{\boldsymbol{\mathcal{K}}}_{ab}^{(j)}]=-\epsilon_{c(a}\tilde{\boldsymbol{\mathcal{K}}}_{b)c}^{(i\diamond
j)},$
$\displaystyle[\boldsymbol{k}^{(i)},\tilde{\boldsymbol{\Sigma}}_{ab}^{(j)}]=-\epsilon_{c(a}\tilde{\boldsymbol{\Sigma}}_{b)c}^{(i\diamond
j)},$
$\displaystyle[\boldsymbol{k}_{a}^{(i)},\tilde{\boldsymbol{\mathcal{K}}}_{b}^{(j)}]=-\epsilon_{a(c}\tilde{\boldsymbol{\Sigma}}_{b)c}^{(i\diamond
j)},$
$\displaystyle[\boldsymbol{k}_{a}^{(i)},\tilde{\boldsymbol{\mathcal{K}}}_{bc}^{(j)}]=-\epsilon_{a(b}\tilde{\boldsymbol{\mathcal{K}}}_{c)}^{(i\diamond
j)},$
$\displaystyle[\boldsymbol{\sigma}^{(i)},\tilde{\boldsymbol{\mathcal{K}}}_{ab}^{(j)}]=-\epsilon_{c(a}\tilde{\boldsymbol{\Sigma}}_{b)c}^{(i\diamond
j)},$
$\displaystyle[\tilde{\boldsymbol{\mathcal{K}}}_{ab}^{(i)},\tilde{\boldsymbol{\mathcal{K}}}_{cd}^{(j)}]=\delta_{(a(c}\epsilon_{d)b)}\boldsymbol{k}^{(i\diamond
j)},$
$\displaystyle[\tilde{\boldsymbol{\mathcal{K}}}_{a}^{(i)},\tilde{\boldsymbol{\mathcal{K}}}_{b}^{(j)}]=\epsilon_{ab}\boldsymbol{\sigma}^{(i\diamond
j)},$
$\displaystyle[\tilde{\boldsymbol{\mathcal{K}}}_{a}^{(i)},\tilde{\boldsymbol{\mathcal{K}}}_{bc}^{(j)}]=\delta_{a(b}\epsilon_{c)d}\boldsymbol{k}_{d}^{(i\diamond
j)},$
$\displaystyle[\tilde{\boldsymbol{\mathcal{K}}}_{ab}^{(i)},\tilde{\boldsymbol{\Sigma}}_{cd}^{(j)}]=\delta_{(a(c}\epsilon_{d)b)}\boldsymbol{\sigma}^{(i\diamond
j)}.$

We should note that, as done in (2.18) in the case of the
$\mathfrak{hs_{3}nw2}$ algebra, one can redefine the expanded generator

$\tilde{\boldsymbol{\Sigma}}_{ab}^{(i)}\;\longrightarrow\;\tilde{\boldsymbol{\Sigma}}^{\prime(i)}_{ab}=\tilde{\boldsymbol{\Sigma}}_{ab}^{(i)}+\delta_{ab}\tilde{\boldsymbol{\Sigma}}_{cc}^{(i)},$
(3.12)

and modify the commutator of the generators $\boldsymbol{k}_{a}^{(i)}$ and
$\tilde{\boldsymbol{\mathcal{K}}}_{a}^{(i)}$ in (3.11) as

$[\boldsymbol{k}_{a}^{(i)},\tilde{\boldsymbol{\mathcal{K}}}_{b}^{(j)}]=-\epsilon_{a(c}\tilde{\boldsymbol{\Sigma}}_{b)c}^{(i\diamond
j}\quad\longrightarrow\quad[\boldsymbol{k}_{a}^{(i)},\tilde{\boldsymbol{\mathcal{K}}}_{b}^{(j)}]=-\epsilon_{ac}\tilde{\boldsymbol{\Sigma}}^{\prime(i\diamond
j)}_{bc},$ (3.13)

while leaving all the remaining commutation relations of
$S\times\mathfrak{hs_{3}nw2}$ invariant.

The algebras defined by the commutation relations (3.8) and (3.11) define two
different higher-spin extensions of the expanded Nappi-Witten algebra (3.5).
We know from Ref. [41] that different choices of the semigroup $S$ lead to
expanded Nappi-Witten algebras $S\times\mathfrak{nw}$ that are isomorphic to
NR symmetries in $2+1$ dimensions, such as the Extended Bargmann algebra [60],
the Exotic Newton-Hooke algebra [61], and NR Maxwell [62] algebra. Higher-spin
extension of these NR symmetries can be obtained by considering the same
semigroups in the expanded algebras (3.8) and (3.11).

###  3.2 Invariant bilinear forms

The Inönü-Wigner contractions introduced in Section 2 can be used to define
non-degenerate invariant bilinear forms on the spin-$3$ Nappi-Witten algebras
$\mathfrak{hs_{3}nw1}$ and $\mathfrak{hs_{3}nw2}$. We begin by showing how
this procedure works in the case of the Nappi-Witten algebra (2.4), by looking
again at the contraction (2.2) of the
$\mathfrak{sl}(2,\mathbb{R})\oplus\mathfrak{so}(2)$ algebra, whose invariant
form is given by

$\left\langle\boldsymbol{L}_{A},\boldsymbol{L}_{B}\right\rangle=\varepsilon_{0}\eta_{AB},\qquad\left\langle\boldsymbol{X},\boldsymbol{X}\right\rangle=\varepsilon_{1},$
(3.14)

where $\varepsilon_{0}$ and $\varepsilon_{1}$ are non-vanishing constants.
Introducing the new constants

$\gamma_{0}=\frac{1}{c^{2}}\varepsilon_{0},\qquad\gamma_{1}=\varepsilon_{1}-\varepsilon_{0},$
(3.15)

the generator redefinition (2.3) yields the invariant bilinear form on the
$\mathfrak{nw}$ algebra [59] in the limit $c\rightarrow\infty$, which is non-
degenerate for $\gamma_{0}\neq 0$:

$\left\langle\boldsymbol{k},\boldsymbol{\sigma}\right\rangle=-\gamma_{0},\quad\left\langle\boldsymbol{k}_{a},\boldsymbol{k}_{b}\right\rangle=\gamma_{0}\delta_{ab},\quad\left\langle\boldsymbol{k},\boldsymbol{k}\right\rangle=\gamma_{1}.$
(3.16)

To define invariant tensors on $\mathfrak{hs_{3}nw1}$ and
$\mathfrak{hs_{3}nw2}$, we follow exactly the same procedure. The non-
degenerate invariant bilinear form of the $\mathfrak{sl}(3,\mathbb{R})$ is
[10]

$\left\langle\boldsymbol{L}_{A},\boldsymbol{L}_{B}\right\rangle=\varepsilon_{0}\eta_{AB},\qquad\left\langle\boldsymbol{\mathcal{L}}_{AB}\boldsymbol{\mathcal{L}}_{CD}\right\rangle=\varepsilon_{0}\left(\eta_{A(C}\eta_{D)B}-\frac{2}{3}\eta_{AB}\eta_{CD}\right),$
(3.17)

which, after splitting temporal and spatial components, takes the form

$\left\langle\boldsymbol{L}_{0}\boldsymbol{L}_{0}\right\rangle=-\varepsilon_{0},\quad\left\langle\boldsymbol{L}_{a}\boldsymbol{L}_{b}\right\rangle=-\left\langle\boldsymbol{\mathcal{L}}_{a0}\boldsymbol{\mathcal{L}}_{b0}\right\rangle=\varepsilon_{0}\delta_{ab},\quad\left\langle\boldsymbol{\mathcal{L}}_{ab}\boldsymbol{\mathcal{L}}_{cd}\right\rangle=\varepsilon_{0}\left(\delta_{a(c}\delta_{d)b}-\frac{2}{3}\delta_{ab}\delta_{cd}\right).$
(3.18)

The extensions we are interested in require the introduction of invariant
tensors on $\mathfrak{so}(3)$ and
$\mathfrak{sl}(2,\mathbb{R})\oplus\mathfrak{so}(2)$ before taking the NR
limit.

  * •

The invariant bilinear form on $\mathfrak{so}(3)$ can be written as

$\left\langle\boldsymbol{X},\boldsymbol{X}\right\rangle=\varepsilon_{1},\qquad\left\langle\boldsymbol{X}_{a},\boldsymbol{X}_{b}\right\rangle=\varepsilon_{1}\delta_{ab}$
(3.19)

Putting this together with (3.18) defines the invariant form on
$\mathfrak{sl}(3,\mathbb{R})\oplus\mathfrak{so}(3)$. Using (3.15), (2.10) and
taking the limit $c\rightarrow\infty$, one finds that the invariant tensor for
the generators of the $\mathfrak{hs_{3}nw1}$ algebra (2.11) is given by (3.16)
plus

$\left\langle\boldsymbol{\mathcal{K}}_{a},\boldsymbol{\Sigma}_{b}\right\rangle=-\gamma_{0}\delta_{ab},\quad\left\langle\boldsymbol{\mathcal{K}}_{ab},\boldsymbol{\mathcal{K}}_{cd}\right\rangle=\gamma_{0}\left(\delta_{a(c}\delta_{d)b}-\frac{2}{3}\delta_{ab}\delta_{cd}\right),\quad\left\langle\boldsymbol{\mathcal{K}}_{a},\boldsymbol{\mathcal{K}}_{b}\right\rangle=\gamma_{1}\delta_{ab},$
(3.20)

which is non-degenerate provided $\gamma_{0}\neq 0$.

  * •

The invariant bilinear form on the
$\mathfrak{sl}(2,\mathbb{R})\oplus\mathfrak{so}(2)$ algebra (2.12) is given by

$\left\langle\boldsymbol{\mathcal{A}},\boldsymbol{\mathcal{A}}\right\rangle=\frac{1}{4}\sigma_{0},\qquad\left\langle\boldsymbol{\mathcal{B}},\boldsymbol{\mathcal{C}}\right\rangle=\frac{1}{2}\sigma_{0},\qquad\left\langle\boldsymbol{\mathcal{D}},\boldsymbol{\mathcal{D}}\right\rangle=\frac{1}{4}\sigma_{1},$
(3.21)

which, by redefining $\varepsilon_{1}=-\sigma_{0}$,
$\varepsilon_{2}=-\sigma_{0}+\sigma_{1}$, can be rewritten in terms of the
generators $\boldsymbol{X}$ and $\boldsymbol{X}_{ab}$ of (2.14) as

$\left\langle\boldsymbol{X},\boldsymbol{X}\right\rangle=\varepsilon_{1},\quad\left\langle\boldsymbol{X}_{ab},\boldsymbol{X}_{cd}\right\rangle=-\varepsilon_{1}\delta_{a(c}\delta_{d)b}+\varepsilon_{2}\delta_{ab}\delta_{cd}.$
(3.22)

Thus, we will consider the invariant tensor on
$\mathfrak{sl}(3,\mathbb{R})\oplus\mathfrak{sl}(2,\mathbb{R})\oplus\mathfrak{so}(2)$
defined by (3.18) and (3.22). Using the redefinitions (2.16), (3.15) and
$\gamma_{2}=\varepsilon_{2}-\frac{2}{3}\varepsilon_{0}$, the limit
$c\rightarrow\infty$ yields an invariant bilinear form on the
$\mathfrak{hs_{3}nw2}$ algebra (2.17), given by (3.16) plus the relations

$\displaystyle\left\langle\tilde{\boldsymbol{\mathcal{K}}}_{ab},\tilde{\boldsymbol{\Sigma}}_{cd}\right\rangle=\gamma_{0}\left(\delta_{a(c}\delta_{d)b}-\frac{2}{3}\delta_{ab}\delta_{cd}\right),$
(3.23)
$\displaystyle\left\langle\tilde{\boldsymbol{\mathcal{K}}}_{ab},\tilde{\boldsymbol{\mathcal{K}}}_{cd}\right\rangle=-\gamma_{1}\delta_{a(c}\delta_{d)b}+\gamma_{2}\delta_{ab}\delta_{cd},\quad\left\langle\tilde{\boldsymbol{\mathcal{K}}}_{a},\tilde{\boldsymbol{\mathcal{K}}}_{b}\right\rangle=-\gamma_{0}\delta_{ab}.$

Again, this invariant tensor has non-vanishing determinant as long as
$\gamma_{0}\neq 0$. If, instead of $\tilde{\boldsymbol{\Sigma}}_{ab}$, one
uses the generator $\tilde{\boldsymbol{\Sigma}}^{\prime}_{ab}$ defined in
(2.18), the first relation in (3.23) is replaced by

$\left\langle\tilde{\boldsymbol{\mathcal{K}}}_{ab},\tilde{\boldsymbol{\Sigma}}^{\prime}_{cd}\right\rangle=\gamma_{0}\delta_{a(c}\delta_{d)b}.$
(3.24)

The next step is to use these results to define non-degenerate invariant
bilinear forms on the expanded algebras $S\times\mathfrak{hs_{3}nw1}$ and
$S\times\mathfrak{hs_{3}nw2}$ given by (3.8) and (3.11), respectively.
Following [34, 35], the procedure is summarized as follows: given a set of Lie
algebra generators $T_{\alpha}$ together with an invertible invariant bilinear
form $\langle T_{\alpha}T_{\beta}\rangle=g_{\alpha\beta}$, the expanded Lie
algebra spanned by the generators $T^{(i)}_{\alpha}=\lambda_{i}\times
T_{\alpha}$ admits the non-degenerate invariant tensor $\langle
T^{(i)}_{\alpha}T^{(j)}_{\beta}\rangle=h_{ij}g_{\alpha\beta}$, where the
components of $h_{ij}=\langle\lambda_{i}\lambda_{j}\rangle$ include a set of
arbitrary constants. When considering the $S\times\mathfrak{nw}$ algebra.
(3.5), we find [41]

$\left\langle\boldsymbol{k}^{(i)},\boldsymbol{\sigma}^{(j)}\right\rangle=-\mu_{i\diamond
j},\quad\left\langle\boldsymbol{k}_{a}^{(i)},\boldsymbol{k}_{b}^{(j)}\right\rangle=\mu_{i\diamond
j}\delta_{ab},\quad\left\langle\boldsymbol{k}^{(i)},\boldsymbol{k}^{(j)}\right\rangle=\nu_{i\diamond
j},$ (3.25)

where $\mu_{i}$ and $\nu_{i}$ stand for arbitrary constants. The procedure can
be generalized straightforwardly to the case of the
$S\times\mathfrak{hs_{3}nw1}$ and $S\times\mathfrak{hs_{3}nw2}$ algebras:

  * •

The generators of $S\times\mathfrak{hs_{3}nw1}$ are given by (3.4) and (3.7).
Thus, we can use the results (3.16) and (3.20) to obtain the invariant tensor
(3.25) together with the extension

$\displaystyle\left\langle\boldsymbol{\mathcal{K}}_{a}^{(i)},\boldsymbol{\Sigma}_{b}^{(j)}\right\rangle=-\mu_{i\diamond
j}\delta_{ab},$ (3.26)
$\displaystyle\left\langle\boldsymbol{\mathcal{K}}_{ab}^{(i)},\boldsymbol{\mathcal{K}}_{cd}^{(j)}\right\rangle=\mu_{i\diamond
j}\left(\delta_{a(c}\delta_{d)b}-\frac{2}{3}\delta_{ab}\delta_{cd}\right),$
$\displaystyle\left\langle\boldsymbol{\mathcal{K}}_{a}^{(i)},\boldsymbol{\mathcal{K}}_{b}^{(j)}\right\rangle=\nu_{i\diamond
j}\delta_{ab}.$

  * •

For $S\times\mathfrak{hs_{3}nw2}$ we use the invariant form defined by (3.16)
and (3.23) to compute an invariant bilinear form for the expanded generators
(3.4) and (3.10). After the expansion, the invariant bilinear form (3.25) is
extended to include the relations

$\displaystyle\left\langle\tilde{\boldsymbol{\mathcal{K}}}_{ab}^{(i)},\tilde{\boldsymbol{\Sigma}}_{cd}^{(j)}\right\rangle=\mu_{i\diamond
j}\left(\delta_{a(c}\delta_{d)b}-\frac{2}{3}\delta_{ab}\delta_{cd}\right),$
(3.27)
$\displaystyle\left\langle\tilde{\boldsymbol{\mathcal{K}}}_{ab}^{(i)},\tilde{\boldsymbol{\mathcal{K}}}_{cd}^{(j)}\right\rangle=-\nu_{i\diamond
j}\delta_{a(c}\delta_{d)b}+\rho_{i\diamond j}\delta_{ab}\delta_{cd},$
$\displaystyle\left\langle\tilde{\boldsymbol{\mathcal{K}}}_{a}^{(i)},\tilde{\boldsymbol{\mathcal{K}}}_{b}^{(j)}\right\rangle=-\mu_{i\diamond
j}\delta_{ab}.$

In complete analogy to (3.24), one can simplify the first relation in (3.27)
to

$\left\langle\tilde{\boldsymbol{\mathcal{K}}}_{ab}^{(i)},\tilde{\boldsymbol{\Sigma}}^{\prime(j)}_{cd}\right\rangle=\mu_{i\diamond
j}\delta_{a(c}\delta_{d)b}$ (3.28)

by redefining the expanded generator $\tilde{\boldsymbol{\Sigma}}_{ab}^{(i)}$
as in (3.12).

