algebra that provides a semi-simple extension of the Poincaré algebra [86,
87], and whose NR limit has been considered in the context of higher-
dimensional (extended) CS theories [48, 74]. In $2+1$dimensions it admits
three central extensions and its commutation relations are the ones given in
(4.1) and (4.21), plus

$\displaystyle\left[\boldsymbol{Y},\boldsymbol{Z}_{a}\right]=\epsilon_{a}^{\;\;b}\boldsymbol{Z}_{b}\,,$
$\displaystyle\left[\boldsymbol{P}_{a},\boldsymbol{Z}_{b}\right]=-\epsilon_{ab}\boldsymbol{M},$
(4.59)
$\displaystyle\left[\boldsymbol{H},\boldsymbol{Z}_{a}\right]=\left[\boldsymbol{Y},\boldsymbol{P}_{a}\right]=\epsilon_{a}^{\;\;b}\boldsymbol{P}_{b},$
$\displaystyle\left[\boldsymbol{Z}_{a},\boldsymbol{Z}_{b}\right]=-\epsilon_{ab}\boldsymbol{T}.$

As said before, this algebra follows from evaluating the expanded Nappi-Witten
algebra (3.5) for the semigroup $S^{(n)}_{M}$ defined in (3.55), and setting
$n=2$. The same procedure allows one to find the commutation relations of the
higher-spin extensions (4.58).

  * •

The first higher-spin generalization of the NR AdS-Lorentz algebra
$\mathfrak{hs_{3}nr\textnormal{-}adslor1_{2+1}}$ includes the higher-spin
generators (4.3) and (4.24). In this case the commutation relations are the
ones of the $\mathfrak{hs_{3}nr\textnormal{-}max1_{2+1}}$ algebra, (4.1),
(4.4), (4.5), (4.21), (4.25), together with (4.59) plus

$\displaystyle\left[\boldsymbol{Y},\boldsymbol{\mathcal{Y}}_{a}\right]=\epsilon_{ab}\boldsymbol{\mathcal{Y}}_{b},$
$\displaystyle\left[\boldsymbol{Y},\boldsymbol{\mathcal{Z}}_{ab}\right]=-\epsilon_{c(a}\boldsymbol{\mathcal{Z}}_{b)c},$
$\displaystyle\left[\boldsymbol{Y},\boldsymbol{\mathcal{T}}_{a}\right]=\left[\boldsymbol{T},\boldsymbol{\mathcal{Y}}_{a}\right]=\epsilon_{ab}\boldsymbol{\mathcal{T}}_{b},$
(4.60)
$\displaystyle\left[\boldsymbol{Z}_{a},\boldsymbol{\mathcal{Y}}_{b}\right]=-\epsilon_{a(c}\boldsymbol{\mathcal{Z}}_{b)c},$
$\displaystyle\left[\boldsymbol{Z}_{a},\boldsymbol{\mathcal{Z}}_{bc}\right]=-\epsilon_{a(b}\boldsymbol{\mathcal{T}}_{c)},$
$\displaystyle\left[\boldsymbol{\mathcal{Y}}_{a},\boldsymbol{\mathcal{Y}}_{b}\right]=\epsilon_{ab}\boldsymbol{Y},$
$\displaystyle\left[\boldsymbol{\mathcal{Y}}_{a},\boldsymbol{\mathcal{T}}_{b}\right]=\epsilon_{ab}\boldsymbol{T},$
$\displaystyle\left[\boldsymbol{\mathcal{Y}}_{a},\boldsymbol{\mathcal{Z}}_{bc}\right]=\delta_{a(b}\epsilon_{c)d}\boldsymbol{Z}_{d},$
$\displaystyle\left[\boldsymbol{\mathcal{Z}}_{ab},\boldsymbol{\mathcal{Z}}_{cd}\right]=\delta_{(a(c}\epsilon_{d)b)}\boldsymbol{T},$

and

$\displaystyle\left[\boldsymbol{Y},\boldsymbol{\mathcal{H}}_{a}\right]=\left[\boldsymbol{H},\boldsymbol{\mathcal{Y}}_{a}\right]=\epsilon_{ab}\boldsymbol{\mathcal{H}}_{b},$
$\displaystyle\left[\boldsymbol{Z}_{a},\boldsymbol{\mathcal{H}}_{b}\right]=\left[\boldsymbol{P}_{a},\boldsymbol{\mathcal{Y}}_{b}\right]=-\epsilon_{a(c}\boldsymbol{\mathcal{P}}_{b)c},$
(4.61)
$\displaystyle\left[\boldsymbol{Y},\boldsymbol{\mathcal{P}}_{ab}\right]=\left[\boldsymbol{H},\boldsymbol{\mathcal{Z}}_{ab}\right]=-\epsilon_{c(a}\boldsymbol{\mathcal{P}}_{b)c},$
$\displaystyle\left[\boldsymbol{\mathcal{Y}}_{a},\boldsymbol{\mathcal{P}}_{bc}\right]=\left[\boldsymbol{\mathcal{H}}_{a},\boldsymbol{\mathcal{Z}}_{bc}\right]=\delta_{a(b}\epsilon_{c)d}\boldsymbol{P}_{d},$
$\displaystyle\left[\boldsymbol{\mathcal{Y}}_{a},\boldsymbol{\mathcal{H}}_{b}\right]=\epsilon_{ab}\boldsymbol{H},$
$\displaystyle\left[\boldsymbol{Z}_{a},\boldsymbol{\mathcal{P}}_{bc}\right]=\left[\boldsymbol{P}_{a},\boldsymbol{\mathcal{Z}}_{bc}\right]=-\epsilon_{a(b}\boldsymbol{\mathcal{M}}_{c)},$
$\displaystyle\left[\boldsymbol{\mathcal{Y}}_{a},\boldsymbol{\mathcal{M}}_{b}\right]=\left[\boldsymbol{\mathcal{H}}_{a},\boldsymbol{\mathcal{T}}_{b}\right]=\epsilon_{ab}\boldsymbol{M},$
$\displaystyle\left[\boldsymbol{Y},\boldsymbol{\mathcal{M}}_{a}\right]=\left[\boldsymbol{T},\boldsymbol{\mathcal{H}}_{a}\right]=\left[\boldsymbol{H},\boldsymbol{\mathcal{T}}_{a}\right]=\left[\boldsymbol{M},\boldsymbol{\mathcal{Y}}_{a}\right]=\epsilon_{ab}\boldsymbol{\mathcal{M}}_{b}.$
$\displaystyle\left[\boldsymbol{\mathcal{Z}}_{ab},\boldsymbol{\mathcal{P}}_{cd}\right]=\delta_{(a(c}\epsilon_{d)b)}\boldsymbol{M},$