###  3.3 Chern-Simons actions

CS forms provide actions for topological field theories in $2+1$ dimensions.
In the case of NR symmetries, CS theories have been used to define three-
dimensional versions of Newton-Cartan gravity and extensions thereof [60, 63,
64, 65]. In our case, after having defined an non-degenerate invariant
bilinear form on the expanded spin-$3$ algebras $S\times\mathfrak{hs_{3}nw1}$
and $S\times\mathfrak{hs_{3}nw2}$, it is natural to construct topological
higher-spin gravity actions by considering the integral of a CS three-form
over a three-dimensional manifold

$I=\int_{M_{3}}\mathcal{L},\qquad\mathcal{L}=\left\langle\boldsymbol{A}\,\left(\boldsymbol{F}-\frac{1}{6}\left[\boldsymbol{A},\boldsymbol{A}\right]\right)\right\rangle,$
(3.29)

where $\boldsymbol{A}$ is a connection one-form taking values on the Lie
algebra of interest and
$\boldsymbol{F}=\mathrm{d}\boldsymbol{A}+\boldsymbol{A}^{2}$ is the
corresponding curvature two-form777Throughout the analysis, wedge product
between forms is understood.. Up to boundary terms, the theory is invariant
under infinitesimal gauge transformations
$\delta\boldsymbol{A}=d\lambda+[\boldsymbol{A},\lambda]$, with $\lambda$ a
local parameter taking values on the Lie algebra. As done in previous
sections, we start by briefly reviewing the pure Nappi-Witten case, without
higher-spin fields. The gauge connection one-form on the $\mathfrak{nw}$
algebra (2.4) and its curvature read

$\boldsymbol{A}^{\mathfrak{nw}}=\theta^{a}\boldsymbol{k}_{a}+\alpha\boldsymbol{k}+\beta\boldsymbol{\sigma},\quad\boldsymbol{F}^{\mathfrak{nw}}=\bar{R}^{a}\boldsymbol{k}_{a}+\bar{R}\boldsymbol{k}+\bar{F}\boldsymbol{\sigma},$
(3.30)

where the curvature components are given by

$\bar{R}^{a}=\mathrm{d}\theta^{a}+\epsilon^{ab}\,\theta^{b}\,\alpha,\quad\bar{R}=\mathrm{d}\alpha,\quad\bar{F}=\mathrm{d}\beta-\frac{1}{2}\epsilon_{ab}\,\theta^{a}\,\theta^{b}.$
(3.31)

Using the invariant form (3.16), the CS action invariant under the
$\mathfrak{nw}$ algebra takes the form

$\mathcal{L}_{\mathfrak{nw}}=\gamma_{0}\left(\theta^{a}\,\bar{R}^{a}-\alpha\,\mathrm{d}\beta-\beta\,\mathrm{d}\alpha\right)+\gamma_{1}\,\alpha\,\mathrm{d}\alpha.$
(3.32)

The next step is to generalize this action to the case of the expanded Nappi-
Wiiten algebra $S\times\mathfrak{nw}$ given in Eq. (3.5) to introduce the
semigroup elements (3.2) through the expanded generators (3.4). The connection
and the curvature for $S\times\mathfrak{nw}$ are

$\boldsymbol{A}^{S\times\mathfrak{nw}}=\theta^{a}_{(i)}\boldsymbol{k}_{a}^{(i)}+\alpha_{(i)}\boldsymbol{k}^{(i)}+\beta_{(i)}\boldsymbol{\sigma}^{(i)},\quad\boldsymbol{F}^{S\times\mathfrak{nw}}=\bar{R}^{a}_{(i)}\boldsymbol{k}_{a}^{(i)}+\bar{R}_{(i)}\boldsymbol{k}^{(i)}+\bar{F}_{(i)}\boldsymbol{\sigma}^{(i)}\\\
$ (3.33)

where we have defined the expanded curvature components

$\bar{R}^{a}_{(i)}=\mathrm{d}\theta^{a}_{(i)}+\epsilon^{ab}\,\theta^{b}_{(j)}\,\alpha_{(k)}\,\delta^{j\diamond
k}_{i},\quad\bar{R}_{(i)}=\mathrm{d}\alpha_{(i)},\quad\bar{F}_{(i)}=\mathrm{d}\beta_{(i)}-\frac{1}{2}\epsilon_{ab}\,\theta^{a}_{(j)}\,\theta^{b}_{(k)}\,\delta^{j\diamond
k}_{i},$ (3.34)

and now the Einstein summation convention also holds for repeated semigroup
indices. After implementing the expansion mechanism, the CS action (3.32) is
extended to [41]

$\mathcal{L}_{S\times\mathfrak{nw}}=\mu_{i\diamond
j}\left(\theta^{a}_{(i)}\,\bar{R}^{a}_{(j)}-\alpha_{(i)}\,\mathrm{d}\beta_{(j)}-\beta_{(i)}\,\mathrm{d}\alpha_{(j)}\right)+\nu_{i\diamond
j}\,\alpha_{(i)}\,\mathrm{d}\alpha_{(j)}\,,$ (3.35)

where the constants $\gamma_{0}$ and $\gamma_{1}$ are replaced by the
corresponding ones entering in the expanded invariant tensor (3.25), $\mu_{i}$
and $\nu_{i}$.

Spin-$3$ generalization: The procedure previously presented can be further
generalized to define CS actions invariant under the spin-$3$ extension of the
Nappi-Witten algebras presented in Section 2. In order to simplify the
discussion, we make a parallel construction for both symmetries
$\mathfrak{hs_{3}nw1}$ and $\mathfrak{hs_{3}nw2}$. We thus consider two
connection one-forms, each one taking values on one of the spin-$3$ extensions
of the Nappi-Witten algebra

$\displaystyle\boldsymbol{A}^{\mathfrak{hs_{3}nw1}}$
$\displaystyle=\boldsymbol{A}^{\mathfrak{nw}}+\Theta^{ab}\boldsymbol{\mathcal{K}}_{ab}+A^{a}\boldsymbol{\mathcal{K}}_{a}+B^{a}\boldsymbol{\Sigma}_{a},$
(3.36) $\displaystyle\boldsymbol{A}^{\mathfrak{hs_{3}nw2}}$
$\displaystyle=\boldsymbol{A}^{\mathfrak{nw}}+\Theta^{a}\tilde{\boldsymbol{\mathcal{K}}}_{a}+A^{ab}\tilde{\boldsymbol{\mathcal{K}}}_{ab}+B^{ab}\tilde{\boldsymbol{\Sigma}}_{ab},$

with associated curvatures

$\displaystyle\boldsymbol{F}^{\mathfrak{hs_{3}nw1}}$
$\displaystyle=R^{a}\boldsymbol{k}_{a}+R\boldsymbol{k}+F\boldsymbol{\sigma}+\mathcal{R}^{ab}\boldsymbol{\mathcal{K}}_{ab}+\mathcal{R}^{a}\boldsymbol{\mathcal{K}}_{a}+\mathcal{F}^{a}\boldsymbol{\Sigma}_{a},$
(3.37) $\displaystyle\boldsymbol{F}^{\mathfrak{hs_{3}nw2}}$
$\displaystyle=\tilde{R}^{a}\boldsymbol{k}_{a}+\tilde{R}\boldsymbol{k}+\tilde{F}\boldsymbol{\sigma}+\tilde{\mathcal{R}}^{a}\tilde{\boldsymbol{\mathcal{K}}}_{a}+\tilde{\mathcal{R}}^{ab}\tilde{\boldsymbol{\mathcal{K}}}_{ab}+\tilde{\mathcal{F}}^{ab}\tilde{\boldsymbol{\Sigma}}_{ab}.$

Notice that, since the higher-spin generators in each expanded algebra have
different index structure, the corresponding higher-spin gauge field content
is different in each case as well. This manifests in the form of the
components of the curvatures. Indeed, for
$\boldsymbol{F}^{\mathfrak{hs_{3}nw1}}$ we find

$\displaystyle R^{a}=\bar{R}^{a}+2\epsilon^{ab}\Theta^{bc}\,A^{c},$ (3.38)
$\displaystyle R=\bar{R}+\frac{1}{2}\epsilon_{ab}A^{a}\,A^{b},$ $\displaystyle
F=\bar{F}+\epsilon_{ab}\left(A^{a}B^{b}-2\Theta^{ac}\,\Theta^{cb}\right),$
$\displaystyle\mathcal{R}^{ab}=\mathrm{d}\Theta^{ab}+\epsilon^{c(a}\left(\frac{1}{2}A^{b)}\,\theta^{c}-\Theta^{b)c}\,\alpha\right)-\delta^{ab}\epsilon_{cd}\theta^{c}\,A^{d},$
$\displaystyle\mathcal{R}^{a}=\mathrm{d}A^{a}+\epsilon^{ab}A^{b}\,\alpha,$
$\displaystyle\mathcal{F}^{a}=\mathrm{d}B^{a}+\epsilon^{ab}\left(B^{b}\,\alpha+A^{b}\,\beta\right)-2\epsilon_{bc}\Theta^{ab}\,\theta^{c},$

whereas the components of the curvature
$\boldsymbol{F}^{\mathfrak{hs_{3}nw2}}$ are

$\displaystyle\tilde{R}^{a}=\bar{R}^{a}+2\epsilon^{ab}A^{bc}\,\Theta^{c},$
(3.39) $\displaystyle\tilde{R}=\bar{R}-2\epsilon_{ab}A^{ac}\,A^{cb},$
$\displaystyle\tilde{F}=\bar{F}+\epsilon_{ab}\left(\frac{1}{2}\Theta^{a}\,\Theta^{b}-4A^{ac}B^{cb}\right),$
$\displaystyle\tilde{\mathcal{R}}^{a}=\mathrm{d}\Theta^{a}+\epsilon^{ab}\Theta^{b}\,\alpha-2\epsilon_{bc}A^{ab}\,\theta^{c},$
$\displaystyle\tilde{\mathcal{R}}^{ab}=\mathrm{d}A^{ab}-\epsilon^{c(a}A^{b)c}\,\alpha,$
$\displaystyle\tilde{\mathcal{F}}^{ab}=\mathrm{d}B^{ab}+\epsilon^{c(a}\left(\frac{1}{2}\Theta^{b)}\,\theta^{c}-A^{b)c}\,\beta-B^{b)c}\,\alpha\right)-\delta^{ab}\epsilon_{cd}\theta^{c}\,\Theta^{d}.$

Naturally, both curvatures possess the same terms $\bar{R}^{a}$, $\bar{R}$ and
$\bar{F}$, defined in Eq. (3.31), associated to the Nappi-Witten subalgebra
that both spin-$3$ symmetries share. The CS forms invariant under
$\mathfrak{hs_{3}nw1}$ and $\mathfrak{hs_{3}nw2}$ can be computed by
evaluating (3.29) for the corresponding connections and curvatures and later
using the invariant tensors given in (3.20) and (3.23). This yields two
inequivalent spin-$3$ extensions of the Nappi-Witten CS action (3.32),

$\displaystyle\mathcal{L}_{\mathfrak{hs_{3}nw1}}$
$\displaystyle=\mathcal{L}_{\mathfrak{nw}}+\gamma_{0}\left(2\Theta^{ab}\,\mathcal{R}^{ab}-\frac{2}{3}\Theta^{aa}\,\mathcal{R}^{bb}-A^{a}\,\mathcal{F}^{a}-B^{a}\,\mathcal{R}^{a}\right)+\gamma_{1}A^{a}\,\mathcal{R}^{a},$
(3.40) $\displaystyle\mathcal{L}_{\mathfrak{hs_{3}nw2}}$
$\displaystyle=\mathcal{L}_{\mathfrak{nw}}+\gamma_{0}\left(2A^{ab}\,\tilde{\mathcal{F}}^{ab}+2B^{ab}\,\tilde{\mathcal{R}}^{ab}-\frac{2}{3}A^{aa}\,\tilde{\mathcal{F}}^{bb}-\frac{2}{3}B^{aa}\,\tilde{\mathcal{R}}^{bb}-\Theta^{a}\,\tilde{\mathcal{R}}^{a}\right)$
$\displaystyle-2\gamma_{1}A^{ab}\,\tilde{\mathcal{R}}^{ab}+\gamma_{2}A^{aa}\,\tilde{\mathcal{R}}^{bb}.$

Expansion: Now we turn our attention to the expanded algebras
$S\times\mathfrak{hs_{3}nw1}$ and $S\times\mathfrak{hs_{3}nw2}$. In order to
obtain the corresponding expanded CS actions, we follow the same procedure as
in the $\mathfrak{nw}$ case. Therefore, we define two expanded connections

$\displaystyle\boldsymbol{A}^{S\times\mathfrak{hs_{3}nw1}}$
$\displaystyle=\boldsymbol{A}^{S\times\mathfrak{nw}}+\Theta_{(i)}^{ab}\boldsymbol{\mathcal{K}}_{ab}^{(i)}+A^{a}_{(i)}\boldsymbol{\mathcal{K}}_{a}^{(i)}+B_{(i)}^{a}\boldsymbol{\Sigma}_{a}^{(i)},$
(3.41) $\displaystyle\boldsymbol{A}^{S\times\mathfrak{hs_{3}nw2}}$
$\displaystyle=\boldsymbol{A}^{S\times\mathfrak{nw}}+\Theta_{(i)}^{a}\tilde{\boldsymbol{\mathcal{K}}}_{a}^{(i)}+A^{ab}_{(i)}\tilde{\boldsymbol{\mathcal{K}}}_{ab}^{(i)}+B_{(i)}^{ab}\tilde{\boldsymbol{\Sigma}}_{ab}^{(i)},$

where $\boldsymbol{A}^{S\times\mathfrak{nw}}$ is the connection that takes
values on the Expanded Nappi-Witten algebra (3.33). The corresponding
curvatures have the form

$\displaystyle\boldsymbol{F}^{S\times\mathfrak{hs_{3}nw1}}$
$\displaystyle=R^{a}_{(i)}\boldsymbol{k}_{a}^{(i)}+R_{(i)}\boldsymbol{k}^{(i)}+F_{(i)}\boldsymbol{\sigma}^{(i)}+\mathcal{R}^{ab}_{(i)}\boldsymbol{\mathcal{K}}_{ab}^{(i)}+\mathcal{R}^{a}_{(i)}\boldsymbol{\mathcal{K}}_{a}^{(i)}+\mathcal{F}^{a}_{(i)}\boldsymbol{\Sigma}_{a}^{(i)},$
(3.42) $\displaystyle\boldsymbol{F}^{S\times\mathfrak{hs_{3}nw2}}$
$\displaystyle=\tilde{R}^{a}_{(i)}\boldsymbol{k}_{a}^{(i)}+\tilde{R}_{(i)}\boldsymbol{k}^{(i)}+\tilde{F}_{(i)}\boldsymbol{\sigma}^{(i)}+\tilde{\mathcal{R}}^{a}_{(i)}\tilde{\boldsymbol{\mathcal{K}}}_{a}^{(i)}+\tilde{\mathcal{R}}^{ab}_{(i)}\tilde{\boldsymbol{\mathcal{K}}}_{ab}^{(i)}+\tilde{\mathcal{F}}^{ab}_{(i)}\tilde{\boldsymbol{\Sigma}}_{ab}^{(i)},$

where the components of the curvature two-form on
$S\times\mathfrak{hs_{3}nw1}$ are given by

$\displaystyle
R^{a}_{(i)}=\bar{R}^{a}_{(i)}+2\epsilon^{ab}\Theta^{bc}_{(j)}\,A^{c}_{(k)}\,\delta^{j\diamond
k}_{i},$ (3.43) $\displaystyle
R_{(i)}=\bar{R}_{(i)}+\frac{1}{2}\epsilon_{ab}A^{a}_{(j)}\,A^{b}_{(k)}\,\delta^{j\diamond
k}_{i},$ $\displaystyle
F_{(i)}=\bar{F}_{(i)}+\epsilon_{ab}\left(A^{a}_{(j)}\,B^{b}_{(k)}-2\Theta^{ac}_{(j)}\,\Theta^{cb}_{(k)}\right)\delta^{j\diamond
k}_{i},$
$\displaystyle\mathcal{R}^{ab}_{(i)}=\mathrm{d}\Theta^{ab}_{(i)}+\epsilon^{c(a}\left(\frac{1}{2}A^{b)}_{(j)}\,\theta^{c}_{(k)}-\Theta^{b)c}_{(j)}\,\alpha_{(k)}\right)\delta^{j\diamond
k}_{i}-\delta^{ab}\epsilon_{cd}\theta^{c}_{(j)}\,A^{d}_{(k)}\,\delta^{j\diamond
k}_{i},$
$\displaystyle\mathcal{R}^{a}_{(i)}=\mathrm{d}A^{a}_{(i)}+\epsilon^{ab}A^{b}_{(j)}\,\alpha_{(k)}\,\delta^{j\diamond
k}_{i},$
$\displaystyle\mathcal{F}^{a}_{(i)}=\mathrm{d}B^{a}_{(i)}+\epsilon^{ab}\left(B^{b}_{(j)}\,\alpha_{(k)}+A^{b}_{(j)}\,\beta_{(k)}\right)\delta^{j\diamond
k}_{i}-2\epsilon_{bc}\Theta^{ab}_{(j)}\,\theta^{c}_{(k)}\,\delta^{j\diamond
k}_{i},$

while for the components of the curvature on the second expanded algebra
$\boldsymbol{F}^{S\times\mathfrak{hs_{3}nw2}}$ we get

$\displaystyle\tilde{R}^{a}_{(i)}=\bar{R}^{a}_{(i)}+2\epsilon^{ab}A^{bc}_{(j)}\,\Theta^{c}_{(k)}\,\delta^{j\diamond
k}_{i},$ (3.44)
$\displaystyle\tilde{R}_{(i)}=\bar{R}_{(i)}-2\epsilon_{ab}A^{ac}_{(j)}\,A^{cb}_{(k)}\,\delta^{j\diamond
k}_{i},$
$\displaystyle\tilde{F}_{(i)}=\bar{F}_{(i)}+\epsilon_{ab}\left(\frac{1}{2}\Theta^{a}_{(j)}\,\Theta^{b}_{(k)}-4A^{ac}_{(j)}B^{cb}_{(k)}\right)\delta^{j\diamond
k}_{i},$
$\displaystyle\tilde{\mathcal{R}}^{a}_{(i)}=\mathrm{d}\Theta^{a}_{(i)}+\epsilon^{ab}\Theta^{b}_{(j)}\,\alpha_{(k)}\,\delta^{j\diamond
k}_{i}-2\epsilon_{bc}A^{ab}_{(j)}\,\theta^{c}_{(k)}\,\delta^{j\diamond
k}_{i},$
$\displaystyle\tilde{\mathcal{R}}^{ab}_{(i)}=\mathrm{d}A^{ab}_{(i)}-\epsilon^{c(a}A^{b)c}_{(j)}\,\alpha_{(k)}\,\delta^{j\diamond
k}_{i},$
$\displaystyle\tilde{\mathcal{F}}^{ab}_{(i)}=\mathrm{d}B^{ab}_{(i)}+\epsilon^{c(a}\left(\frac{1}{2}\Theta^{b)}_{(j)}\,\theta^{c}_{(k)}-A^{b)c}_{(j)}\,\beta_{(k)}-B^{b)c}_{(j)}\,\alpha_{(k)}\right)\delta^{j\diamond
k}_{i}-\delta^{ab}\epsilon_{cd}\theta^{c}_{(j)}\,\Theta^{d}_{(k)}\,\delta^{j\diamond
k}_{i}.$

Note that, as before, the components of the curvature two-forms along the
expanded generators $\boldsymbol{k}_{a}^{(i)}$, $\boldsymbol{k}^{(i)}$ and
$\boldsymbol{\sigma}^{(i)}$ contain the terms given in (3.34) plus
contributions of the higher-spin fields that are different in each case due to
the structure of the commutation relations involving the higher-spin
generators. The expanded CS actions can be obtained in a straightforward way
by applying the same procedure used for the action (3.32) to obtain (3.35): i)
take the actions given in (3.40) and replace
$\gamma_{0}\rightarrow\mu_{i\diamond j}$, $\gamma_{1}\rightarrow\nu_{i\diamond
j}$, $\gamma_{2}\rightarrow\rho_{i\diamond j}$, ii) insert the semigroup
indices $i$ and $j$ in the gauge fields and curvatures, contracted with the
invariant bilinear form constants in the proper way, iii) take care of using
Kronecker deltas of the form $\delta^{j\diamond k}_{i}$ in the terms that come
from the commutator $[\boldsymbol{A},\boldsymbol{A}]$. We find the resulting
actions

$\displaystyle\mathcal{L}_{S\times\mathfrak{hs_{3}nw1}}=\;\mathcal{L}_{S\times\mathfrak{nw}}+\mu_{i\diamond
j}\left(2\Theta^{ab}_{(i)}\,\mathcal{R}^{ab}_{(j)}-\dfrac{2}{3}\Theta^{aa}_{(i)}\,\mathcal{R}^{bb}_{(j)}-A^{a}_{(i)}\,\mathcal{F}^{a}_{(j)}-B^{a}_{(i)}\,\mathcal{R}^{a}_{(j)}\right)+\nu_{i\diamond
j}\;A^{a}_{(i)}\,\mathcal{R}^{a}_{(j)},$ (3.45)
$\displaystyle\mathcal{L}_{S\times\mathfrak{hs_{3}nw2}}$
$\displaystyle=\;\mathcal{L}_{S\times\mathfrak{nw}}+\mu_{i\diamond
j}\left(2A^{ab}_{(i)}\,\tilde{\mathcal{F}}^{ab}_{(j)}+2B^{ab}_{(i)}\,\tilde{\mathcal{R}}^{ab}_{(j)}-\dfrac{2}{3}A^{aa}_{(i)}\,\tilde{\mathcal{F}}^{bb}_{(j)}-\dfrac{2}{3}B^{aa}_{(i)}\,\tilde{\mathcal{R}}^{bb}_{(j)}-\Theta^{a}_{(i)}\,\tilde{\mathcal{R}}^{a}_{(j)}\right)$
(3.46) $\displaystyle\hskip 14.22636pt-2\nu_{i\diamond
j}\;A^{ab}_{(i)}\,\tilde{\mathcal{R}}^{ab}_{(j)}+\rho_{i\diamond
j}\;A^{aa}_{(i)}\,\tilde{\mathcal{R}}^{bb}_{(j)},$

where one should remark that, in the same way as for the spatial indices,
there is an implicit sum over repeated semigroup indices $i$ and $j$.
Naturally, these actions can alternatively be obtained by evaluating (3.29)
using the connections and curvatures given in (3.41) and (3.42) together with
the invariant bilinear forms given in Eqs. (3.25)-(3.27). As we have seen,
however, once the CS action invariant under some to-be-expanded algebra is
known, the expansion method provides a quite efficient and direct way to
obtain the CS actions invariant under the corresponding expanded algebra.

###  3.4 Non-relativistic limit of expanded $\mathfrak{sl}(3,\mathbb{R})$
algebras

It is natural to expect that the NR spin-$3$ symmetries
$S\times\mathfrak{hs_{3}nw1}$ and $S\times\mathfrak{hs_{3}nw2}$, respectively
defined by (3.8) and (3.11), are not only expansions of a NR algebra, but also
NR contractions of some relativistic symmetry. In [49], spin-$3$ extensions of
relativistic algebras in $2+1$ dimensions were obtained as expansions of
$\mathfrak{sl}(3,\mathbb{R})$. Combining the notation introduced in [49] and
the definition (3.2), an expanded algebra of the form
$S\times\mathfrak{sl}(3,\mathbb{R})$ has the commutation relations

$\left[\boldsymbol{L}_{A}^{(i)},\boldsymbol{L}_{B}^{(j)}\right]=\epsilon_{AB}^{\quad
C}\boldsymbol{L}_{C}^{(i\diamond
j)}\,\quad\left[\boldsymbol{L}_{A}^{(i)},\boldsymbol{\mathcal{L}}_{BC}^{(j)}\right]=\epsilon_{A(B}^{\quad\;D}\boldsymbol{\mathcal{L}}_{C)D}^{(i\diamond
j)}\,\quad\left[\boldsymbol{\mathcal{L}}_{AB}^{(i)},\boldsymbol{\mathcal{L}}_{CD}^{(j)}\right]=-\eta_{(A(C}\epsilon_{D)B)}^{\quad\;E}\boldsymbol{L}_{E}^{(i\diamond
j)}.$ (3.47)

Depending on the chosen semigroup, this structure can reproduce spin-$3$
extensions of the Poincaré, AdS, Maxwell and AdS-Lorentz algebras (see
(3.64)). The method was also extended to define spin-$3$ extensions of the
Generalized Poincaré (also known as $\mathfrak{B}_{m}$) [51] and Generalized
AdS (also known as $\mathfrak{C}_{m}$) [50] algebras in three space-time
dimensions.

It is interesting to point out that the contractions introduced in Section 2.1
reveal the way to extend (3.47) in order to define expanded relativistic
symmetries whose NR limits are precisely the expanded algebras
$S\times\mathfrak{hs_{3}nw1}$ and $S\times\mathfrak{hs_{3}nw2}$.

  * •

By expanding the $\mathfrak{so}(3)$ algebra (2.8) we find

$\displaystyle\left[\boldsymbol{X}^{(i)},\boldsymbol{X}_{a}^{(j)}\right]=$
$\displaystyle\epsilon_{ab}\boldsymbol{X}_{b}^{(i\diamond
j)},\qquad\left[\boldsymbol{X}_{a}^{(i)},\boldsymbol{X}_{b}^{(j)}\right]=$
$\displaystyle\epsilon_{ab}\boldsymbol{X}^{(i\diamond j)},$ (3.48)

which can be put together with (3.47) to define the contraction

$\left[S\times\mathfrak{sl}(3,\mathbb{R})\right]\oplus\left[S\times\mathfrak{so}(3)\right]\quad\underset{c\rightarrow\infty}{\longrightarrow}\qquad
S\times\mathfrak{hs_{3}nw1}$ (3.49)

by means of the generator redefinition

$\displaystyle\boldsymbol{L}_{0}^{(i)}=\dfrac{1}{2}\boldsymbol{k}^{(i)}+c^{2}\boldsymbol{\sigma}^{(i)},$
$\displaystyle\boldsymbol{L}_{a}^{(i)}=c\,\boldsymbol{k}_{a}^{(i)},$
$\displaystyle\boldsymbol{X}^{(i)}=\dfrac{1}{2}\boldsymbol{k}-c^{2}\boldsymbol{\sigma}^{(i)},$
(3.50)
$\displaystyle\boldsymbol{\mathcal{L}}_{a0}^{(i)}=\dfrac{1}{2}\boldsymbol{\mathcal{K}}_{a}^{(i)}+c^{2}\boldsymbol{\Sigma}_{a}^{(i)},$
$\displaystyle\boldsymbol{\mathcal{L}}_{ab}^{(i)}=c\,\boldsymbol{\mathcal{K}}_{ab}^{(i)},$
$\displaystyle\boldsymbol{X}_{a}^{(i)}=\dfrac{1}{2}\boldsymbol{\mathcal{K}}_{a}^{(i)}-c^{2}\boldsymbol{\Sigma}_{a}^{(i)}.$

In the limit $c\rightarrow\infty$ this contraction yields exactly the
commutation relations (3.8) of the algebra $S\times\mathfrak{hs_{3}nw1}$. What
we have done here, oppositely to the _first contract and later expand_
procedure given in the previous sections, is to expand the relativistic
relations (3.47) and (3.48) before applying the contraction.

  * •

The expansion and the contraction also commute if we try to obtain the
$S\times\mathfrak{hs_{3}nw2}$ algebra. One can use (3.47) together with the
expansion of the $\mathfrak{sl}(2,\mathbb{R})\oplus\mathfrak{so}(2)$ algebra
(2.14)

$\displaystyle\left[\boldsymbol{X}^{(i)},\boldsymbol{X}_{ab}^{(j)}\right]=$
$\displaystyle-\epsilon_{c(a}\boldsymbol{X}_{b)c}^{(i\diamond
j)},\qquad\left[\boldsymbol{X}_{ab}^{(i)},\boldsymbol{X}_{cd}^{(j)}\right]=$
$\displaystyle\delta_{(a(c}\epsilon_{d)b)}\boldsymbol{X}^{(i\diamond j)}$
(3.51)

to define a NR limit

$\left[S\times\mathfrak{sl}(3,\mathbb{R})\right]\oplus\left[S\times\mathfrak{sl}(2,\mathbb{R})\right]\oplus\left[S\times\mathfrak{so}(2)\right]\quad\underset{c\rightarrow\infty}{\longrightarrow}\qquad
S\times\mathfrak{hs_{3}nw2}.$ (3.52)

Thus, by defining relations analog to (2.15),

$\displaystyle\boldsymbol{L}_{0}^{(i)}=\dfrac{1}{2}\boldsymbol{k}^{(i)}+c^{2}\boldsymbol{\sigma}^{(i)},$
$\displaystyle\boldsymbol{L}_{a}^{(i)}=c\,\boldsymbol{k}_{a}^{(i)},$
$\displaystyle\boldsymbol{X}^{(i)}=\dfrac{1}{2}\boldsymbol{k}-c^{2}\boldsymbol{\sigma}^{(i)},$
(3.53)
$\displaystyle\boldsymbol{\mathcal{L}}_{ab}^{(i)}=\dfrac{1}{2}\tilde{\boldsymbol{\mathcal{K}}}_{ab}^{(i)}+c^{2}\tilde{\boldsymbol{\Sigma}}_{ab}^{(i)},$
$\displaystyle\boldsymbol{\mathcal{L}}_{a0}^{(i)}=c\,\tilde{\boldsymbol{\mathcal{K}}}_{a}^{(i)},$
$\displaystyle\boldsymbol{X}_{ab}^{(i)}=\dfrac{1}{2}\tilde{\boldsymbol{\mathcal{K}}}_{ab}^{(i)}-c^{2}\tilde{\boldsymbol{\Sigma}}_{ab}^{(i)},$

the limit $c\rightarrow\infty$ leads to the $S\times\mathfrak{hs_{3}nw2}$
commutation relations (3.11), as expected. This construction provides an
alternative method to generate the expanded algebras that will be presented in
the next section, which is based on the fact that (for the type of expansions
here considered) the expansion and contraction procedures commute. The
construction of the invariant bilinear forms (3.25)-(3.27), as well as the CS
actions (3.45) and (3.46) that are invariant under these expanded NR
symmetries can be obtained by means of this alternative method as well.

###  3.5 Flat limit

In the following section, we will use two particular semigroups to construct
higher-spin NR symmetries, namely, the $S_{E}^{(n)}$ and $S_{M}^{(n)}$
semigroups [35], defined by the following multiplication rules (3.2).

$\hskip-65.44142ptS_{E}^{(n)}=\left\\{\lambda_{0},\lambda_{1},\dots,\lambda_{n+1}\right\\},\qquad
i\diamond j=\left\\{\begin{array}[]{lll}i+j&\mathrm{if}\quad i+j\leq n\,,&\\\
n+1&\mathrm{if}\quad i+j>n\,.&\end{array}\right.$ (3.54)
$S_{M}^{(n)}=\\{\lambda_{0},\lambda_{1},\dots,\lambda_{n}\\},\qquad\hskip
14.22636pti\diamond j=\left\\{\begin{array}[]{lll}i+j&\mathrm{if}\,\,\,\,\
i+j\leq n\,,&\\\
i+j-2\left[\frac{n+1}{2}\right]&\mathrm{if}\,\,\,\,i+j>n\,.&\end{array}\right.$
(3.55)

Note that $S_{E}^{(n)}$ has an extra _zeroth_ element $\lambda_{n+1}$
($\lambda_{i}\cdot\lambda_{n+1}=\lambda_{n+1}\;\forall\;\lambda_{i}\in
S_{E}^{(n)}$), not present in $S_{M}^{(n)}$. When defining an expanded Lie
algebra, the fact that the semigroup $S_{E}^{(n)}$ has a zero requires to
impose the _reduction condition_ $T^{(n+1)}_{\alpha}\equiv 0$ on the expanded
generators, together with $\mu_{n+1}\equiv 0$ and $\nu_{n+1}\equiv 0$ (and
additionally $\rho_{n+1}\equiv 0$ for $S\times\mathfrak{hs_{3}nw2}$) on the
invariant tensor constants introduced in (3.25)-(3.27). This condition is not
necessary when defining expanded algebras by means of the semigroup
$S_{M}^{(n)}$.

In [41], it was shown that the expanded Nappi-Witten algebras
$S_{E}^{(n)}\times\mathfrak{nw}$ define _Generalized Bargmann algebras_ , i.e.
an infinite family of extensions of the Bargmann algebra in $2+$1 dimensions.
On the other hand, the expanded algebras $S_{M}^{(n)}\times\mathfrak{nw}$ were
named _Generalized Newton-Hooke algebras_ since they provide generalizations
of the $S_{E}^{(n)}\times\mathfrak{nw}$ algebras that include a cosmological
constant. By using the same semigroups in the definition of the expanded
higher-spin Nappi-Witten algebras $\mathfrak{hs_{3}nw1}$ and
$\mathfrak{hs_{3}nw2}$, we can define the following families of spin-$3$
symmetries

$\displaystyle\textnormal{Generalized spin-}3\;\textnormal{Bargmann
algebras}:$ $\displaystyle
S_{E}^{(n)}\times\mathfrak{hs_{3}nw1},\;S_{E}^{(n)}\times\mathfrak{hs_{3}nw2},$
(3.56) $\displaystyle\textnormal{Generalized spin-}3\;\textnormal{Newton-Hooke
algebras}:$ $\displaystyle
S_{M}^{(n)}\times\mathfrak{hs_{3}nw1},\;S_{M}^{(n)}\times\mathfrak{hs_{3}nw2}.$
(3.57)

These symmetries are related by an Inöü-Wigner contraction, which can be
interpreted as a vanishing cosmological constant limit or, equivalently, as
the limit when the AdS radius, say $\ell$, becomes infinite. The constant
$\ell$ can be made explicit in the symmetry algebras
$S^{(n)}_{M}\times\mathfrak{hs_{3}nw1}$ and
$S^{(n)}_{M}\times\mathfrak{hs_{3}nw2}$ previously presented by rescaling the
expanded generators (3.4), (3.7), (3.10) in the form

$\displaystyle\boldsymbol{k}_{a}^{(i)}\rightarrow\ell^{i}\,\boldsymbol{k}_{a}^{(i)},$
$\displaystyle\boldsymbol{k}^{(i)}\rightarrow\ell^{i}\,\boldsymbol{k}^{(i)},$
$\displaystyle\boldsymbol{\sigma}^{(i)}\rightarrow\ell^{i}\,\boldsymbol{\sigma}^{(i)},$
(3.58)
$\displaystyle\boldsymbol{\mathcal{K}}_{ab}^{(i)}\rightarrow\ell^{i}\,\boldsymbol{\mathcal{K}}_{ab}^{(i)},$
$\displaystyle\boldsymbol{\mathcal{K}}_{a}^{(i)}\rightarrow\ell^{i}\,\boldsymbol{\mathcal{K}}_{a}^{(i)},$
$\displaystyle\boldsymbol{\Sigma}_{a}^{(i)}\rightarrow\ell^{i}\,\boldsymbol{\Sigma}_{a}^{(i)},$
$\displaystyle\boldsymbol{\mathcal{K}}_{a}^{(i)}\rightarrow\ell^{i}\,\boldsymbol{\mathcal{K}}_{a}^{(i)},$
$\displaystyle\boldsymbol{\mathcal{K}}_{ab}^{(i)}\rightarrow\ell^{i}\,\boldsymbol{\mathcal{K}}_{ab}^{(i)},$
$\displaystyle\boldsymbol{\Sigma}_{ab}^{(i)}\rightarrow\ell^{i}\,\boldsymbol{\Sigma}_{ab}^{(i)},$

and similarly for the invariant tensor constants given in (3.25)-(3.27), i.e.