  * •

The second higher-spin extension of the NR AdS-Lorentz algebra
$\mathfrak{hs_{3}nr\textnormal{-}adslor2_{2+1}}$ includes the higher-spin
generators (4.6) and (4.26) and has the commutation relations of the
$\mathfrak{hs_{3}nr\textnormal{-}max2_{2+1}}$ algebra, (4.1), (4.7), (4.8),
(4.21), (4.27), together with (4.59) plus

$\displaystyle[\boldsymbol{Y},\boldsymbol{\mathcal{Z}}_{a}]=\epsilon_{ab}\boldsymbol{\mathcal{Z}}_{b},$
$\displaystyle[\boldsymbol{Y},\boldsymbol{\mathcal{Y}}_{ab}]=-\epsilon_{c(a}\boldsymbol{\mathcal{Y}}_{b)c},$
$\displaystyle[\boldsymbol{Y},\boldsymbol{\mathcal{T}}_{ab}]=[\boldsymbol{T},\boldsymbol{\mathcal{Y}}_{ab}]=-\epsilon_{c(a}\boldsymbol{\mathcal{T}}_{b)c},$
(4.62)
$\displaystyle[\boldsymbol{Z}_{a},\boldsymbol{\mathcal{Z}}_{b}]=-\epsilon_{a(c}\boldsymbol{\mathcal{T}}_{b)c},$
$\displaystyle[\boldsymbol{Z}_{a},\boldsymbol{\mathcal{Y}}_{bc}]=-\epsilon_{a(b}\boldsymbol{\mathcal{Z}}_{c)},$
$\displaystyle[\boldsymbol{\mathcal{Y}}_{ab},\boldsymbol{\mathcal{Y}}_{cd}]=\delta_{(a(c}\epsilon_{d)b)}\boldsymbol{Y}.$
$\displaystyle[\boldsymbol{\mathcal{Z}}_{a},\boldsymbol{\mathcal{Z}}_{b}]=\epsilon_{ab}\boldsymbol{T},$
$\displaystyle[\boldsymbol{\mathcal{Z}}_{a},\boldsymbol{\mathcal{Y}}_{bc}]=\delta_{a(b}\epsilon_{c)d}\boldsymbol{Z}_{d},$
$\displaystyle[\boldsymbol{\mathcal{Y}}_{ab},\boldsymbol{\mathcal{T}}_{cd}]=\delta_{(a(c}\epsilon_{d)b)}\boldsymbol{T},$

and

$\displaystyle[\boldsymbol{Y},\boldsymbol{\mathcal{P}}_{a}]=[\boldsymbol{H},\boldsymbol{\mathcal{Z}}_{a}]=\epsilon_{ab}\boldsymbol{\mathcal{P}}_{b},$
$\displaystyle[\boldsymbol{Z}_{a},\boldsymbol{\mathcal{H}}_{bc}]=[\boldsymbol{P}_{a},\boldsymbol{\mathcal{Y}}_{bc}]=-\epsilon_{a(b}\boldsymbol{\mathcal{P}}_{c)},$
(4.63)
$\displaystyle[\boldsymbol{Y},\boldsymbol{\mathcal{H}}_{ab}]=[\boldsymbol{H},\boldsymbol{\mathcal{Y}}_{ab}]=-\epsilon_{c(a}\boldsymbol{\mathcal{H}}_{b)c},$
$\displaystyle[\boldsymbol{\mathcal{Z}}_{a},\boldsymbol{\mathcal{H}}_{bc}]=[\boldsymbol{\mathcal{P}}_{a},\boldsymbol{\mathcal{Y}}_{bc}]=\delta_{a(b}\epsilon_{c)d}\boldsymbol{P}_{d},$
$\displaystyle[\boldsymbol{\mathcal{Y}}_{ab},\boldsymbol{\mathcal{H}}_{cd}]=\delta_{(a(c}\epsilon_{d)b)}\boldsymbol{H},$
$\displaystyle[\boldsymbol{\mathcal{Y}}_{ab},\boldsymbol{\mathcal{M}}_{cd}]=[\boldsymbol{\mathcal{H}}_{ab},\boldsymbol{\mathcal{T}}_{cd}]=\delta_{(a(c}\epsilon_{d)b)}\boldsymbol{M},$
$\displaystyle[\boldsymbol{\mathcal{Z}}_{a},\boldsymbol{\mathcal{P}}_{b}]=\epsilon_{ab}\boldsymbol{M},$
$\displaystyle[\boldsymbol{Y},\boldsymbol{\mathcal{M}}_{ab}]=[\boldsymbol{H},\boldsymbol{\mathcal{T}}_{ab}]=[\boldsymbol{T},\boldsymbol{\mathcal{H}}_{ab}]$
$\displaystyle[\boldsymbol{Z}_{a},\boldsymbol{\mathcal{P}}_{b}]=[\boldsymbol{P}_{a},\boldsymbol{\mathcal{Z}}_{b}]=-\epsilon_{a(c}\boldsymbol{\mathcal{M}}_{b)c},$
$\displaystyle\hskip
48.36958pt=[\boldsymbol{M},\boldsymbol{\mathcal{Y}}_{ab}]=-\epsilon_{c(a}\boldsymbol{\mathcal{M}}_{b)c}.$