$\mu_{i}\rightarrow\ell^{i}\mu_{i},\quad\nu_{i}\rightarrow\ell^{i}\rho_{i},\quad\rho_{i}\rightarrow\ell^{i}\rho_{i}.$
(3.59)

For a fixed value of $n$, the contraction $\ell\rightarrow\infty$ takes an
expanded algebra (5.3) into the corresponding expanded algebra (5.2), i.e.

$S_{M}^{(n)}\times\mathfrak{hs_{3}nw1}\xrightarrow[\ell\rightarrow\infty]{\,}S_{E}^{(n)}\times\mathfrak{hs_{3}nw1},\qquad
S_{M}^{(n)}\times\mathfrak{hs_{3}nw2}\xrightarrow[\ell\rightarrow\infty]{\,}S_{E}^{(n)}\times\mathfrak{hs_{3}nw2}.$
(3.60)

The rescaling (3.58) implies that the gauge fields in the connection one forms
(3.36) and (3.41) should be rescaled with the opposite powers of $\ell$,

$\displaystyle\theta^{a}_{(i)}\rightarrow\dfrac{1}{\ell^{i}}\theta^{a}_{(i)},$
$\displaystyle\alpha_{(i)}\rightarrow\dfrac{1}{\ell^{i}}\alpha_{(i)},$
$\displaystyle\beta_{(i)}\rightarrow\dfrac{1}{\ell^{i}}\beta_{(i)},$ (3.61)
$\displaystyle\Theta^{ab}_{(i)}\rightarrow\dfrac{1}{\ell^{i}}\Theta^{ab}_{(i)},$
$\displaystyle A^{a}_{(i)}\rightarrow\dfrac{1}{\ell^{i}}A^{a}_{(i)},$
$\displaystyle B^{a}_{(i)}\rightarrow\dfrac{1}{\ell^{i}}B^{a}_{(i)},$
$\displaystyle\Theta^{a}_{(i)}\rightarrow\dfrac{1}{\ell^{i}}\Theta^{a}_{(i)},$
$\displaystyle A^{ab}_{(i)}\rightarrow\dfrac{1}{\ell^{i}}A^{ab}_{(i)},$
$\displaystyle B^{ab}_{(i)}\rightarrow\dfrac{1}{\ell^{i}}B^{a}_{(i)},$

and similarly for the corresponding curvatures (3.42). In this way, one can
apply the flat limit $\ell\rightarrow\infty$ to the expanded CS actions (3.45)
and (3.46), reducing higher-spin gravity actions with $S^{(n)}_{M}$-expanded
invariance to the corresponding $S^{(n)}_{E}$-expanded ones.

NR and flat limits: As a final remark, the flat limit $\ell\rightarrow\infty$
can also be used at the level of the relativistic symmetries introduced in
Section 3.4. In particular, one can relate the NR limit given in (3.49) for
$S=S^{(n)}_{M}$ with the corresponding ones for $S=S^{(n)}_{E}$.

  * •

We first look at the $\mathfrak{sl}(3,\mathbb{R})\oplus\mathfrak{so}(3)$ case
(3.49), where one can extend the diagram (3.60) as

$\displaystyle\left[S^{(m)}_{M}\times\mathfrak{sl}(3,\mathbb{R})\right]\oplus\left[S^{(m)}_{M}\times\mathfrak{so}(3)\right]\;$
$\displaystyle\underset{c\rightarrow\infty}{\longrightarrow}$
$\displaystyle\qquad S^{(m)}_{M}\times\mathfrak{hs_{3}nw1}$ (3.62)
$\displaystyle\hskip
65.44142pt{\scriptstyle\ell\rightarrow\infty}\;\Big{\downarrow}$
$\displaystyle\hskip
25.6073pt{\scriptstyle\ell\rightarrow\infty}\;\Big{\downarrow}$
$\displaystyle\left[S^{(m)}_{E}\times\mathfrak{sl}(3,\mathbb{R})\right]\oplus\left[S^{(m)}_{E}\times\mathfrak{so}(3)\right]\;$
$\displaystyle\underset{c\rightarrow\infty}{\longrightarrow}$
$\displaystyle\qquad S^{(m)}_{E}\times\mathfrak{hs_{3}nw1}.$

To see this, it is necessary to extend the rescaling (3.63) to the case of the
relativistic algebras (3.49) and (3.52) for $S=S^{(n)}_{M}$,

$\boldsymbol{L}_{A}^{(i)}\rightarrow\ell^{i}\,\boldsymbol{L}_{A}^{(i)},\quad\boldsymbol{\mathcal{L}}_{AB}^{(i)}\rightarrow\ell^{i}\,\boldsymbol{\mathcal{L}}_{AB}^{(i)},$
(3.63)

and establish the following relations between relativistic spin-$3$ symmetries
[49]

$\displaystyle
S^{(1)}_{M}\times\mathfrak{sl}(3,\mathbb{R})\simeq\mathfrak{hs_{3}ads_{2+1}}$
$\displaystyle\xrightarrow[\ell\rightarrow\infty]{\,}$ $\displaystyle\qquad
S^{(1)}_{E}\times\mathfrak{sl}(3,\mathbb{R})\simeq\mathfrak{hs_{3}poin_{2+1}}$
(3.64) $\displaystyle
S^{(2)}_{M}\times\mathfrak{sl}(3,\mathbb{R})\simeq\mathfrak{hs_{3}adslor_{2+1}}$
$\displaystyle\xrightarrow[\ell\rightarrow\infty]{\,}$ $\displaystyle\qquad
S^{(2)}_{E}\times\mathfrak{sl}(3,\mathbb{R})\simeq\mathfrak{hs_{3}max_{2+1}}$
$\displaystyle\vdots$ $\displaystyle
S^{(n)}_{M}\times\mathfrak{sl}(3,\mathbb{R})$
$\displaystyle\xrightarrow[\ell\rightarrow\infty]{\,}$ $\displaystyle\qquad
S^{(n)}_{E}\times\mathfrak{sl}(3,\mathbb{R}),$

where the first line in (3.64) relates the spin-$3$ extensions of the AdS and
the Poincaré algebras in $2+1$ dimensions, whereas the second line does so for
the spin-$3$ extensions of the Maxwell and the AdS-Lorentz algebras. However,
in order to make contact with (3.62), one has to introduce the expanded
algebra (3.48) and similarly rescale the generators of
$S^{(n)}_{M}\times\mathfrak{so}(3)$ as

$\boldsymbol{X}^{(i)}\rightarrow\ell^{i}\,\boldsymbol{X}^{(i)},\quad\boldsymbol{X}_{a}^{(i)}\rightarrow\ell^{i}\,\boldsymbol{X}_{a}^{(i)},\\\
$ (3.65)

which leads to the contractions

$\displaystyle S^{(1)}_{M}\times\mathfrak{so}(3)\simeq\mathfrak{so}(4)$
$\displaystyle\xrightarrow[\ell\rightarrow\infty]{\,}$ $\displaystyle\qquad
S^{(1)}_{E}\times\mathfrak{so}(3)\simeq\mathfrak{iso}(3)$ (3.66)
$\displaystyle
S^{(2)}_{M}\times\mathfrak{so}(3)\simeq\mathfrak{e{\textnormal{-}}adslor_{3}}$
$\displaystyle\xrightarrow[\ell\rightarrow\infty]{\,}$ $\displaystyle\qquad
S^{(2)}_{E}\times\mathfrak{so}(3)\simeq\mathfrak{e{\textnormal{-}}max_{3}}$
$\displaystyle\vdots$ $\displaystyle S^{(m)}_{M}\times\mathfrak{so}(3)$
$\displaystyle\xrightarrow[\ell\rightarrow\infty]{\,}$ $\displaystyle\qquad
S^{(m)}_{E}\times\mathfrak{so}(3),$

where $\mathfrak{e{\textnormal{-}}max_{3}}$ and
$\mathfrak{e{\textnormal{-}}adslor_{3}}$ denote the Euclidean versions of the
Maxwell algebra and the AdS-Lorentz algebra in three spatial dimensions,
respectively. We have related the first two examples in (3.66) to known
algebras since these cases will be explicitly used in the next section.

  * •

When considering the expansion of
$\mathfrak{sl}(3,\mathbb{R})\oplus\mathfrak{sl}(2,\mathbb{R})\oplus\mathfrak{so}(2)$
introduced in (3.52), one can also combine the $c\rightarrow\infty$ and
$\ell\rightarrow\infty$ limits and extend the diagram (3.60)in the form

$\displaystyle\left[S^{(m)}_{M}\times\mathfrak{sl}(3,\mathbb{R})\right]\oplus\left[S^{(m)}_{M}\times\mathfrak{sl}(2,\mathbb{R})\right]\oplus\left[S^{(m)}_{M}\times\mathfrak{so}(2)\right]\;$
$\displaystyle\underset{c\rightarrow\infty}{\longrightarrow}$
$\displaystyle\qquad S^{(m)}_{M}\times\mathfrak{hs_{3}nw2}$ (3.67)
$\displaystyle\hskip
108.12054pt{\scriptstyle\ell\rightarrow\infty}\;\Big{\downarrow}$
$\displaystyle\hskip
25.6073pt{\scriptstyle\ell\rightarrow\infty}\;\Big{\downarrow}$
$\displaystyle\left[S^{(m)}_{E}\times\mathfrak{sl}(3,\mathbb{R})\right]\oplus\left[S^{(m)}_{E}\times\mathfrak{sl}(2,\mathbb{R})\right]\oplus\left[S^{(m)}_{E}\times\mathfrak{so}(2)\right]\;$
$\displaystyle\underset{c\rightarrow\infty}{\longrightarrow}$
$\displaystyle\qquad S^{(m)}_{E}\times\mathfrak{hs_{3}nw2}.$

The expansion of $\mathfrak{sl}(3,\mathbb{R})$ in (3.72) is again (3.64),
which now is to be supplemented with expansions of
$\mathfrak{sl}(2,\mathbb{R})\oplus\mathfrak{so}(2)$. Expanding the generators
of $\mathfrak{sl}(2,\mathbb{R})$ introduced in (2.12) and using the rescaling

$\boldsymbol{\mathcal{A}}^{(i)}\rightarrow\ell^{i}\,\boldsymbol{\mathcal{A}}^{(i)},\quad\boldsymbol{\mathcal{B}}^{(i)}\rightarrow\ell^{i}\,\boldsymbol{\mathcal{B}}^{(i)},\quad\boldsymbol{\mathcal{C}}^{(i)}\rightarrow\ell^{i}\,\boldsymbol{\mathcal{C}}^{(i)},$
(3.68)

allows one to relate the following expanded symmetries in the flat limit
$\ell\rightarrow\infty$

$\displaystyle
S^{(1)}_{M}\times\mathfrak{sl}(2,\mathbb{R})\simeq\mathfrak{so}(2,2)$
$\displaystyle\xrightarrow[\ell\rightarrow\infty]{\,}$ $\displaystyle\qquad
S^{(1)}_{E}\times\mathfrak{sl}(2,\mathbb{R})\simeq\mathfrak{isl}(2,\mathbb{R})$
(3.69) $\displaystyle
S^{(2)}_{M}\times\mathfrak{sl}(2,\mathbb{R})\simeq\mathfrak{adslor_{2+1}}$
$\displaystyle\xrightarrow[\ell\rightarrow\infty]{\,}$ $\displaystyle\qquad
S^{(2)}_{E}\times\mathfrak{sl}(2,\mathbb{R})\simeq\mathfrak{max_{2+1}}$
$\displaystyle\vdots$ $\displaystyle
S^{(m)}_{M}\times\mathfrak{sl}(2,\mathbb{R})$
$\displaystyle\xrightarrow[\ell\rightarrow\infty]{\,}$ $\displaystyle\qquad
S^{(m)}_{E}\times\mathfrak{sl}(2,\mathbb{R}).$

Due to the isomorphism $\mathfrak{so}(2,1)\simeq\mathfrak{sl}(2,\mathbb{R})$,
these relations correspond to the Euclidean continuation of the ones given in
(3.66). In the first case, the expanded algebras are the AdS and the Poincaré
algebras in three space-time dimensions, whereas in the second case one finds
the Maxwell and the AdS-Lorentz algebras in $2+1$ dimensions, which are known
to be related by an Inöü-Wigner contraction. Note that (3.68) can be
equivalently defined in the basis (2.14) where the expanded generators (3.51)
are rescaled as

$\boldsymbol{X}^{(i)}\rightarrow\ell^{i}\,\boldsymbol{X}^{(i)},\quad\boldsymbol{X}_{ab}^{(i)}\rightarrow\ell^{i}\,\boldsymbol{X}_{ab}^{(i)}.$
(3.70)

Finally, to complete the expansion of (2.12), we need to expand the
$\mathfrak{so}(2)$ generator $\boldsymbol{\mathcal{D}}$, which simply produces
$n$ copies of $\mathfrak{so}(2)$. In this case, the corresponding rescaling of
$\boldsymbol{\mathcal{D}}^{(i)}\rightarrow\ell^{i}\,\boldsymbol{\mathcal{D}}^{(i)}$
and flat limit does not modify the algebra, i.e.

$S^{(n)}_{M}\times\mathfrak{so}(2)\simeq
S^{(n)}_{E}\times\mathfrak{so}(2)\simeq\mathfrak{so}(2)\oplus\underset{n\;{\rm
times}}{\cdots}\oplus\mathfrak{so}(2).$ (3.71)

Putting it all together allows one to construct the left-hand side of (3.72).

The contractions here presented also relate the CS gravity theories invariant
under the corresponding symmetry algebras. It is thus a direct but involved
computation to obtain the CS actions invariant under the spin-$3$ NR algebras
(3.45) and (3.45), from the corresponding CS theories based on the
relativistic symmetries considered in (3.49) and (3.49), respectively, which
requires one to introduce rescalings in the relativistic gauge fields in
complete analogy with Eq. (3.61) in the NR case.

Pure Nappi-Witten case: It is worth remarking that, since the generators
$\boldsymbol{L}_{A}^{(i)}$ in (3.47) define an expansion of the
$\mathfrak{sl}(2,\mathbb{R})$ subalgebra of $\mathfrak{sl}(3,\mathbb{R})$, a
diagram of the form (3.69) is also obtained from (3.64) when eliminating the
higher-spin generators $\boldsymbol{\mathcal{L}}_{AB}^{(i)}$, reducing the
analysis to the pure gravity case. Dropping the higher-spin generators
$\boldsymbol{X}_{a}^{(i)}$ and $\boldsymbol{X}_{ab}^{(i)}$ in (3.48) and
(3.51) leaves in each case only the expanded generator $\boldsymbol{X}^{(i)}$
for which the expansion can be summarized in a diagram equivalent to (3.71).
Thus, in the pure gravity case, both diagrams (3.62) and (3.72) coincide and
reduce to.

$\displaystyle\left[S^{(m)}_{M}\times\mathfrak{sl}(2,\mathbb{R})\right]\oplus\left[S^{(m)}_{M}\times\mathfrak{so}(2)\right]\;$
$\displaystyle\underset{c\rightarrow\infty}{\longrightarrow}$
$\displaystyle\qquad S^{(m)}_{M}\times\mathfrak{nw}$ (3.72)
$\displaystyle\hskip
108.12054pt{\scriptstyle\ell\rightarrow\infty}\;\Big{\downarrow}$
$\displaystyle\hskip
25.6073pt{\scriptstyle\ell\rightarrow\infty}\;\Big{\downarrow}$
$\displaystyle\left[S^{(m)}_{E}\times\mathfrak{sl}(2,\mathbb{R})\right]\oplus\left[S^{(m)}_{E}\times\mathfrak{so}(2)\right]\;$
$\displaystyle\underset{c\rightarrow\infty}{\longrightarrow}$
$\displaystyle\qquad S^{(m)}_{E}\times\mathfrak{nw}.$

This means that for a given $n$, an expanded Nappi-Witten algebra (3.3) is the
NR contractions of the semidirect sum of the corresponding expanded
$\mathfrak{sl}(2,\mathbb{R})$ algebra and $n$ Abelian generators. As we will
see in the following section, this reproduces the known contractions that lead
to the Extended Bargmann algebra [63] and the Extended Electric NR Maxwell
algebra [62], and extends the result to expansions with arbitrary $n$.

##  4 Examples

Let us consider explicit examples of the higher-spin symmetries presented in
the previous section. In the following, we explicitly evaluate expanded
symmetries with the semigroups $S_{E}^{(n)}$ and $S_{M}^{(n)}$ for $n=1$ and
$n=2$. The $n=1$ case reproduces the higher-spin extensions of the extended
Bargmann and Newton-Hooke algebras in $2+1$ dimensions presented in [13]. The
$n=2$ case yields novel higher-spin extensions of the NR Maxwell algebra and
the NR AdS-Lorentz algebra in $2+1$ dimensions. The explicit construction of
the expanded algebras for $n\geq 3$ can be straightforwardly done along the
same lines and they will not be given here.