Direct sum structure: The AdS-Lorentz symmetry in $d+1$ dimensions was so
dubbed because it is isomorphic to the direct
$\mathfrak{so}(d,2)\oplus\mathfrak{so}(d,1)$ [87]. Along the same lines, the
NR AdS-Lorentz algebra is isomorphic to the direct sum of the Extended Newton-
Hooke algebra and the Nappi-Witten algebra, which in turn is equivalent to
three copies of the $\mathfrak{nw}$ algebra. The spin-$3$ generalization of
this statement reads

$\displaystyle\mathfrak{hs_{3}nr\textnormal{-}adslor1_{2+1}}\backsimeq\mathfrak{hs_{3}nh1_{2+1}}\oplus\mathfrak{hs_{3}nw1}\backsimeq\mathfrak{hs_{3}nw1}\oplus\mathfrak{hs_{3}nw1}\oplus\mathfrak{hs_{3}nw1},$
(4.64)
$\displaystyle\mathfrak{hs_{3}nr\textnormal{-}adslor2_{2+1}}\backsimeq\mathfrak{hs_{3}nh2_{2+1}}\oplus\mathfrak{hs_{3}nw2}\backsimeq\mathfrak{hs_{3}nw2}\oplus\mathfrak{hs_{3}nw2}\oplus\mathfrak{hs_{3}nw2},$

where, on the right-hand side of this relation, we have used (LABEL:hsnhiso)
(This is the NR analog of the fact that, in $2+1$ dimensions, the AdS-Lorentz
algebra is isomorphic to the direct sum of three copies of
$\mathfrak{sl}(2,\mathbb{R})$ [88]). This can be shown by considering the
change of basis

$\displaystyle\hat{\boldsymbol{J}}=\boldsymbol{Y},$
$\displaystyle\hat{\boldsymbol{G}}_{a}=\boldsymbol{Z}_{a},$
$\displaystyle\hat{\boldsymbol{S}}=\boldsymbol{T},$
$\displaystyle\hat{\boldsymbol{\mathcal{J}}}_{a}=\boldsymbol{\mathcal{Y}}_{a},$
$\displaystyle\hat{\boldsymbol{\mathcal{G}}}_{ab}=\boldsymbol{\mathcal{Z}}_{ab},$
$\displaystyle\hat{\boldsymbol{\mathcal{S}}}_{a}=\boldsymbol{\mathcal{T}}_{a},$
(4.65)
$\displaystyle\hat{\boldsymbol{\mathcal{J}}}_{ab}=\boldsymbol{\mathcal{Y}}_{ab},$
$\displaystyle\hat{\boldsymbol{\mathcal{G}}}_{a}=\boldsymbol{\mathcal{Z}}_{a},$
$\displaystyle\hat{\boldsymbol{\mathcal{S}}}_{ab}=\boldsymbol{\mathcal{T}}_{ab},$
$\displaystyle\hat{\boldsymbol{H}}=\boldsymbol{H},$
$\displaystyle\hat{\boldsymbol{P}}_{a}=\boldsymbol{P}_{a},$
$\displaystyle\hat{\boldsymbol{M}}=\boldsymbol{M},$
$\displaystyle\hat{\boldsymbol{\mathcal{H}}}_{a}=\boldsymbol{\mathcal{H}}_{a},$
$\displaystyle\hat{\boldsymbol{\mathcal{P}}}_{ab}=\boldsymbol{\mathcal{P}}_{ab},$
$\displaystyle\hat{\boldsymbol{\mathcal{M}}}_{a}=\boldsymbol{\mathcal{M}}_{a},$
(4.66)
$\displaystyle\hat{\boldsymbol{\mathcal{H}}}_{ab}=\boldsymbol{\mathcal{H}}_{ab},$
$\displaystyle\hat{\boldsymbol{\mathcal{P}}}_{a}=\boldsymbol{\mathcal{P}}_{a},$
$\displaystyle\hat{\boldsymbol{\mathcal{M}}}_{ab}=\boldsymbol{\mathcal{M}}_{ab},$
$\displaystyle\hat{\boldsymbol{k}}=\boldsymbol{J}-\boldsymbol{Y},$
$\displaystyle\hat{\boldsymbol{k}}_{a}=\boldsymbol{G}_{a}-\boldsymbol{Z}_{a},$
$\displaystyle\hat{\boldsymbol{\sigma}}=\boldsymbol{S}-\boldsymbol{T},$
$\displaystyle\hat{\boldsymbol{\mathcal{K}}}_{a}=\boldsymbol{\mathcal{J}}_{a}-\boldsymbol{\mathcal{Y}}_{a},$
$\displaystyle\hat{\boldsymbol{\mathcal{K}}}_{ab}=\boldsymbol{\mathcal{G}}_{ab}-\boldsymbol{\mathcal{Z}}_{ab},$
$\displaystyle\hat{\boldsymbol{\Sigma}}_{a}=\boldsymbol{\mathcal{S}}_{a}-\boldsymbol{\mathcal{T}}_{a},$
(4.67)
$\displaystyle\hat{\tilde{\boldsymbol{\mathcal{K}}}}_{ab}=\boldsymbol{\mathcal{J}}_{ab}-\boldsymbol{\mathcal{Y}}_{ab},$
$\displaystyle\hat{\tilde{\boldsymbol{\mathcal{K}}}}_{a}=\boldsymbol{\mathcal{G}}_{a}-\boldsymbol{\mathcal{Z}}_{a},$
$\displaystyle\hat{\tilde{\boldsymbol{\Sigma}}}_{ab}=\boldsymbol{\mathcal{S}}_{ab}-\boldsymbol{\mathcal{T}}_{ab}.$