###  4.1 Spin-$3$ Bargmann algebras

The Bargmann algebra is the universal central extension of the Galilei algebra
[66] (for a recent review on non-Lorentzian symmetries, see also [52]). In
$2+1$ dimensions, it admits a second non-trivial central extension888In fact,
the Galilei algebra in three space-time dimensions admits a third central
extension, which is non-physical [67] and will not be considered here., that
allows defining a non-degenerate invariant bilinear form [60]. As a first
example, we reconstruct the spin-$3$ extensions of the Extended Bargmann
algebra in $2+1$ dimensions [13], as particular expansions of our spin-$3$
Nappi-Witten algebras. As shown in [41], the extended Bargmann algebra in
$2+1$ dimensions

$\displaystyle\left[\boldsymbol{J},\boldsymbol{G}_{a}\right]=\epsilon_{ab}\boldsymbol{G}_{b},$
$\displaystyle\left[\boldsymbol{G}_{a},\boldsymbol{G}_{b}\right]=-\epsilon_{ab}\boldsymbol{S},$
(4.1)
$\displaystyle\left[\boldsymbol{J},\boldsymbol{P}_{a}\right]=[\boldsymbol{H},\boldsymbol{G}_{a}]=\epsilon_{ab}\boldsymbol{P}_{b},$
$\displaystyle\left[\boldsymbol{G}_{a},\boldsymbol{P}_{b}\right]=-\epsilon_{ab}\boldsymbol{M},$

is the expanded Nappi-Witten algebra $S^{(1)}_{E}\times\mathfrak{nw}$,
generated by

$\displaystyle\boldsymbol{J}=\boldsymbol{k}^{(0)},$
$\displaystyle\boldsymbol{G}_{a}=\boldsymbol{k}_{a}^{(0)},$
$\displaystyle\boldsymbol{S}=\boldsymbol{\sigma}^{(0)},$ (4.2)
$\displaystyle\boldsymbol{H}=\boldsymbol{k}^{(1)},$
$\displaystyle\boldsymbol{P}_{a}=\boldsymbol{k}_{a}^{(1)},$
$\displaystyle\boldsymbol{M}=\boldsymbol{\sigma}^{(1)},$

where the semigroup $S^{(1)}_{E}$ is obtained by replacing $n=1$ in (3.54).
Here $\boldsymbol{J}$, $\boldsymbol{H}$, $\boldsymbol{G}_{a}$ and
$\boldsymbol{P}_{a}$ stand respectively for the generators of spatial
rotations, temporal translations, Galilean boosts and spatial translations,
whereas $\boldsymbol{S}$ and $\boldsymbol{M}$ are central extensions
associated to spin and mass999While the mass generator $\boldsymbol{M}$ is
present in higher-dimensional versions of the Bargmann algebra and its
physical meaning is understood (see, for example, [66]), the generator
$\boldsymbol{S}$ is particular to the three-dimensional case and its
interpretation as a spin angular momentum generator has not yet been fully
established [68, 69].. In the same way, the spin-$3$ extensions of the
Bargmann algebra can be defined by evaluating the expanded higher-spin
symmetries (3.8) and (3.11) for $S=S_{E}^{(1)}$.

  * •

The first spin-$3$ extension of the Extended Bargmann algebra in $2+1$
dimensions, $\mathfrak{hs_{3}barg1_{2+1}}$, is given by
$S_{E}^{(1)}\times\mathfrak{hs_{3}nw1}$. It can be obtained by defining the
higher-spin generators

$\displaystyle\boldsymbol{\mathcal{J}}_{a}=\boldsymbol{\mathcal{K}}_{a}^{(0)},$
$\displaystyle\boldsymbol{\mathcal{G}}_{ab}=\boldsymbol{\mathcal{K}}_{ab}^{(0)},$
$\displaystyle\boldsymbol{\mathcal{S}}_{a}=\boldsymbol{\Sigma}_{a}^{(0)},$
(4.3)
$\displaystyle\boldsymbol{\mathcal{H}}_{a}=\boldsymbol{\mathcal{K}}_{a}^{(1)},$
$\displaystyle\boldsymbol{\mathcal{P}}_{ab}=\boldsymbol{\mathcal{K}}_{ab}^{(1)},$
$\displaystyle\boldsymbol{\mathcal{M}}_{a}=\boldsymbol{\Sigma}_{a}^{(1)},$

and using the relations (3.8) and (3.54) for $n=1$. The commutation relations
of the expanded generators (4.2) and (4.3) are given by (4.1) together with

$\displaystyle\left[\boldsymbol{J},\boldsymbol{\mathcal{J}}_{a}\right]=\epsilon_{ab}\boldsymbol{\mathcal{J}}_{b},$
$\displaystyle\left[\boldsymbol{J},\boldsymbol{\mathcal{G}}_{ab}\right]=-\epsilon_{c(a}\boldsymbol{\mathcal{G}}_{b)c},$
$\displaystyle\left[\boldsymbol{J},\boldsymbol{\mathcal{S}}_{a}\right]=\left[\boldsymbol{S},\boldsymbol{\mathcal{J}}_{a}\right]=\epsilon_{ab}\boldsymbol{\mathcal{S}}_{b},$
(4.4)
$\displaystyle\left[\boldsymbol{G}_{a},\boldsymbol{\mathcal{J}}_{b}\right]=-\epsilon_{a(c}\boldsymbol{\mathcal{G}}_{b)c},$
$\displaystyle\left[\boldsymbol{G}_{a},\boldsymbol{\mathcal{G}}_{bc}\right]=-\epsilon_{a(b}\boldsymbol{\mathcal{S}}_{c)},$
$\displaystyle\left[\boldsymbol{\mathcal{J}}_{a},\boldsymbol{\mathcal{J}}_{b}\right]=\epsilon_{ab}\boldsymbol{J},$
$\displaystyle\left[\boldsymbol{\mathcal{J}}_{a},\boldsymbol{\mathcal{S}}_{b}\right]=\epsilon_{ab}\boldsymbol{S},$
$\displaystyle\left[\boldsymbol{\mathcal{J}}_{a},\boldsymbol{\mathcal{G}}_{bc}\right]=\delta_{a(b}\epsilon_{c)d}\boldsymbol{G}_{d},$
$\displaystyle\left[\boldsymbol{\mathcal{G}}_{ab},\boldsymbol{\mathcal{G}}_{cd}\right]=\delta_{(a(c}\epsilon_{d)b)}\boldsymbol{S},$

together with

$\displaystyle\left[\boldsymbol{J},\boldsymbol{\mathcal{H}}_{a}\right]=\left[\boldsymbol{H},\boldsymbol{\mathcal{J}}_{a}\right]=\epsilon_{ab}\boldsymbol{\mathcal{H}}_{b},$
$\displaystyle\left[\boldsymbol{G}_{a},\boldsymbol{\mathcal{H}}_{b}\right]=\left[\boldsymbol{P}_{a},\boldsymbol{\mathcal{J}}_{b}\right]=-\epsilon_{a(c}\boldsymbol{\mathcal{P}}_{b)c},$
(4.5)
$\displaystyle\left[\boldsymbol{J},\boldsymbol{\mathcal{P}}_{ab}\right]=\left[\boldsymbol{H},\boldsymbol{\mathcal{G}}_{ab}\right]=-\epsilon_{c(a}\boldsymbol{\mathcal{P}}_{b)c},$
$\displaystyle\left[\boldsymbol{\mathcal{J}}_{a},\boldsymbol{\mathcal{P}}_{bc}\right]=\left[\boldsymbol{\mathcal{H}}_{a},\boldsymbol{\mathcal{G}}_{bc}\right]=\delta_{a(b}\epsilon_{c)d}\boldsymbol{P}_{d},$
$\displaystyle\left[\boldsymbol{\mathcal{J}}_{a},\boldsymbol{\mathcal{H}}_{b}\right]=\epsilon_{ab}\boldsymbol{H},$
$\displaystyle\left[\boldsymbol{G}_{a},\boldsymbol{\mathcal{P}}_{bc}\right]=\left[\boldsymbol{P}_{a},\boldsymbol{\mathcal{G}}_{bc}\right]=-\epsilon_{a(b}\boldsymbol{\mathcal{M}}_{c)},$
$\displaystyle\left[\boldsymbol{\mathcal{J}}_{a},\boldsymbol{\mathcal{M}}_{b}\right]=\left[\boldsymbol{\mathcal{H}}_{a},\boldsymbol{\mathcal{S}}_{b}\right]=\epsilon_{ab}\boldsymbol{M},$
$\displaystyle\left[\boldsymbol{J},\boldsymbol{\mathcal{M}}_{a}\right]=\left[\boldsymbol{S},\boldsymbol{\mathcal{H}}_{a}\right]=\left[\boldsymbol{H},\boldsymbol{\mathcal{S}}_{a}\right]=\left[\boldsymbol{M},\boldsymbol{\mathcal{J}}_{a}\right]=\epsilon_{ab}\boldsymbol{\mathcal{M}}_{b}.$
$\displaystyle\left[\boldsymbol{\mathcal{G}}_{ab},\boldsymbol{\mathcal{P}}_{cd}\right]=\delta_{(a(c}\epsilon_{d)b)}\boldsymbol{M},$

Notice that, when including the higher-spin extension, the generators
$\boldsymbol{S}$ and $\boldsymbol{M}$ are not central anymore. The
$\mathfrak{hs_{3}barg1_{2+1}}$ algebra was obtained in [13] from a double
contraction (large-speed-of-light and large-AdS-raduis limit) of the spin-$3$
extension of the AdS algebra in three dimensions $\mathfrak{hs_{3}ads_{2+1}}$.

  * •

One can define a second spin-$3$ extension of the Extended Bargman algebra in
$2+1$ dimensions, $\mathfrak{hs_{3}barg2_{2+1}}$, as the expanded algebra
$S_{E}^{(1)}\times\mathfrak{hs_{3}nw2}$. Now, instead of (4.3), the
alternative generators

$\displaystyle\boldsymbol{\mathcal{J}}_{ab}=\tilde{\boldsymbol{\mathcal{K}}}_{ab}^{(0)},$
$\displaystyle\boldsymbol{\mathcal{G}}_{a}=\tilde{\boldsymbol{\mathcal{K}}}_{a}^{(0)},$
$\displaystyle\boldsymbol{\mathcal{S}}_{ab}=\tilde{\boldsymbol{\Sigma}}_{ab}^{(0)},$
(4.6)
$\displaystyle\boldsymbol{\mathcal{H}}_{ab}=\tilde{\boldsymbol{\mathcal{K}}}_{ab}^{(1)},$
$\displaystyle\boldsymbol{\mathcal{P}}_{a}=\tilde{\boldsymbol{\mathcal{K}}}_{a}^{(1)},$
$\displaystyle\boldsymbol{\mathcal{M}}_{ab}=\tilde{\boldsymbol{\Sigma}}_{ab}^{(1)},$

are introduced. Putting together the Extended Bargmann generators (4.2) with
(4.6), and using the commutation relations of the $\mathfrak{hs_{3}nw2}$
algebra (3.11) for the $S_{E}^{(1)}$ semigroup (3.54), one finds the
commutation relations (4.1) plus

$\displaystyle[\boldsymbol{J},\boldsymbol{\mathcal{G}}_{a}]=\epsilon_{ab}\boldsymbol{\mathcal{G}}_{b},$
$\displaystyle[\boldsymbol{J},\boldsymbol{\mathcal{J}}_{ab}]=-\epsilon_{c(a}\boldsymbol{\mathcal{J}}_{b)c},$
$\displaystyle[\boldsymbol{J},\boldsymbol{\mathcal{S}}_{ab}]=[\boldsymbol{S},\boldsymbol{\mathcal{J}}_{ab}]=-\epsilon_{c(a}\boldsymbol{\mathcal{S}}_{b)c},$
(4.7)
$\displaystyle[\boldsymbol{G}_{a},\boldsymbol{\mathcal{G}}_{b}]=-\epsilon_{a(c}\boldsymbol{\mathcal{S}}_{b)c},$
$\displaystyle[\boldsymbol{G}_{a},\boldsymbol{\mathcal{J}}_{bc}]=-\epsilon_{a(b}\boldsymbol{\mathcal{G}}_{c)},$
$\displaystyle[\boldsymbol{\mathcal{J}}_{ab},\boldsymbol{\mathcal{J}}_{cd}]=\delta_{(a(c}\epsilon_{d)b)}\boldsymbol{J},$
$\displaystyle[\boldsymbol{\mathcal{G}}_{a},\boldsymbol{\mathcal{G}}_{b}]=\epsilon_{ab}\boldsymbol{S},$
$\displaystyle[\boldsymbol{\mathcal{G}}_{a},\boldsymbol{\mathcal{J}}_{bc}]=\delta_{a(b}\epsilon_{c)d}\boldsymbol{G}_{d},$
$\displaystyle[\boldsymbol{\mathcal{J}}_{ab},\boldsymbol{\mathcal{S}}_{cd}]=\delta_{(a(c}\epsilon_{d)b)}\boldsymbol{S},$

and

$\displaystyle[\boldsymbol{J},\boldsymbol{\mathcal{P}}_{a}]=[\boldsymbol{H},\boldsymbol{\mathcal{G}}_{a}]=\epsilon_{ab}\boldsymbol{\mathcal{P}}_{b},$
$\displaystyle[\boldsymbol{G}_{a},\boldsymbol{\mathcal{H}}_{bc}]=[\boldsymbol{P}_{a},\boldsymbol{\mathcal{J}}_{bc}]=-\epsilon_{a(b}\boldsymbol{\mathcal{P}}_{c)},$
(4.8)
$\displaystyle[\boldsymbol{J},\boldsymbol{\mathcal{H}}_{ab}]=[\boldsymbol{H},\boldsymbol{\mathcal{J}}_{ab}]=-\epsilon_{c(a}\boldsymbol{\mathcal{H}}_{b)c},$
$\displaystyle[\boldsymbol{\mathcal{G}}_{a},\boldsymbol{\mathcal{H}}_{bc}]=[\boldsymbol{\mathcal{P}}_{a},\boldsymbol{\mathcal{J}}_{bc}]=\delta_{a(b}\epsilon_{c)d}\boldsymbol{P}_{d},$
$\displaystyle[\boldsymbol{\mathcal{J}}_{ab},\boldsymbol{\mathcal{H}}_{cd}]=\delta_{(a(c}\epsilon_{d)b)}\boldsymbol{H},$
$\displaystyle[\boldsymbol{\mathcal{J}}_{ab},\boldsymbol{\mathcal{M}}_{cd}]=[\boldsymbol{\mathcal{H}}_{ab},\boldsymbol{\mathcal{S}}_{cd}]=\delta_{(a(c}\epsilon_{d)b)}\boldsymbol{M},$
$\displaystyle[\boldsymbol{\mathcal{G}}_{a},\boldsymbol{\mathcal{P}}_{b}]=\epsilon_{ab}\boldsymbol{M},$
$\displaystyle[\boldsymbol{J},\boldsymbol{\mathcal{M}}_{ab}]=[\boldsymbol{H},\boldsymbol{\mathcal{S}}_{ab}]=[\boldsymbol{S},\boldsymbol{\mathcal{H}}_{ab}]$
$\displaystyle[\boldsymbol{G}_{a},\boldsymbol{\mathcal{P}}_{b}]=[\boldsymbol{P}_{a},\boldsymbol{\mathcal{G}}_{b}]=-\epsilon_{a(c}\boldsymbol{\mathcal{M}}_{b)c},$
$\displaystyle\hskip
45.52458pt=[\boldsymbol{M},\boldsymbol{\mathcal{J}}_{ab}]=-\epsilon_{c(a}\boldsymbol{\mathcal{M}}_{b)c}.$

Even though this algebra is not explicitly equivalent to the
$\mathfrak{hs_{3}barg2_{2+1}}$ algebra obtained in [13] by means of the
Medina-Revoy theorem, one can show they are isomorphic by using the
redefinition (3.12) for the higher-spin generators
$\boldsymbol{\mathcal{S}}_{ab}$ and $\boldsymbol{\mathcal{M}}_{ab}$.

Thus, we have explicitly constructed the spin-$3$ NR algebras

$\mathfrak{hs_{3}barg1_{2+1}}=S^{(1)}_{E}\times\mathfrak{hs_{3}nw1},\qquad\mathfrak{hs_{3}barg2_{2+1}}=S^{(1)}_{E}\times\mathfrak{hs_{3}nw2}.$
(4.9)

As argued in [13], two additional higher-spin extensions of the Extended
Bargmann algebra can be obtained by interchanging

$\displaystyle\boldsymbol{\mathcal{J}}_{a}\leftrightarrow\boldsymbol{\mathcal{H}}_{a},$
$\displaystyle\boldsymbol{\mathcal{G}}_{ab}\leftrightarrow\boldsymbol{\mathcal{P}}_{ab},$
$\displaystyle\boldsymbol{\mathcal{S}}_{a}\leftrightarrow\boldsymbol{\mathcal{M}}_{a},$
(4.10)
$\displaystyle\boldsymbol{\mathcal{J}}_{ab}\leftrightarrow\boldsymbol{\mathcal{H}}_{ab},$
$\displaystyle\boldsymbol{\mathcal{G}}_{a}\leftrightarrow\boldsymbol{\mathcal{P}}_{b},$
$\displaystyle\boldsymbol{\mathcal{S}}_{ab}\leftrightarrow\boldsymbol{\mathcal{M}}_{ab},$

in (4.4) and (4.5), and in (4.7) and (4.8), respectively.