The first and second lines in Eqs. (4.65)-(4.67) take the
$\mathfrak{hs_{3}nr\textnormal{-}adslor1_{2+1}}$ algebra into the direct sum
of $\mathfrak{hs_{3}nh1_{2+1}}$ and $\mathfrak{hs_{3}nw1}$, whereas the first
and third lines in Eqs. (4.65)-(4.67) do the analog job for
$\mathfrak{hs_{3}nr\textnormal{-}adslor2_{2+1}}$. Clearly, one can apply the
change of basis (4.47)-(4.49) to the hatted generators to end up with three
copies of a spin-$3$ Nappi-Witten algebra in each case. The previous results
can be extended to the case of a positive cosmological constant by using the
redefinition (4.50) together with

$\displaystyle\boldsymbol{Y}\rightarrow-\boldsymbol{Y},$
$\displaystyle\boldsymbol{Z}_{a}\rightarrow-\boldsymbol{Z}_{a},$
$\displaystyle\boldsymbol{T}\rightarrow-\boldsymbol{T},$ (4.68)
$\displaystyle\boldsymbol{\mathcal{Y}}_{a}\rightarrow-\boldsymbol{\mathcal{Y}}_{a},$
$\displaystyle\boldsymbol{\mathcal{Z}}_{ab}\rightarrow-\boldsymbol{\mathcal{Z}}_{ab},$
$\displaystyle\boldsymbol{\mathcal{T}}_{a}\rightarrow-\boldsymbol{T}_{a},$
$\displaystyle\boldsymbol{\mathcal{Y}}_{ab}\rightarrow-\boldsymbol{\mathcal{Y}}_{ab},$
$\displaystyle\boldsymbol{\mathcal{Z}}_{a}\rightarrow-\boldsymbol{\mathcal{Z}}_{a},$
$\displaystyle\boldsymbol{\mathcal{T}}_{ab}\rightarrow-\boldsymbol{T}_{ab}.$

NR and flat limits: We can construct relativistic symmetries that reproduce
(4.58) in the limit $c\rightarrow\infty$ starting from the higher-spin
extension of the AdS-Lorentz symmetry in $2+1$ dimensions [49]

$\mathfrak{hs_{3}adslor_{2+1}}=S^{(2)}_{M}\times\mathfrak{sl}(3,\mathbb{R}).$
(4.69)

The commutation relations of (4.69) can be read off from Eq. (3.47) with the
same definition (4.15) and (4.36) for the expanded generators as the
Maxwellian case. We extend this symmetry to define the algebras (3.49) and
(3.52) for $S=S^{(2)}_{M}$, which requires introducing

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

The NR contractions have the same form as the one found in the Maxwell case,
which follows from the general expressions (3.50) and (3.53), and are
explicitly given by (4.17), (4.18), (4.38), (4.39), and (4.17), (4.19),
(4.38), (4.40), respectively.

The previous results can be connected with the ones found in Section 4.2 for
the Maxwell case by means of a contraction analog to the vanishing
cosmological limit that relates the Extended Newton-Hooke and the Bargmann
symmetry. The flat limit in this case extends the relations (4.54) by
introducing the extra rescalings

$\displaystyle\boldsymbol{Y}\rightarrow\ell^{2}\,\boldsymbol{Y},$
$\displaystyle\boldsymbol{Z}_{a}\rightarrow\ell^{2}\,\boldsymbol{Z}_{a},$
$\displaystyle\boldsymbol{T}\rightarrow\ell^{2}\,\boldsymbol{T},$ (4.71)
$\displaystyle\boldsymbol{\mathcal{Y}}_{a}\rightarrow\ell^{2}\,\boldsymbol{\mathcal{Y}}_{a},$
$\displaystyle\boldsymbol{\mathcal{Z}}_{ab}\rightarrow\ell^{2}\,\boldsymbol{\mathcal{Z}}_{ab},$
$\displaystyle\boldsymbol{\mathcal{T}}_{a}\rightarrow\ell^{2}\,\boldsymbol{\mathcal{T}}_{a},$
$\displaystyle\boldsymbol{\mathcal{Y}}_{ab}\rightarrow\ell^{2}\,\boldsymbol{\mathcal{Y}}_{ab},$
$\displaystyle\boldsymbol{\mathcal{Z}}_{a}\rightarrow\ell^{2}\,\boldsymbol{\mathcal{Z}}_{a},$
$\displaystyle\boldsymbol{\mathcal{T}}_{ab}\rightarrow\ell^{2}\,\boldsymbol{\mathcal{T}}_{ab},$

as defined in (3.58), which sets all the commutators given in Eqs.
(4.59)-(4.63) to zero in the limit $\ell^{2}\rightarrow\infty$. For the
relativistic generators we use the rescaling (3.63), (3.65) and (3.70), which
lead to (LABEL:reslcaleP'sRel) plus

$\displaystyle\boldsymbol{Z}_{A}^{\rm
rel}\rightarrow\ell^{2}\,\boldsymbol{Z}_{A}^{\rm rel},$
$\displaystyle\boldsymbol{\mathcal{Z}}_{AB}^{\rm
rel}\rightarrow\ell^{2}\,\boldsymbol{\mathcal{Z}}_{AB}^{\rm rel},$ (4.72)
$\displaystyle\boldsymbol{X}^{(2)}\rightarrow\ell^{2}\,\boldsymbol{X}^{(2)},$
$\displaystyle\boldsymbol{X}_{a}^{(2)}\rightarrow\ell^{2}\,\boldsymbol{X}_{a}^{(2)},\quad\boldsymbol{X}_{ab}^{(2)}\rightarrow\ell^{2}\,\boldsymbol{X}_{ab}^{(2)}.$

Thus, in the case of the $\mathfrak{hs_{3}nr\textnormal{-}adslor1_{2+1}}$
algebra we find the following diagram analog to (4.56)

$\displaystyle\mathfrak{hs_{3}adslor_{2+1}}\oplus\mathfrak{so}(3)\oplus\mathfrak{so}(3)\oplus\mathfrak{so}(3)\;$
$\displaystyle\underset{c\rightarrow\infty}{\longrightarrow}$
$\displaystyle\qquad\mathfrak{hs_{3}nr\textnormal{-}adslor1_{2+1}}$ (4.73)
$\displaystyle\hskip
42.67912pt{\scriptstyle\ell\rightarrow\infty}\;\Big{\downarrow}$
$\displaystyle\hskip
25.6073pt{\scriptstyle\ell\rightarrow\infty}\;\Big{\downarrow}$
$\displaystyle\mathfrak{hs_{3}max_{2+1}}\oplus\mathfrak{e\textnormal{-}max_{3}}\;$
$\displaystyle\underset{c\rightarrow\infty}{\longrightarrow}$
$\displaystyle\qquad\mathfrak{hs_{3}nr\textnormal{-}max1_{2+1}},$

while in the case of $\mathfrak{hs_{3}nr\textnormal{-}adslor2_{2+1}}$ the
corresponding diagram, analog to (4.57), has the form

$\displaystyle\mathfrak{hs_{3}adslor_{2+1}}\oplus\mathfrak{adslor_{2+1}}\oplus\mathfrak{so}(2)\oplus\mathfrak{so}(2)\oplus\mathfrak{so}(2)\;$
$\displaystyle\underset{c\rightarrow\infty}{\longrightarrow}$
$\displaystyle\qquad\mathfrak{hs_{3}nr\textnormal{-}adslor2_{2+1}}$ (4.74)
$\displaystyle\hskip
65.44142pt{\scriptstyle\ell\rightarrow\infty}\;\Big{\downarrow}$
$\displaystyle\hskip
25.6073pt{\scriptstyle\ell\rightarrow\infty}\;\Big{\downarrow}$
$\displaystyle\mathfrak{hs_{3}max_{2+1}}\oplus\mathfrak{max_{2+1}}\oplus\mathfrak{so}(2)\oplus\mathfrak{so}(2)\oplus\mathfrak{so}(2)\;$
$\displaystyle\underset{c\rightarrow\infty}{\longrightarrow}$
$\displaystyle\qquad\mathfrak{hs_{3}nr\textnormal{-}max2_{2+1}}.$

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).

###  4.5 Higher-spin gravity theories

Higher-spin gravity actions invariant under the spin-$3$ symmetries presented
above can be constructed using the definitions (3.41) and (3.42) in (3.29),
together with the invariant tensors defined by (3.25), (3.26) and (3.25),
(3.27), which correspond to the expansions $S\times\mathfrak{hs_{3}nw1}$ and
$S\times\mathfrak{hs_{3}nw2}$, respectively. A considerable easier task is to
use the results (3.45) and (3.46) to simply evaluate the semigroup product law
$i\diamond j$ for the semigroups under consideration. Here we adopt the latter
strategy and describe the steps to be followed in order to find the result.
Since the form of the resulting CS forms turns out to be rather cumbersome,
the explicit form of the Lagrangians and the details of the derivations are
given in Appendix A.

  * •

Spin-$3$ Bargmann algebras: the action (3.35) for $S=S_{E}^{(1)}$ is the one
of Extended Bargmann gravity [60, 63, 64, 37] with gauge symmetry (4.1), and
thus, the following gauge fields are identified

$\displaystyle\alpha_{(0)}=w,$ $\displaystyle\theta_{(0)}^{a}=\omega^{a},$
$\displaystyle\beta_{(0)}=s,$ (4.75) $\displaystyle\alpha_{(1)}=\tau,$
$\displaystyle\theta_{(1)}^{a}=e^{a},$ $\displaystyle\beta_{(1)}=m,$

where $w$ and $\omega^{a}$ are the spin connections associated with rotations
and Galilean boosts, respectively, $\tau$ is the clock form, $e^{a}$ is the
spatial dreibein, and $m$ and $s$ are the gauge fields that go along the mass
and the spin generators. The higher-spin extensions (3.45) and (3.46)
introduce a spin-$3$ version of each generator given in Eq. (4.75). It is
therefore natural to adopt the notation

$\displaystyle A^{a}_{(0)}=W^{a},$
$\displaystyle\Theta_{(0)}^{ab}=\Omega^{ab},$ $\displaystyle
B^{a}_{(0)}=S^{a},$ (4.76) $\displaystyle A^{a}_{(1)}=V^{a},$
$\displaystyle\Theta_{(1)}^{ab}=E^{ab},$ $\displaystyle B^{a}_{(1)}=M^{a},$

when defining the gauge fields associated to the
$\mathfrak{hs_{3}barg1_{2+1}}$ algebra. Similarly, when considering the
$\mathfrak{hs_{3}barg2_{2+1}}$ symmetry, one can define the spin-$3$ extension
of (4.75) as

$\displaystyle A^{ab}_{(0)}=W^{ab},$
$\displaystyle\Theta_{(0)}^{a}=\Omega^{a},$ $\displaystyle
B^{ab}_{(0)}=S^{ab},$ (4.77) $\displaystyle A^{ab}_{(1)}=V^{ab},$
$\displaystyle\Theta_{(1)}^{a}=E^{a},$ $\displaystyle B^{ab}_{(1)}=M^{ab}.$

The explicit form of the spin-$3$ CS gravity Lagrangians (3.45) and (3.46) for
$S^{(1)}_{E}$ follows from Eqs. (A.4) and (A.6) of Appendix A.1 after taking
the limit $\ell\rightarrow\infty$.

  * •

Spin-$3$ Maxwell algebras: considering $S^{(2)}_{E}$ in the CS form (3.35)
yields a NR Maxwellian CS gravity theory [62, 41] for the gravitational gauge
fields (4.75) and the Maxwellian gauge fields

$\alpha_{(2)}=y,\quad\theta_{(2)}^{a}=k^{a},\quad\beta_{(2)}=t.$ (4.78)

One can then extend such theory by evaluating the Lagrangian (3.46) for
$S^{(2)}_{E}$, which yields a CS action with
$\mathfrak{hs_{3}nr\textnormal{-}max1_{2+1}}$ gauge symmetry that includes the
spin-$3$ gauge fields (4.77) plus

$A^{a}_{(2)}=Y^{a},\quad\Theta_{(2)}^{ab}=K^{ab},\quad B^{a}_{(2)}=T^{a}.$
(4.79)

Similarly, one can write the
$\mathfrak{hs_{3}nr\textnormal{-}max2_{2+1}}$-invariant CS theory coming from
(3.46) in terms of (4.77) and

$A^{ab}_{(2)}=Y^{ab},\quad\Theta_{(2)}^{a}=K^{a},\quad B^{ab}_{(2)}=T^{ab}.$
(4.80)

These are respectively given in Eqs. (A.13) and (A.15) of Appendix A.2 when
the limit $\ell\rightarrow\infty$ is taken.