Semidirect sum structure: The algebras $\mathfrak{hs_{3}barg1_{2+1}}$ and
$\mathfrak{hs_{3}barg2_{2+1}}$ are semidirect sums. This is to be expected
since they are higher-spin extensions of the NR limit of the Poincaré algebra,
which is a semidirect sum itself. Indeed, the first line of (4.1) together
with Eq. (4.4) (Eq. (4.7)) is a subalgebra isomorphic to
$\mathfrak{hs_{3}nw1}$ ($\mathfrak{hs_{3}nw2}$). The remaining generators form
an Abelian ideal that can be identified with $\mathfrak{hs_{3}nw1}$
($\mathfrak{hs_{3}nw2}$) seen as an Abelian vector space, which can be denoted
by $\mathfrak{hs_{3}nw1}^{({\rm ab})}$ ($\mathfrak{hs_{3}nw2}^{({\rm ab})}$),
and on which $\mathfrak{hs_{3}nw1}$ ($\mathfrak{hs_{3}nw2}$) acts with the
adjoint representation [70]. Thus, the $\mathfrak{hs_{3}barg1_{2+1}}$
($\mathfrak{hs_{3}barg2_{2+1}}$) algebra can be written in a compact form as

$\left[(X,X^{\ast}),(Y,Y^{\ast})\right]=\left([X,Y],[X,Y^{\ast}]-[Y,X^{\ast}]\right),$
(4.11)

where $X$ is an element on the $\mathfrak{hs_{3}nw1}$ ($\mathfrak{hs_{3}nw2}$)
algebra and $X^{\ast}$ is an element on $\mathfrak{hs_{3}nw1}^{({\rm ab})}$
($\mathfrak{hs_{3}nw2}^{({\rm ab})}$). Therefore, the spin-$3$ extensions of
the Extended Bargmann algebra in $2+1$ dimensions can be defined as the
semidirect sums

$\displaystyle\mathfrak{hs_{3}barg1_{2+1}}$
$\displaystyle\simeq\mathfrak{hs_{3}nw1}\oright_{\rm
ad}\mathfrak{hs_{3}nw1}^{({\rm ab})},$ (4.12)
$\displaystyle\mathfrak{hs_{3}barg2_{2+1}}$
$\displaystyle\simeq\mathfrak{hs_{3}nw2}\oright_{\rm
ad}\mathfrak{hs_{3}nw2}^{({\rm ab})}.$

This is the NR analog of the semidirect sum [71]

$\mathfrak{hs_{3}poin_{2+1}}\simeq\mathfrak{sl}(3,\mathbb{R})\oright_{\rm
ad}\mathfrak{sl}(3,\mathbb{R})^{({\rm ab})},$ (4.13)

which corresponds to the spin-$3$ extension of the Poincaré algebra in three
space-time dimensions [11, 12].

NR limit: The spin-$3$ extensions of the Bargman algebra (4.9) can
alternatively be obtained as Inöü-Wigner contractions of relativistic
symmetries. In [49], it was shown that the spin-$3$ extension of the Poincaré
algebra in three space-time dimensions (4.13) is the expansion

$\mathfrak{hs_{3}poin_{2+1}}\simeq
S^{(1)}_{E}\times\mathfrak{sl}(3,\mathbb{R}),$ (4.14)

with expanded generators

$\boldsymbol{J}_{A}^{\rm
rel}=\boldsymbol{L}^{(0)}_{A},\quad\boldsymbol{P}_{A}^{\rm
rel}=\boldsymbol{L}^{(1)}_{A},\quad\boldsymbol{\mathcal{J}}_{AB}^{\rm
rel}=\boldsymbol{\mathcal{L}}^{(0)}_{AB},\quad\boldsymbol{\mathcal{P}}_{AB}^{\rm
rel}=\boldsymbol{\mathcal{L}}^{(1)}_{AB},$ (4.15)

and commutation relations given by (3.47) evaluated for the semigroup product
law of $S^{(1)}_{E}$ given in (3.54). In order to define the NR limits (3.49)
and (3.52), we need to introduce the expanded algebras (3.48) and (3.51). For
$S=S^{(1)}_{E}$, these are respectively isomorphic to the Euclidean algebra in
three spatial dimensions $\mathfrak{iso}(3)$ (spanned by
$\\{\boldsymbol{X}^{(0)},\boldsymbol{X}^{(1)},\boldsymbol{X}_{a}^{(0)},\boldsymbol{X}_{a}^{(1)}\\}$
in Eq. (3.48)) and the Poincaré algebra in three space-time dimensions,
$\mathfrak{isl}(2,\mathbb{R})$, supplemented with two Abelian
generators101010The isomorphism
$S^{(1)}_{E}\times\Big{(}\mathfrak{sl}(2,\mathbb{R})\oplus\mathfrak{so}(2)\Big{)}\simeq\mathfrak{isl}(2,\mathbb{R})\oplus\mathfrak{so}(2)\oplus\mathfrak{so}(2)$
is more transparent at the level of the commutation relations when expanding
$\mathfrak{sl}(2,\mathbb{R})\oplus\mathfrak{so}(2)$ in the basis (2.12) rather
than (2.14). (spanned by
$\\{\boldsymbol{X}^{(0)},\boldsymbol{X}^{(1)},\boldsymbol{X}_{ab}^{(0)},\boldsymbol{X}_{ab}^{(1)}\\}$
in Eq. (3.51)), i.e.

$\displaystyle S^{(1)}_{E}\times\mathfrak{so}(3)\simeq\mathfrak{iso}(3),$
(4.16) $\displaystyle
S^{(1)}_{E}\times\Big{(}\mathfrak{sl}(2,\mathbb{R})\oplus\mathfrak{so}(2)\Big{)}\simeq\mathfrak{isl}(2,\mathbb{R})\oplus\mathfrak{so}(2)\oplus\mathfrak{so}(2).$

These isomorphisms also follow from the diagrams (3.66), (3.69) and (3.71).
Inverting (3.50), one finds the relation between the Poincaré and Bargman
generators

$\displaystyle\boldsymbol{J}$ $\displaystyle=\boldsymbol{J}_{0}^{\rm
rel}+\boldsymbol{X}^{(0)},$ $\displaystyle\boldsymbol{J}_{a}$
$\displaystyle=\frac{1}{c}\boldsymbol{J}_{a}^{\rm rel},$
$\displaystyle\boldsymbol{S}$
$\displaystyle=\frac{1}{2c^{2}}\left(\boldsymbol{J}_{0}^{\rm
rel}-\boldsymbol{X}^{(0)}\right),$ (4.17) $\displaystyle\boldsymbol{H}$
$\displaystyle=\boldsymbol{P}_{0}^{\rm rel}+\boldsymbol{X}^{(1)},$
$\displaystyle\boldsymbol{P}_{a}$
$\displaystyle=\frac{1}{c}\boldsymbol{P}_{a}^{\rm rel},$
$\displaystyle\boldsymbol{M}$
$\displaystyle=\frac{1}{2c^{2}}\left(\boldsymbol{P}_{0}^{\rm
rel}-\boldsymbol{X}^{(1)}\right),$

and the corresponding relations between NR and relativistic spin-$3$
generators

$\displaystyle\boldsymbol{\mathcal{J}}_{a}$
$\displaystyle=\boldsymbol{\mathcal{J}}_{a0}^{\rm
rel}+\boldsymbol{X}_{a}^{(0)},$ $\displaystyle\boldsymbol{\mathcal{G}}_{ab}$
$\displaystyle=\frac{1}{c}\boldsymbol{\mathcal{J}}_{ab}^{\rm rel},$
$\displaystyle\boldsymbol{\mathcal{S}}_{a}$
$\displaystyle=\frac{1}{2c^{2}}\left(\boldsymbol{\mathcal{J}}_{a0}^{\rm
rel}-\boldsymbol{X}_{a}^{(0)}\right),$ (4.18)
$\displaystyle\boldsymbol{\mathcal{H}}_{a}$
$\displaystyle=\boldsymbol{\mathcal{P}}_{a0}^{\rm
rel}+\boldsymbol{X}_{a}^{(1)},$ $\displaystyle\boldsymbol{\mathcal{P}}_{ab}$
$\displaystyle=\frac{1}{c}\boldsymbol{\mathcal{P}}_{ab}^{\rm rel},$
$\displaystyle\boldsymbol{\mathcal{M}}_{a}$
$\displaystyle=\frac{1}{2c^{2}}\left(\boldsymbol{\mathcal{P}}_{a0}^{\rm
rel}-\boldsymbol{X}_{a}^{(1)}\right).$

Similarly, using (3.53) one gets (4.17) and

$\displaystyle\boldsymbol{\mathcal{J}}_{ab}$
$\displaystyle=\boldsymbol{\mathcal{J}}_{ab}^{\rm
rel}+\boldsymbol{X}_{ab}^{(0)},$ $\displaystyle\boldsymbol{\mathcal{G}}_{a}$
$\displaystyle=\frac{1}{c}\boldsymbol{\mathcal{J}}_{a0}^{\rm rel},$
$\displaystyle\boldsymbol{\mathcal{S}}_{ab}$
$\displaystyle=\frac{1}{2c^{2}}\left(\boldsymbol{\mathcal{J}}_{ab}^{\rm
rel}-\boldsymbol{X}_{ab}^{(0)}\right),$ (4.19)
$\displaystyle\boldsymbol{\mathcal{H}}_{ab}$
$\displaystyle=\boldsymbol{\mathcal{P}}_{ab}^{\rm
rel}+\boldsymbol{X}_{ab}^{(1)},$ $\displaystyle\boldsymbol{\mathcal{P}}_{a}$
$\displaystyle=\frac{1}{c}\boldsymbol{\mathcal{P}}_{a0}^{\rm rel},$
$\displaystyle\boldsymbol{\mathcal{M}}_{ab}$
$\displaystyle=\frac{1}{2c^{2}}\left(\boldsymbol{\mathcal{P}}_{ab}^{\rm
rel}-\boldsymbol{X}_{ab}^{(1)}\right).$

Thus, putting it all together one finds that $\mathfrak{hs_{3}barg1_{2+1}}$
can be obtained from the contraction defined by (4.17) and (4.18), while
$\mathfrak{hs_{3}barg2_{2+1}}$ follows from taking $c\rightarrow\infty$ in
(4.17) and (4.19). Schematically, the NR limits (3.49) and (3.52) for
$S=S^{(1)}_{E}$ can be written as

$\displaystyle\mathfrak{hs_{3}poin_{2+1}}\oplus\mathfrak{iso}(3)\;$
$\displaystyle\underset{c\rightarrow\infty}{\longrightarrow}$
$\displaystyle\qquad\mathfrak{hs_{3}barg1_{2+1}}$ (4.20)
$\displaystyle\mathfrak{hs_{3}poin_{2+1}}\oplus\mathfrak{isl}(2,\mathbb{R})\oplus\mathfrak{so}(2)\oplus\mathfrak{so}(2)\;$
$\displaystyle\underset{c\rightarrow\infty}{\longrightarrow}$
$\displaystyle\qquad\mathfrak{hs_{3}barg2_{2+1}}$

Moreover, one can see from (4.17) that dropping the higher-spin generators
precisely corresponds to the contraction relating the Poincaré and the
Extended Bargmann algebras in three space-time dimensions [63]. In other
words, when removing the spin-$3$ generators, the left-hand sides of the
contractions in (4.20) boil down to
$\mathfrak{iso}(2,1)\oplus\mathfrak{so}(2)\oplus\mathfrak{so}(2)$ whereas the
right-hand sides reduce to the Extended Bargman algebra in $2+1$ dimensions.

###  4.2 Spin-$3$ non-relativistic Maxwell algebras

The Maxwell algebra is an extension of the Poincaré algebra that describes
particle systems in the presence of a constant electromagnetic field [72, 73].
In $2+1$ dimensions, the electric NR limit of the Maxwell algebra [48, 74, 75]
admits three central extensions [62] (there dubbed _exotic_), in complete
analogy with the two central extensions appearing in the Extended Bargmann
algebra. This particular extended NR limit111111Other NR limits of the Maxwell
algebra have been considered in [76, 77, 78, 75]., which from now on in this
paper will be simply referred to as NR Maxwell algebra in $2+1$ dimensions, is
isomorphic to the expanded Nappi-Witten algebra
$S_{E}^{(2)}\times\mathfrak{nw}$ [41], and its commutation relations are (4.1)
plus

$\displaystyle\left[\boldsymbol{J},\boldsymbol{Z}_{a}\right]=\left[\boldsymbol{H},\boldsymbol{P}_{a}\right]=\left[\boldsymbol{Y},\boldsymbol{G}_{a}\right]=\epsilon_{a}^{\;\;b}\boldsymbol{Z}_{b},\qquad\left[\boldsymbol{G}_{a},\boldsymbol{Z}_{b}\right]=\left[\boldsymbol{P}_{a},\boldsymbol{P}_{b}\right]=-\epsilon_{ab}\boldsymbol{T},$
(4.21)

where the expanded Nappi-Witten generators are the ones given in (4.2) and

$\boldsymbol{Y}=\boldsymbol{k}^{(2)},\quad\boldsymbol{Z}_{a}=\boldsymbol{k}_{b}^{(2)},\quad\boldsymbol{T}=\boldsymbol{\sigma}^{(2)}.$
(4.22)

In the NR Maxwell algebra, the generators $\boldsymbol{Y}$ and
$\boldsymbol{Z}_{a}$ are associated to the magnetic and the electric field
that define the constant electromagnetic background, while $\boldsymbol{T}$ is
a third central term (which, as in the previous case, ceases being central
when the higher-spin generators are introduced). One can define two
inequivalent spin-$3$ extensions of the NR Maxwell algebra in three dimensions
by evaluating the expanded algebras (3.8) and (3.11) for $S=S^{(2)}_{E}$, i.e.

$\mathfrak{hs_{3}nr\textnormal{-}max1_{2+1}}=S^{(2)}_{E}\times\mathfrak{hs_{3}nw1},\qquad\mathfrak{hs_{3}nr\textnormal{-}max2_{2+1}}=S^{(2)}_{E}\times\mathfrak{hs_{3}nw2}.$
(4.23)

  * •

In order to define the first higher-spin extension
$\mathfrak{hs_{3}nr\textnormal{-}max1_{2+1}}$, we evaluate the commutation
relations of $S^{(2)}_{E}\times\mathfrak{hs_{3}nw1}$ using (3.8) and the
semigroup product law (3.54) for $n=2$. We define

$\boldsymbol{\mathcal{Y}}_{a}=\boldsymbol{\mathcal{K}}_{a}^{(2)},\quad\boldsymbol{\mathcal{Z}}_{ab}=\boldsymbol{\mathcal{K}}_{ab}^{(2)},\quad\boldsymbol{\mathcal{T}}_{a}=\boldsymbol{\Sigma}_{a}^{(2)},$
(4.24)

which, together with the set of expanded generators appearing in Eqs. (4.2),
(4.3) and (4.22), satisfy the commutation relations (4.1), (4.4), (4.5) (4.21)
and

$\displaystyle\left[\boldsymbol{J},\boldsymbol{\mathcal{Y}}_{a}\right]=\left[\boldsymbol{Y},\boldsymbol{\mathcal{J}}_{a}\right]=\left[\boldsymbol{H},\boldsymbol{\mathcal{H}}_{a}\right]=\epsilon_{ab}\boldsymbol{\mathcal{Y}}_{b},$
(4.25)
$\displaystyle\left[\boldsymbol{J},\boldsymbol{\mathcal{Z}}_{ab}\right]=\left[\boldsymbol{Y},\boldsymbol{\mathcal{G}}_{ab}\right]=\left[\boldsymbol{H},\boldsymbol{\mathcal{P}}_{ab}\right]=-\epsilon_{c(a}\boldsymbol{\mathcal{Z}}_{b)c},$
$\displaystyle\left[\boldsymbol{\mathcal{J}}_{a},\boldsymbol{\mathcal{Y}}_{b}\right]=\left[\boldsymbol{\mathcal{H}}_{a},\boldsymbol{\mathcal{H}}_{b}\right]=\epsilon_{ab}\boldsymbol{Y},$
$\displaystyle\left[\boldsymbol{\mathcal{J}}_{a},\boldsymbol{\mathcal{T}}_{b}\right]=\left[\boldsymbol{\mathcal{Y}}_{a},\boldsymbol{\mathcal{S}}_{b}\right]=\left[\boldsymbol{\mathcal{H}}_{a},\boldsymbol{\mathcal{M}}_{b}\right]=\epsilon_{ab}\boldsymbol{T},$
$\displaystyle\left[\boldsymbol{\mathcal{G}}_{ab},\boldsymbol{\mathcal{Z}}_{cd}\right]=\left[\boldsymbol{\mathcal{P}}_{ab},\boldsymbol{\mathcal{P}}_{cd}\right]=\delta_{(a(c}\epsilon_{d)b)}\boldsymbol{T},$
$\displaystyle\left[\boldsymbol{G}_{a},\boldsymbol{\mathcal{Y}}_{b}\right]=\left[\boldsymbol{Z}_{a},\boldsymbol{\mathcal{J}}_{b}\right]=\left[\boldsymbol{P}_{a},\boldsymbol{\mathcal{H}}_{b}\right]=-\epsilon_{a(c}\boldsymbol{\mathcal{Z}}_{b)c},$
$\displaystyle\left[\boldsymbol{\mathcal{J}}_{a},\boldsymbol{\mathcal{Z}}_{bc}\right]=\left[\boldsymbol{\mathcal{Y}}_{a},\boldsymbol{\mathcal{G}}_{bc}\right]=\left[\boldsymbol{\mathcal{H}}_{a},\boldsymbol{\mathcal{P}}_{bc}\right]=\delta_{a(b}\epsilon_{c)d}\boldsymbol{Z}_{d},$
$\displaystyle\left[\boldsymbol{G}_{a},\boldsymbol{\mathcal{Z}}_{bc}\right]=\left[\boldsymbol{Z}_{a},\boldsymbol{\mathcal{G}}_{bc}\right]=\left[\boldsymbol{P}_{a},\boldsymbol{\mathcal{P}}_{bc}\right]=-\epsilon_{a(b}\boldsymbol{\mathcal{T}}_{c)},$
$\displaystyle\left[\boldsymbol{J},\boldsymbol{\mathcal{T}}_{a}\right]=\left[\boldsymbol{S},\boldsymbol{\mathcal{Y}}_{a}\right]=\left[\boldsymbol{Y},\boldsymbol{\mathcal{S}}_{a}\right]=\left[\boldsymbol{T},\boldsymbol{\mathcal{J}}_{a}\right]=\left[\boldsymbol{H},\boldsymbol{\mathcal{M}}_{a}\right]=\left[\boldsymbol{M},\boldsymbol{\mathcal{H}}_{a}\right]=\epsilon_{ab}\boldsymbol{\mathcal{T}}_{b}.$