  * •

Spin-$3$ Newton-Hooke algebras: replacing $S=S^{(1)}_{M}$ in the CS form
(3.35) yields Extended Newton-Hooke CS gravity [89, 64, 37]. When considering
the extensions $\mathfrak{hs_{3}nh1_{2+1}}$ and $\mathfrak{hs_{3}nh2_{2+1}}$,
the CS forms (3.45) and (3.46) provide actions involving the same fields
(4.75)-(4.77) as the Bargmann case. One can implement the rescaling (3.61),
which now looks like

$\displaystyle\tau\rightarrow\ell^{-1}\,\tau,$ $\displaystyle
e^{a}\rightarrow\ell^{-1}\,e^{a},$ $\displaystyle m\rightarrow\ell^{-1}\,m,$
(4.81) $\displaystyle V^{a}\rightarrow\ell^{-1}\,V^{a},$ $\displaystyle
E^{ab}\rightarrow\ell^{-1}\,E^{ab},$ $\displaystyle
M^{a}\rightarrow\ell^{-1}\,M^{a},$ $\displaystyle
V^{ab}\rightarrow\ell^{-1}\,V^{ab},$ $\displaystyle
E^{a}\rightarrow\ell^{-1}\,E^{a},$ $\displaystyle
M^{ab}\rightarrow\ell^{-1}\,M^{ab},$

to find the CS forms given in Eqs. (A.4) and (A.6) of Appendix A.1.

  * •

Spin-$3$ AdS-Lorentz algebras: evaluating the action (3.35) for $S^{(2)}_{M}$
yields the NR AdS-Lorentz CS gravity theory first considered in Refs. [90,
41]. The spin-$3$ extensions defined by and (3.45) and (3.46) lead to CS
theories invariant under $\mathfrak{hs_{3}nr\textnormal{-}adslor1_{2+1}}$ and
$\mathfrak{hs_{3}nr\textnormal{-}adslor2_{2+1}}$, respectively. These involve
the gauge fields (4.75)-(4.80) appearing in the Maxwellian case that we
previously considered, and reduces to it when applying the flat limit
$\ell\rightarrow\infty$, provided the fields are rescaled as in Eq. (3.61),
i.e.

$\displaystyle k^{a}\rightarrow\ell^{-2}\,k^{a},$ $\displaystyle
y\rightarrow\ell^{-2}\,y,$ $\displaystyle t\rightarrow\ell^{-2}\,t,$ (4.82)
$\displaystyle K^{ab}\rightarrow\ell^{-2}\,K^{ab},$ $\displaystyle
Y^{a}\rightarrow\ell^{-2}\,Y^{a},$ $\displaystyle
T^{a}\rightarrow\ell^{-2}\,T^{a},$ $\displaystyle
K^{a}\rightarrow\ell^{-2}\,K^{a},$ $\displaystyle
Y^{ab}\rightarrow\ell^{-2}\,Y^{ab},$ $\displaystyle
T^{ab}\rightarrow\ell^{-2}\,T^{ab}.$

The explicit form of the CS forms is given in Eqs. (A.13) and (A.15) of
Appendix A.2.

Naturally, all the CS forms here considered can be obtained from the
corresponding relativistic CS forms constructed out of the expanded/extended
$\mathfrak{sl}(3,\mathbb{R})$ algebras appearing in (3.69), where the
rescaling of the relativistic gauge fields in terms of $c$ can be easily read
off from (3.50) and (3.53) by demanding that the relativistic gauge
connections match the NR ones given in Eq. (3.41) .

##  5 Concluding Remarks

We have constructed NR higher-spin theories of Galilean type in $2+1$ space-
time dimensions as expansions of particular higher-spin extensions of the
Nappi-Witten algebras, namely, $\mathfrak{hs_{3}nw1}$ and
$\mathfrak{hs_{3}nw2}$. These extended Nappi-Witten symmetries were obtained
after applying Inönü-Wigner contractions to suitable extensions of
$\mathfrak{sl}(3,\mathbb{R})$:

$\displaystyle\mathfrak{sl}(3,\mathbb{R})\oplus\mathfrak{so}(3)\;$
$\displaystyle\underset{c\rightarrow\infty}{\longrightarrow}$
$\displaystyle\qquad\mathfrak{hs_{3}nw1}$ (5.1)
$\displaystyle\mathfrak{sl}(3,\mathbb{R})\oplus\mathfrak{sl}(2,\mathbb{R})\oplus\mathfrak{so}(2)\;$
$\displaystyle\underset{c\rightarrow\infty}{\longrightarrow}$
$\displaystyle\qquad\mathfrak{hs_{3}nw2}.$

The expansion method applied to these algebras gave rise to two infinite
families of NR spin-$3$ symmetries, which are labeled by the coefficient $n$
in the semigroups $S^{(n)}_{E}$ and $S^{(n)}_{M}$ used in the expansion
mechanism. These are

$\displaystyle\textnormal{Generalized spin-}3\;\textnormal{Bargmann
algebras}:$ $\displaystyle S_{E}^{(n)}\times\mathfrak{hs_{3}nw1,2}$ (5.2)
$\displaystyle\textnormal{Generalized spin-}3\;\textnormal{Newton-Hooke
algebras}:$ $\displaystyle S_{M}^{(n)}\times\mathfrak{hs_{3}nw1,2}.\;$ (5.3)

The first level of these families of algebras reproduces the higher-spin
extensions of the Extended Bargmann and Newton-Hooke algebras in three space-
time dimensions previously found in [13], whereas the $n=2$ case yields novel
spin-$3$ NR symmetries in $2+1$-dimensions, which come to extend the (exotic)
NR Maxwell algebra [62] and the NR AdS-Lorentz algebra [41]. Greater values of
$n$ correspond to spin-$3$ extensions of Generalized Galilean symmetries
algebras [48, 41]. We have shown that these two families of NR spin-$3$
algebras are related by a vanishing cosmological constant limit, i.e.

$\displaystyle
S^{(1)}_{M}\times\mathfrak{hs_{3}nw1,2}\simeq\mathfrak{hs_{3}nh1,2_{2+1}}$
$\displaystyle\xrightarrow[\ell\rightarrow\infty]{\,}$ $\displaystyle\qquad
S^{(1)}_{E}\times\mathfrak{hs_{3}nw1,2}\simeq\mathfrak{hs_{3}barg1,2_{2+1}}$
(5.4) $\displaystyle
S^{(2)}_{M}\times\mathfrak{hs_{3}nw1,2}\simeq\mathfrak{hs_{3}nr\textnormal{-}adslor1,2_{2+1}}$
$\displaystyle\xrightarrow[\ell\rightarrow\infty]{\,}$ $\displaystyle\qquad
S^{(2)}_{E}\times\mathfrak{hs_{3}nw1,2}\simeq\mathfrak{hs_{3}nr\textnormal{-}max1,2_{2+1}}$
$\displaystyle\vdots$ $\displaystyle S^{(n)}_{M}\times\mathfrak{hs_{3}nw1,2}$
$\displaystyle\xrightarrow[\ell\rightarrow\infty]{\,}$ $\displaystyle\qquad
S^{(n)}_{E}\times\mathfrak{hs_{3}nw1,2}.$

Moreover, we have shown that all the algebras of this form define a NR version
of an expanded $\mathfrak{sl}(3,\mathbb{R})$ symmetry. The relation between
relativistic and non-relativistic symmetries is however non-trivial. Indeed,
the relativistic algebras needed to define a $c\rightarrow\infty$ contraction
leading to a non-degenerate invariant tensor are semidirect sum extensions of
the expanded $\mathfrak{sl}(3,\mathbb{R})$ algebras considered in [49]. Unlike
the known contractions giving rise to double and triple central extension in
the spin-$2$ case (see for example [63, 62, 90]), these extensions do not add
only Abelian generators to the relativistic symmetries under consideration,
but include Euclidean and Lorentzian symmetries in three dimensions (see
(4.56), (4.57) and (4.73), (4.74)), which are expansions of the
$\mathfrak{so}(3)$ or the $\mathfrak{sl}(2,\mathbb{R})\oplus\mathfrak{so}(2)$
algebras. The addition of such non-Abelian generators to the relativistic
spin-$3$ algebras constructed in [49] was not to be expected and provides a
non-trivial generalization of the known spin-$2$ constructions. In this way,
the expansion mechanism produces a non-degenerate invariant bilinear form for
every NR expanded algebra, together with a CS higher-spin gravity action. The
particular cases $n=1,2$ were explicitly worked out.

An interesting question for future research is the generalization of our
results to the case of NR spin-$N$ symmetries for $N>3$. Indeed the fact that
CS forms based on the $\mathfrak{sl}(N)\times\mathfrak{sl}(N)$ algebra define
more general higher-spin gravity theories in $2+1$ dimensions [10] admitting a
flat limit [71], together with the structure of the algebras
$\mathfrak{sl}(3,\mathbb{R})\oplus\mathfrak{so}(3)$ and
$\mathfrak{sl}(3,\mathbb{R})\oplus\mathfrak{sl}(2,\mathbb{R})\oplus\mathfrak{so}(2)$
suggest that spin-$N$ extensions of the Nappi-Witten algebra $\mathfrak{nw}$
could be obtained as contractions of suitable extensions of
$\mathfrak{sl}(N)$. A direct generalization of the structure (5.1) signals the
symmetries

$\mathfrak{sl}(N,\mathbb{R})\oplus\mathfrak{so}(N),\qquad\mathfrak{sl}(N,\mathbb{R})\oplus\mathfrak{sl}(N-1,\mathbb{R})\oplus\cdots\oplus\mathfrak{sl}(3,\mathbb{R})\oplus\mathfrak{sl}(2,\mathbb{R})\oplus\mathfrak{so}(2),$
(5.5)

as potential candidates to be expanded. One could as well explore extensions
of $\mathfrak{sl}(N,\mathbb{R})$ that look like

$\displaystyle\mathfrak{sl}(N,\mathbb{R})\oplus\mathfrak{sl}(N-1,\mathbb{R})\oplus\cdots\oplus\mathfrak{sl}(N-\mathfrak{m},\mathbb{R})\oplus\mathfrak{so}(N-\mathfrak{m})$
$\displaystyle\underset{c\rightarrow\infty}{\longrightarrow}$
$\displaystyle\quad\mathfrak{hs}_{N}\mathfrak{nw}(\mathfrak{1+m}),$ (5.6)

and thus interpolate between the ones given in (5.5), which could lead to
several possible higher-spin extensions of the Nappi-Witten algebra. We hope
to explore this problem elsewhere.