  * •

The second spin-$3$ extension of the NR Maxwell symmetry,
$\mathfrak{hs_{3}nr\textnormal{-}max2_{2+1}}$, follows from defining

$\boldsymbol{\mathcal{Y}}_{ab}=\tilde{\boldsymbol{\mathcal{K}}}_{ab}^{(2)},\quad\boldsymbol{\mathcal{Z}}_{a}=\tilde{\boldsymbol{\mathcal{K}}}_{a}^{(2)},\quad\boldsymbol{\mathcal{T}}_{ab}=\tilde{\boldsymbol{\Sigma}}_{ab}^{(2)},$
(4.26)

in addition to (4.2), (4.6) and (4.22). Using again the multiplication law of
the semigroup $S^{(2)}_{E}$ the algebra
$\mathfrak{hs_{3}nr\textnormal{-}max2_{2+1}}$ can be read off from (3.11) and
is given by (4.1), (4.7), (4.8), (4.21) together with the commutation
realations

$\displaystyle[\boldsymbol{J},\boldsymbol{\mathcal{Z}}_{a}]=[\boldsymbol{Y},\boldsymbol{\mathcal{G}}_{a}]=[\boldsymbol{H},\boldsymbol{\mathcal{P}}_{a}]=\epsilon_{ab}\boldsymbol{\mathcal{Z}}_{b},$
(4.27)
$\displaystyle[\boldsymbol{J},\boldsymbol{\mathcal{Y}}_{ab}]=[\boldsymbol{Y},\boldsymbol{\mathcal{J}}_{ab}]=[\boldsymbol{H},\boldsymbol{\mathcal{H}}_{ab}]=-\epsilon_{c(a}\boldsymbol{\mathcal{Y}}_{b)c},$
$\displaystyle[\boldsymbol{\mathcal{J}}_{ab},\boldsymbol{\mathcal{Y}}_{cd}]=[\boldsymbol{\mathcal{H}}_{ab},\boldsymbol{\mathcal{H}}_{cd}]=\delta_{(a(c}\epsilon_{d)b)}\boldsymbol{Y},$
$\displaystyle[\boldsymbol{\mathcal{G}}_{a},\boldsymbol{\mathcal{Z}}_{b}]=[\boldsymbol{\mathcal{P}}_{a},\boldsymbol{\mathcal{P}}_{b}]=\epsilon_{ab}\boldsymbol{T},$
$\displaystyle[\boldsymbol{G}_{a},\boldsymbol{\mathcal{Z}}_{b}]=[\boldsymbol{Z}_{a},\boldsymbol{\mathcal{G}}_{b}]=[\boldsymbol{P}_{a},\boldsymbol{\mathcal{P}}_{b}]=-\epsilon_{a(c}\boldsymbol{\mathcal{T}}_{b)c},$
$\displaystyle[\boldsymbol{G}_{a},\boldsymbol{\mathcal{Y}}_{bc}]=[\boldsymbol{Z}_{a},\boldsymbol{\mathcal{J}}_{bc}]=[\boldsymbol{P}_{a},\boldsymbol{\mathcal{H}}_{bc}]=-\epsilon_{a(b}\boldsymbol{\mathcal{Z}}_{c)},$
$\displaystyle[\boldsymbol{\mathcal{G}}_{a},\boldsymbol{\mathcal{Y}}_{bc}]=[\boldsymbol{\mathcal{Z}}_{a},\boldsymbol{\mathcal{J}}_{bc}]=[\boldsymbol{\mathcal{P}}_{a},\boldsymbol{\mathcal{H}}_{bc}]=\delta_{a(b}\epsilon_{c)d}\boldsymbol{Z}_{d},$
$\displaystyle[\boldsymbol{\mathcal{J}}_{ab},\boldsymbol{\mathcal{T}}_{cd}]=[\boldsymbol{\mathcal{Y}}_{ab},\boldsymbol{\mathcal{S}}_{cd}]=[\boldsymbol{\mathcal{H}}_{ab},\boldsymbol{\mathcal{M}}_{cd}]=\delta_{(a(c}\epsilon_{d)b)}\boldsymbol{T},$
$\displaystyle[\boldsymbol{J},\boldsymbol{\mathcal{T}}_{ab}]=[\boldsymbol{Y},\boldsymbol{\mathcal{S}}_{ab}]=[\boldsymbol{S},\boldsymbol{\mathcal{Y}}_{ab}]=[\boldsymbol{T},\boldsymbol{\mathcal{J}}_{ab}]=[\boldsymbol{H},\boldsymbol{\mathcal{M}}_{ab}]=[\boldsymbol{M},\boldsymbol{\mathcal{H}}_{ab}]=-\epsilon_{c(a}\boldsymbol{\mathcal{T}}_{b)c}.$

In the same way as in the Bargmann case, one can rename the higher-spin
generators in the algebras given above either as

$\displaystyle\boldsymbol{\mathcal{H}}_{a}\leftrightarrow\boldsymbol{\mathcal{Y}}_{a},$
$\displaystyle\boldsymbol{\mathcal{P}}_{ab}\leftrightarrow\boldsymbol{\mathcal{Z}}_{ab},$
$\displaystyle\boldsymbol{\mathcal{M}}_{a}\leftrightarrow\boldsymbol{\mathcal{T}}_{a},$
(4.28)
$\displaystyle\boldsymbol{\mathcal{H}}_{ab}\leftrightarrow\boldsymbol{\mathcal{Y}}_{ab},$
$\displaystyle\boldsymbol{\mathcal{P}}_{a}\leftrightarrow\boldsymbol{\mathcal{Z}}_{b},$
$\displaystyle\boldsymbol{\mathcal{M}}_{ab}\leftrightarrow\boldsymbol{\mathcal{T}}_{ab},$

or as

$\displaystyle\boldsymbol{\mathcal{J}}_{a}\leftrightarrow\boldsymbol{\mathcal{Y}}_{a},$
$\displaystyle\boldsymbol{\mathcal{G}}_{ab}\leftrightarrow\boldsymbol{\mathcal{Z}}_{ab},$
$\displaystyle\boldsymbol{\mathcal{S}}_{a}\leftrightarrow\boldsymbol{\mathcal{T}}_{a},$
(4.29)
$\displaystyle\boldsymbol{\mathcal{J}}_{ab}\leftrightarrow\boldsymbol{\mathcal{Y}}_{ab},$
$\displaystyle\boldsymbol{\mathcal{G}}_{a}\leftrightarrow\boldsymbol{\mathcal{Z}}_{b},$
$\displaystyle\boldsymbol{\mathcal{S}}_{ab}\leftrightarrow\boldsymbol{\mathcal{T}}_{ab},$

giving isomorphic spin-$3$ extensions of the NR Maxwell algebra in $2+1$
dimensions, which can be physically different depending on how the higher-spin
generators are physically interpreted. Similarly one can try the combinations
(4.10), (4.28) and (4.10), (4.29).

It is interesting to note that the expanded algebras
$S^{(2)}_{E}\times\mathfrak{hs_{3}nw1}$ and
$S^{(2)}_{E}\times\mathfrak{hs_{3}nw2}$ also define spin-$3$ extensions of a
NR version of the Hietarinta algebra in $2+1$ dimensions [79, 80] if one
interchanges the generators

$\boldsymbol{H}\leftrightarrow\boldsymbol{Y},\quad\boldsymbol{P}_{a}\leftrightarrow\boldsymbol{Z}_{a},\quad\boldsymbol{M}\leftrightarrow\boldsymbol{T},\\\
$ (4.30)

in the NR Maxwell sublagebra defined by Eqs. (4.1) and (4.21).

Extended semidirect sum structure: The relativistic Maxwell algebra has the
structure of an extended semidirect sum, where the Poincaré translations
become non-Abelian. On the other hand, the higher-spin symmetry
$\mathfrak{hs_{3}nr\textnormal{-}max1_{2+1}}$
($\mathfrak{hs_{3}nr\textnormal{-}max2_{2+1}}$) extends the algebra
$\mathfrak{hs_{3}barg1_{2+1}}$ ($\mathfrak{hs_{3}barg2_{2+1}}$) found in
Section 4.1 by rendering its Abelian ideal of space and time translations non-
Abelian. Since the extra generators in this extension form, in turn, a new
Abelian ideal that can be identified with $\mathfrak{hs_{3}nw1}^{({\rm ab})}$
($\mathfrak{hs_{3}nw1}^{({\rm ab})}$), one can conclude that
$\mathfrak{hs_{3}nr\textnormal{-}max1_{2+1}}$
($\mathfrak{hs_{3}nr\textnormal{-}max2_{2+1}}$) is an extended semidirect sum
algebra of the same type as the original Maxwell algebra in three space-time
dimensions. Indeed, the commutation relations of the
$\mathfrak{hs_{3}nr\textnormal{-}max1_{2+1}}$
($\mathfrak{hs_{3}nr\textnormal{-}max2_{2+1}}$) algebra can be written in a
compact form as [81]

$\left[(X,X^{\ast},X^{\prime}),(Y,Y^{\ast},Y^{\prime})\right]=\left([X,Y],[X,Y^{\ast}]-[Y,X^{\ast}],[X,Y^{\prime}]-[Y,X^{\prime}]+[X^{\ast},Y^{\ast}]\right),$
(4.31)

with $X$ an element of $\mathfrak{hs_{3}nw1}$ ($\mathfrak{hs_{3}nw2}$), and
$X^{\ast}$ and $X^{\prime}$ belonging to $\mathfrak{hs_{3}nw1}^{({\rm ab})}$
($\mathfrak{hs_{3}nw2}^{({\rm ab})}$). Therefore, the spin-$3$ extensions of
the NR Maxwell algebra in $2+1$ dimensions can be expressed as

$\displaystyle\mathfrak{hs_{3}nr\textnormal{-}max1_{2+1}}$
$\displaystyle=\mathfrak{hs_{3}nw1}\oright_{\rm ad}\mathfrak{hs_{3}nw1}^{({\rm
ab})}_{\rm ext},$ (4.32)
$\displaystyle\mathfrak{hs_{3}nr\textnormal{-}max2_{2+1}}$
$\displaystyle=\mathfrak{hs_{3}nw2}\oright_{\rm ad}\mathfrak{hs_{3}nw2}^{({\rm
ab})}_{\rm ext},$

where $\mathfrak{hs_{3}nw1}^{({\rm ab})}_{\rm ext}$
($\mathfrak{hs_{3}nw2}^{({\rm ab})}_{\rm ext}$) denotes the extended group
$\mathfrak{hs_{3}nw1}\otimes\mathfrak{hs_{3}nw1}$
($\mathfrak{hs_{3}nw2}\otimes\mathfrak{hs_{3}nw2}$) endowed with the product

$\left(X^{\ast},X^{\prime}\right)\hat{+}\left(Y^{\ast},Y^{\prime}\right)=\left(X^{\ast}+Y^{\ast}\,,\;X^{\prime}+Y^{\prime}+\frac{1}{2}\,[X^{\ast},Y^{\ast}]\right).$
(4.33)

The relativistic analog of (4.32) is the extended semidirect sum that defines
the spin-$3$ extension of the Maxwell algebra121212A higher-spin extension of
the Maxwell algebra in higher dimensions has been considered in [82]. in $2+1$
dimensions [81]

$\mathfrak{hs_{3}max_{2+1}}\simeq\mathfrak{sl}(3,\mathbb{R})\oright_{\rm
ad}\mathfrak{sl}(3,\mathbb{R})^{({\rm ab})}_{\rm ext}.$ (4.34)

NR limit: As shown in the general case treated in Section 3.4, the higher-spin
NR symmetries (4.23) can be obtained as NR limits applied to relativistic
symmetries. The algebra (4.34) was constructed in [49] as the expansion

$\mathfrak{hs_{3}max_{2+1}}\simeq
S^{(2)}_{E}\times\mathfrak{sl}(3,\mathbb{R}),$ (4.35)

with the expanded generators (4.15) plus

$\boldsymbol{Z}_{A}^{\rm
rel}=\boldsymbol{L}^{(2)}_{A},\quad\boldsymbol{\mathcal{Z}}_{AB}^{\rm
rel}=\boldsymbol{\mathcal{L}}^{(2)}_{AB},$ (4.36)

and commutation relations (3.47) for $S=S^{(2)}_{E}$. Now we look at the
expanded algebras (3.48) and (3.51), necessary to define the NR limits (3.50)
and (3.53). When using the semigroup product law of $S^{(2)}_{E}$ given in
(3.54), these can be shown to be isomorphic to the Euclidean Maxwell algebra
in three dimensions $\mathfrak{e\textnormal{-}max_{3}}$ (spanned by
$\\{\boldsymbol{X}^{(0)},\boldsymbol{X}^{(1)},\boldsymbol{X}_{a}^{(0)},\boldsymbol{X}_{a}^{(1)},\boldsymbol{X}_{a}^{(2)}\\}$
in Eq. (3.48)) and of the direct sum of the Maxwell algebra in $2+1$
dimensions and three Abelian generators (spanned by
$\\{\boldsymbol{X}^{(0)},\boldsymbol{X}^{(1)},\boldsymbol{X}_{ab}^{(0)},\boldsymbol{X}_{ab}^{(1)},\boldsymbol{X}_{ab}^{(2)}\\}$
in Eq. (3.51)),

$\displaystyle
S^{(2)}_{E}\times\mathfrak{so}(3)\simeq\mathfrak{e\textnormal{-}max_{3}},$
(4.37) $\displaystyle
S^{(2)}_{E}\times\Big{(}\mathfrak{sl}(2,\mathbb{R})\oplus\mathfrak{so}(2)\Big{)}\simeq\mathfrak{max_{2+1}}\oplus\mathfrak{so}(2)\oplus\mathfrak{so}(2)\oplus\mathfrak{so}(2).$

Using the relation (3.50) to relate relativistic and NR generators, we find
for the NR Maxwell generators the relations (4.17) plus

$\displaystyle\boldsymbol{Y}$ $\displaystyle=\boldsymbol{Z}_{0}^{\rm
rel}+\boldsymbol{X}^{(2)},$ $\displaystyle\boldsymbol{Z}_{a}$
$\displaystyle=\frac{1}{c}\boldsymbol{Z}_{a}^{\rm rel},$
$\displaystyle\boldsymbol{T}$
$\displaystyle=\frac{1}{2c^{2}}\left(\boldsymbol{Z}_{0}^{\rm
rel}-\boldsymbol{X}^{(2)}\right),$ (4.38)

whereas the higher-spin generators of
$\mathfrak{hs_{3}nr\textnormal{-}max1_{2+1}}$ satisfy (4.18) and

$\displaystyle\boldsymbol{\mathcal{Y}}_{a}$
$\displaystyle=\boldsymbol{\mathcal{Z}}_{a0}^{\rm
rel}+\boldsymbol{X}_{a}^{(2)},$ $\displaystyle\boldsymbol{\mathcal{Z}}_{ab}$
$\displaystyle=\frac{1}{c}\boldsymbol{\mathcal{Z}}_{ab}^{\rm rel},$
$\displaystyle\boldsymbol{\mathcal{T}}_{a}$
$\displaystyle=\frac{1}{2c^{2}}\left(\boldsymbol{\mathcal{Z}}_{a0}^{\rm
rel}-\boldsymbol{X}_{a}^{(2)}\right).$ (4.39)

The second possible contraction (3.53), yields once again (4.17) and (4.38),
but the higher-spin extension is given by (4.19) together with

$\displaystyle\boldsymbol{\mathcal{Y}}_{ab}$
$\displaystyle=\boldsymbol{\mathcal{Z}}_{ab}^{\rm
rel}+\boldsymbol{X}_{ab}^{(2)},$ $\displaystyle\boldsymbol{\mathcal{Z}}_{a}$
$\displaystyle=\frac{1}{c}\boldsymbol{\mathcal{Z}}_{a0}^{\rm rel},$
$\displaystyle\boldsymbol{\mathcal{T}}_{ab}$
$\displaystyle=\frac{1}{2c^{2}}\left(\boldsymbol{\mathcal{Z}}_{ab}^{\rm
rel}-\boldsymbol{X}_{ab}^{(2)}\right),$ (4.40)

and leads to $\mathfrak{hs_{3}nr\textnormal{-}max2_{2+1}}$ in the limit
$c\rightarrow\infty$. Therefore, the spin-$3$ algebras (4.23) follow from the
NR limits

$\displaystyle\mathfrak{hs_{3}max_{2+1}}\oplus\mathfrak{e\textnormal{-}max_{3}}\;$
$\displaystyle\underset{c\rightarrow\infty}{\longrightarrow}$
$\displaystyle\qquad\mathfrak{hs_{3}nr\textnormal{-}max1_{2+1}}$ (4.41)
$\displaystyle\mathfrak{hs_{3}max_{2+1}}\oplus\mathfrak{max_{2+1}}\oplus\mathfrak{so}(2)\oplus\mathfrak{so}(2)\oplus\mathfrak{so}(2)\;$
$\displaystyle\underset{c\rightarrow\infty}{\longrightarrow}$
$\displaystyle\qquad\mathfrak{hs_{3}nr\textnormal{-}max2_{2+1}}.$

The relations (4.17) and (4.38) show that, in absence of spin-$3$ generators,
the standard contraction of the Maxwell algebra in three space-time dimensions
and three Abelian generators that yields the NR Maxwell algebra [62] is
recovered.

###  4.3 Spin-$3$ Newton-Hooke algebras

One can include a cosmological constant in the Bargman algebra by considering
a NR contraction of the (anti-)deSitter algebra [66]. This leads to the
Newton-Hooke algebra, which in three space-time dimensions admits two central
extensions [64] and is given by the extended Bargmann commutation relations
(4.1) plus

$\left[\boldsymbol{H},\boldsymbol{P}_{a}\right]=\epsilon_{ab}\boldsymbol{G}_{b},\quad\left[\boldsymbol{P}_{a},\boldsymbol{P}_{b}\right]=-\epsilon_{ab}\;\boldsymbol{S}.$
(4.42)

The Extended Newton-Hooke algebra is isomorphic to the expanded Nappi-Witten
algebra $S_{M}^{(1)}\times\mathfrak{nw}$. Thus, we can construct the
corresponding spin-$3$ extensions by evaluating expanded algebras

$\mathfrak{hs_{3}nh1_{2+1}}=S^{(1)}_{M}\times\mathfrak{hs_{3}nw1},\qquad\mathfrak{hs_{3}nh2_{2+1}}=S^{(1)}_{M}\times\mathfrak{hs_{3}nw2}.$
(4.43)

We find the explicit form of their commutation relations by using the product
law of the semigroup $S^{(1)}_{M}\backsimeq\mathbb{Z}_{2}$ given in (3.55) for
$n=1$.

  * •

We define the first higher-spin extension of the Extended Newton-Hooke algebra
in $2+1$ dimensions algebra, $\mathfrak{hs_{3}nh1_{2+1}}$, by defining the
higher-spin generators (4.3), which yields the commutation relations (4.1),
(4.4), (4.5), (4.42) and

$\displaystyle\left[\boldsymbol{H},\boldsymbol{\mathcal{H}}_{a}\right]=\epsilon_{ab}\boldsymbol{\mathcal{J}}_{b},$
$\displaystyle\left[\boldsymbol{H},\boldsymbol{\mathcal{P}}_{ab}\right]=-\epsilon_{c(a}\boldsymbol{\mathcal{G}}_{b)c},$
$\displaystyle\left[\boldsymbol{H},\boldsymbol{\mathcal{M}}_{a}\right]=\left[\boldsymbol{M},\boldsymbol{\mathcal{H}}_{a}\right]=\epsilon_{ab}\boldsymbol{\mathcal{S}}_{b},$
(4.44)
$\displaystyle\left[\boldsymbol{P}_{a},\boldsymbol{\mathcal{H}}_{b}\right]=-\epsilon_{a(c}\boldsymbol{\mathcal{G}}_{b)c},$
$\displaystyle\left[\boldsymbol{P}_{a},\boldsymbol{\mathcal{P}}_{bc}\right]=-\epsilon_{a(b}\boldsymbol{\mathcal{S}}_{c)},$
$\displaystyle\left[\boldsymbol{\mathcal{H}}_{a},\boldsymbol{\mathcal{H}}_{b}\right]=\epsilon_{ab}\boldsymbol{J},$
$\displaystyle\left[\boldsymbol{\mathcal{H}}_{a},\boldsymbol{\mathcal{M}}_{b}\right]=\epsilon_{ab}\boldsymbol{S},$
$\displaystyle\left[\boldsymbol{\mathcal{H}}_{a},\boldsymbol{\mathcal{P}}_{bc}\right]=\delta_{a(b}\epsilon_{c)d}\boldsymbol{G}_{d},$
$\displaystyle\left[\boldsymbol{\mathcal{P}}_{ab},\boldsymbol{\mathcal{P}}_{cd}\right]=\delta_{(a(c}\epsilon_{d)b)}\boldsymbol{S}.$

  * •

The commutation relations of the second higher-spin extension
$\mathfrak{hs_{3}nh2_{2+1}}$ are obtained by considering the expanded
generators (4.6). The result is the relations (4.1), (4.7), (4.8), (4.42) plus

$\displaystyle[\boldsymbol{H},\boldsymbol{\mathcal{P}}_{a}]=\epsilon_{ab}\boldsymbol{\mathcal{G}}_{b},$
$\displaystyle[\boldsymbol{H},\boldsymbol{\mathcal{H}}_{ab}]=-\epsilon_{c(a}\boldsymbol{\mathcal{J}}_{b)c},$
$\displaystyle[\boldsymbol{H},\boldsymbol{\mathcal{M}}_{ab}]=[\boldsymbol{M},\boldsymbol{\mathcal{H}}_{ab}]=-\epsilon_{c(a}\boldsymbol{\mathcal{S}}_{b)c},$
(4.45)
$\displaystyle[\boldsymbol{P}_{a},\boldsymbol{\mathcal{P}}_{b}]=-\epsilon_{a(c}\boldsymbol{\mathcal{S}}_{b)c},$
$\displaystyle[\boldsymbol{P}_{a},\boldsymbol{\mathcal{H}}_{bc}]=-\epsilon_{a(b}\boldsymbol{\mathcal{G}}_{c)},$
$\displaystyle[\boldsymbol{\mathcal{H}}_{ab},\boldsymbol{\mathcal{H}}_{cd}]=\delta_{(a(c}\epsilon_{d)b)}\boldsymbol{J},$
$\displaystyle[\boldsymbol{\mathcal{P}}_{a},\boldsymbol{\mathcal{P}}_{b}]=\epsilon_{ab}\boldsymbol{S},$
$\displaystyle[\boldsymbol{\mathcal{P}}_{a},\boldsymbol{\mathcal{H}}_{bc}]=\delta_{a(b}\epsilon_{c)d}\boldsymbol{G}_{d},$
$\displaystyle[\boldsymbol{\mathcal{H}}_{ab},\boldsymbol{\mathcal{M}}_{cd}]=\delta_{(a(c}\epsilon_{d)b)}\boldsymbol{S}.$

Direct sum structure: The Extended Newton-Hooke algebra defined by (4.1) and
(4.42) is isomorphic to a direct product of two copies of the Nappi-Witten
algebra [83, 84, 85], which is a NR analog of the isomorphism
$\mathfrak{so}(2,2)\backsimeq\mathfrak{sl}(2,\mathbb{R})\oplus\mathfrak{sl}(2,\mathbb{R})$.
Similarly, in the spin-$3$ case, the following isomorphisms hold

$\displaystyle\mathfrak{hs_{3}nh1_{2+1}}\backsimeq\mathfrak{hs_{3}nw1}\oplus\mathfrak{hs_{3}nw1},$
(4.46)
$\displaystyle\mathfrak{hs_{3}nh2_{2+1}}\backsimeq\mathfrak{hs_{3}nw2}\oplus\mathfrak{hs_{3}nw2}.$

Indeed, the combinations

$\boldsymbol{k}^{\pm}=\frac{1}{2}\left(\boldsymbol{J}\pm\boldsymbol{H}\right),\quad\boldsymbol{k}^{\pm}_{a}=\frac{1}{2}\left(\boldsymbol{G}_{a}\pm\boldsymbol{P}_{a}\right),\quad\boldsymbol{\sigma}^{\pm}=\frac{1}{2}\left(\boldsymbol{S}\pm\boldsymbol{M}\right),$
(4.47)

satisfy the $\mathfrak{nw}$ commutation relations (2.4). The isomorphism
(LABEL:hsnhiso) can be easily proven by using change of basis (4.47) and the
higher-spin generalization

$\boldsymbol{\mathcal{K}}^{\pm}_{a}=\frac{1}{2}\left(\boldsymbol{\mathcal{J}}_{a}\pm\boldsymbol{\mathcal{H}}_{a}\right),\quad\boldsymbol{\mathcal{K}}^{\pm}_{ab}=\frac{1}{2}\left(\boldsymbol{\mathcal{G}}_{ab}\pm\boldsymbol{\mathcal{P}}_{ab}\right),\quad\boldsymbol{\Sigma}^{\pm}_{a}=\frac{1}{2}\left(\boldsymbol{\mathcal{S}}_{a}\pm\boldsymbol{\mathcal{M}}_{a}\right),$
(4.48)

in the $\mathfrak{hs_{3}nh1_{2+1}}$ algebra. Instead, supplementing (2.4) with

$\tilde{\boldsymbol{\mathcal{K}}}^{\pm}_{ab}=\frac{1}{2}\left(\boldsymbol{\mathcal{J}}_{ab}\pm\boldsymbol{\mathcal{H}}_{ab}\right),\quad\tilde{\boldsymbol{\mathcal{K}}}^{\pm}_{a}=\frac{1}{2}\left(\boldsymbol{\mathcal{G}}_{a}\pm\boldsymbol{\mathcal{P}}_{a}\right),\quad\tilde{\boldsymbol{\Sigma}}^{\pm}_{ab}=\frac{1}{2}\left(\boldsymbol{\mathcal{S}}_{ab}\pm\boldsymbol{\mathcal{M}}_{ab}\right),$
(4.49)

does the job for the $\mathfrak{hs_{3}nh2_{2+1}}$ algebra. It is important to
remark that all the previous results consider a negative cosmological constant
(the Newton-Hooke symmetry in our analysis is strictly speaking Newton-
Hooke-). The results can be generalized to the case of a positive cosmological
constant by redefining the generators in the form

$\displaystyle\boldsymbol{H}\rightarrow i\boldsymbol{H},$
$\displaystyle\boldsymbol{P}_{a}\rightarrow i\boldsymbol{P}_{a},$
$\displaystyle\boldsymbol{M}\rightarrow i\boldsymbol{M},$ (4.50)
$\displaystyle\boldsymbol{\mathcal{H}}_{a}\rightarrow
i\boldsymbol{\mathcal{H}}_{a},$
$\displaystyle\boldsymbol{\mathcal{P}}_{ab}\rightarrow
i\boldsymbol{\mathcal{P}}_{ab},$
$\displaystyle\boldsymbol{\mathcal{M}}_{a}\rightarrow
i\boldsymbol{\mathcal{M}}_{a},$
$\displaystyle\boldsymbol{\mathcal{H}}_{ab}\rightarrow
i\boldsymbol{\mathcal{H}}_{ab},$
$\displaystyle\boldsymbol{\mathcal{P}}_{a}\rightarrow
i\boldsymbol{\mathcal{P}}_{a},$
$\displaystyle\boldsymbol{\mathcal{M}}_{ab}\rightarrow
i\boldsymbol{\mathcal{M}}_{ab}.$

NR and flat limits: As in the Bargmann case, we can obtain (4.43) by taking a
NR limit of suitable extensions of the spin-$3$ AdS algebra [10]

$\mathfrak{hs_{3}ads_{2+1}}\simeq\mathfrak{sl}(3,\mathbb{R})\oplus\mathfrak{sl}(3,\mathbb{R}).$
(4.51)

Indeed, it is clear that one can obtain (LABEL:hsnhiso) from (4.51) using two
copies of the contractions developed in Section 2 that take
$\mathfrak{sl}(3,\mathbb{R})$ into the NR algebras $\mathfrak{hs_{3}nw1}$ and
$\mathfrak{hs_{3}nw2}$. Moreover, (4.51) is equivalent to the expansion

$\mathfrak{hs_{3}ads_{2+1}}\simeq
S^{(1)}_{M}\times\mathfrak{sl}(3,\mathbb{R}),$ (4.52)

introduced in [49], where the commutation relations can be read off from
(3.47) using the product law of the semigroup $S^{(1)}_{M}$ and expanded
generators (4.15) defined in the Poincaré case. Thus, one can define the
expanded algebras (3.49) and (3.52) for $S=S^{(1)}_{M}$ and take the NR limit
to find (LABEL:hsnhiso). This requires to extend (4.52) with the algebras
(3.48) and (3.51), which we can identify using the following isomorphisms (see
(3.66), (3.69) and (3.71)):

$\displaystyle
S^{(1)}_{M}\times\mathfrak{so}(3)\simeq\mathfrak{so}(3)\oplus\mathfrak{so}(3)\simeq\mathfrak{so}(4),$
(4.53) $\displaystyle
S^{(1)}_{M}\times\Big{(}\mathfrak{sl}(2,\mathbb{R})\oplus\mathfrak{so}(2)\Big{)}\simeq\mathfrak{so}(2,2)\oplus\mathfrak{so}(2)\oplus\mathfrak{so}(2).$

As shown in Section 3.5, one can define a vanishing cosmological constant
limit that takes (4.43) into (4.9) by means of the rescaling (3.58), i.e. to
redefine (4.2), (4.3) and (4.6) as

$\displaystyle\boldsymbol{H}\rightarrow\ell\,\boldsymbol{H},$
$\displaystyle\boldsymbol{P}_{a}\rightarrow\ell\,\boldsymbol{P}_{a},$
$\displaystyle\boldsymbol{M}\rightarrow\ell\,\boldsymbol{M},$ (4.54)
$\displaystyle\boldsymbol{\mathcal{H}}_{a}\rightarrow\ell\,\boldsymbol{\mathcal{H}}_{a},$
$\displaystyle\boldsymbol{\mathcal{P}}_{ab}\rightarrow\ell\,\boldsymbol{\mathcal{P}}_{ab},$
$\displaystyle\boldsymbol{\mathcal{M}}_{a}\rightarrow\ell\,\boldsymbol{\mathcal{M}}_{a},$
$\displaystyle\boldsymbol{\mathcal{H}}_{ab}\rightarrow\ell\,\boldsymbol{\mathcal{H}}_{ab},$
$\displaystyle\boldsymbol{\mathcal{P}}_{a}\rightarrow\ell\,\boldsymbol{\mathcal{P}}_{a},$
$\displaystyle\boldsymbol{\mathcal{M}}_{ab}\rightarrow\ell\,\boldsymbol{\mathcal{M}}_{ab}.$

This has the effect of adding a factor $1/\ell^{2}$ on the right-hand side of
the commutation relations given in Eqs. (4.42), (4.44) and (4.45), so that
they dissapear in the limit $\ell\rightarrow\infty$. Similarly, the
contraction defined by (3.63), (3.65) and (3.70) in this case takes the form

$\displaystyle\boldsymbol{P}_{A}^{\rm
rel}\rightarrow\ell\,\boldsymbol{P}_{A}^{\rm rel},$
$\displaystyle\boldsymbol{\mathcal{P}}_{AB}^{\rm
rel}\rightarrow\ell\,\boldsymbol{\mathcal{P}}_{AB}^{\rm rel}$ (4.55)
$\displaystyle\boldsymbol{X}^{(1)}\rightarrow\ell\,\boldsymbol{X}^{(1)},$
$\displaystyle\boldsymbol{X}_{a}^{(1)}\rightarrow\ell\,\boldsymbol{X}_{a}^{(1)},\quad\boldsymbol{X}_{ab}^{(1)}\rightarrow\ell\,\boldsymbol{X}_{ab}^{(1)}$

and defines an analog contraction at the relativistic level, taking (4.52) and
(4.16) into (4.14) and (4.53), respectively. Thus, the NR contraction leading
to $\mathfrak{hs_{3}nh1_{2+1}}$ together with the corresponding flat limit is

$\displaystyle\mathfrak{hs_{3}ads_{2+1}}\oplus\mathfrak{so}(4)\;$
$\displaystyle\underset{c\rightarrow\infty}{\longrightarrow}$
$\displaystyle\qquad\mathfrak{hs_{3}nh1_{2+1}}$ (4.56) $\displaystyle\hskip
28.45274pt{\scriptstyle\ell\rightarrow\infty}\;\Big{\downarrow}$
$\displaystyle\hskip
25.6073pt{\scriptstyle\ell\rightarrow\infty}\;\Big{\downarrow}$
$\displaystyle\mathfrak{hs_{3}poin_{2+1}}\oplus\mathfrak{iso}(3)\;$
$\displaystyle\underset{c\rightarrow\infty}{\longrightarrow}$
$\displaystyle\qquad\mathfrak{hs_{3}barg1_{2+1}},$

whereas the corresponding diagram for $\mathfrak{hs_{3}nh2_{2+1}}$ has the
form

$\displaystyle\mathfrak{hs_{3}ads_{2+1}}\oplus\mathfrak{so}(2,2)\oplus\mathfrak{so}(2)\oplus\mathfrak{so}(2)\;$
$\displaystyle\underset{c\rightarrow\infty}{\longrightarrow}$
$\displaystyle\qquad\mathfrak{hs_{3}nh2_{2+1}}$ (4.57) $\displaystyle\hskip
65.44142pt{\scriptstyle\ell\rightarrow\infty}\;\Big{\downarrow}$
$\displaystyle\hskip
25.6073pt{\scriptstyle\ell\rightarrow\infty}\;\Big{\downarrow}$
$\displaystyle\mathfrak{hs_{3}poin_{2+1}}\oplus\mathfrak{isl}(2,\mathbb{R})\oplus\mathfrak{so}(2)\oplus\mathfrak{so}(2)\;$
$\displaystyle\underset{c\rightarrow\infty}{\longrightarrow}$
$\displaystyle\qquad\mathfrak{hs_{3}barg2_{2+1}}.$

In both cases, the explicit relations of the $c\rightarrow\infty$ contractions
are the same as the ones given in the Poincaré case (4.17)-(4.19).

###  4.4 Spin-$3$ non-relativistic AdS-Lorentz algebras

As the last example, we turn our attention to the AdS-Lorentz symmetry. Based
on the fact that NR AdS-Lorentz algebra is the expanded Nappi-Witten algebra
(3.5) for $S=S^{(2)}_{M}$, we define the corresponding spin-$3$ extensions as

$\mathfrak{hs_{3}nr\textnormal{-}adslor1_{2+1}}=S^{(2)}_{M}\times\mathfrak{hs_{3}nw1},\qquad\mathfrak{hs_{3}nr\textnormal{-}adslor2_{2+1}}=S^{(2)}_{M}\times\mathfrak{hs_{3}nw2}.$
(4.58)

The relativistic AdS-Lorentz algebra is a generalization of the Maxwell