Due to the recently discovered relations between NR geometry and condensed
matter systems [27, 28, 29, 30], and in particular, the recent application of
CS theories invariant under extended Nappi-Witten algebras to construct
effective descriptions of the fractional quantum Hall effect [91, 92], it
would be interesting to consider the NR spin-$3$ CS theories here obtained in
the description of collective excitations in quantum Hall systems along the
lines of [93], where NR higher-spin fields in three space-time dimensions
appear in the multipole expansion of the low-energy excitations of a quantum
Hall fluid. Moreover, in [92], a supersymmetric extension of the Nappi-Witten
algebra was considered in the description of the Moore-Read fractional quantum
Hall state. Thus, supersymmetric extensions of the $\mathfrak{hs_{3}nw1}$ and
$\mathfrak{hs_{3}nw2}$ algebras defined along the same lines can be of
interest in the exploration of collective excitations of the fractional
quantum Hall effect in terms of superpartners [94]. Furthermore, a natural
generalization of our results is the inclusion of a space-time boundary and
the construction of (Hamiltonian reductions of) higher-spin WZW theories
(along the lines of, for example, [95]), which could lead to interesting edge
dynamics and could be useful both in the context of the holographic duality in
3D gravity as well as as in condensed matter physics. We expect to consider
this possibility in a future work.

## Acknowledgements

We are grateful to Andrea Cappelli, Giandomenico Palumbo, Simon Pekar,
Konstantinos Sfestos and Utku Zorba for useful comments. P.S-R. thanks Joaquim
Gomis for pointing out Ref. [34], where Lie algebra expansions based on
associative commutative algebras were considered. We specially thank Michelle
Lacoste Adunka for carefully reading the manuscript. R.C. is partially funded
by the National Agency for Research and Development ANID (ex-CONICYT) -
FONDECYT grant N∘1211077 and by the Research Project Code DIREG$\\_$09/2020 of
the Universidad Católica de la Santísima Concepción, Chile. R.C. would like to
thank the Dirección de Investigación and the Vicerrectoría de Investigación of
the Universidad Católica de la Santísima Concepción, Chile, for their constant
support. P.S-R. has received funding from the Norwegian Financial Mechanism
2014-2021 via the National Science Centre (NCN) POLS grant
2020/37/K/ST3/03390.

##  Appendix A Non-relativistic spin-$3$ Chern-Simons theories

In this appendix, we explicitly construct the CS forms associated with the
expanded algebras $S\times\mathfrak{hs_{3}nw1}$ and
$S\times\mathfrak{hs_{3}nw2}$ for the semigroups $S^{(1)}_{M}$ and
$S^{(2)}_{M}$. The corresponding forms for the semigroups $S^{(1)}_{E}$ and
$S^{(2)}_{E}$ can be obtained by considering the flat limit
$\ell\rightarrow\infty$.

###  A.1 Newton-Hooke and Bargmann cases

The CS forms (quasi)invariant under the spin-$3$ extensions of the Extended
Newton-Hooke algebra, $\mathfrak{hs_{3}nh1_{2+1}}$ and
$\mathfrak{hs_{3}nh2_{2+1}}$, defined in (4.43) are obtained by evaluating
Eqs. (3.45) and (3.46) for the semigroup $S^{(1)}_{M}$. Using the product law
(3.55) for $n=1$, we find the following values for the pairs $(i,j)$

$\displaystyle i\diamond j=0\quad\rightarrow$
$\displaystyle(i,j)=\\{(0,0),(1,1)\\},$ (A.1) $\displaystyle i\diamond
j=1\quad\rightarrow$ $\displaystyle(i,j)=\\{(0,1),(1,0)\\},$

which can be used to read off the terms that multiply the constants $\mu_{i}$,
$\nu_{i}$ and $\rho_{i}$ in the expanded actions (3.45) and (3.46). Using
(4.75)-(4.77) and applying the rescaling (4.81), we find for (3.35)

$\displaystyle\mathcal{L}_{S^{(1)}_{M}\times\mathfrak{nw}}$
$\displaystyle=\mu_{0}\left(\omega^{a}\,\bar{R}^{a}_{(0)}-2w\,\mathrm{d}s\right)+\nu_{0}\,w\,\mathrm{d}w+\mu_{1}\left(\omega^{a}\,\bar{R}^{a}_{(1)}+e^{a}\,\bar{R}^{a}_{(0)}-2w\,\mathrm{d}m-2\tau\,\mathrm{d}s\right)$
(A.2)
$\displaystyle+2\nu_{1}\,w\,\mathrm{d}\tau+\frac{\mu_{0}}{\ell^{2}}\left(e^{a}\,\bar{R}^{a}_{(1)}-2\tau\,\mathrm{d}m\right)+\frac{\nu_{0}}{\ell^{2}}\,\tau\,\mathrm{d}\tau,$

where the curvature components that appear in the action can be derived from
(3.34) and read

$\bar{R}^{a}_{(0)}=\mathrm{d}\omega^{a}+\epsilon^{ab}\;\omega^{b}\,w+\frac{1}{\ell^{2}}\epsilon^{ab}\;e^{b}\,\tau,\quad\bar{R}^{a}_{(1)}=\mathrm{d}e^{a}+\epsilon^{ab}\left(\omega^{b}\,\tau+e^{b}\,w\right),$
(A.3)

Eq. (A.2) defines Extended Newton-Hooke CS gravity [89, 64, 37]. In the limit
$\ell\rightarrow\infty$, the second line in (A.2) disappears, leading to
Extended Bargmann CS gravity [60, 63, 64]. Let us look now at the
$\mathfrak{hs_{3}nh1_{2+1}}$ extension of this Lagrangian. Applying the same
steps to the CS form (3.45) yields

$\displaystyle\mathcal{L}_{\mathfrak{hs_{3}nh1_{2+1}}}=\mathcal{L}_{S^{(1)}_{M}\times\mathfrak{nw}}+\mu_{0}\left(2\Omega^{ab}\,\mathcal{R}^{ab}_{(0)}-\dfrac{2}{3}\Omega^{aa}\,\mathcal{R}^{bb}_{(0)}-W^{a}\,\mathcal{F}^{a}_{(0)}-S^{a}\,\mathcal{R}^{a}_{(0)}\right)+\nu_{0}\;W^{a}\,\mathcal{R}^{a}_{(0)}$
(A.4)
$\displaystyle+\mu_{1}\bigg{(}2E^{ab}\,\mathcal{R}^{ab}_{(0)}-\dfrac{2}{3}E^{aa}\,\mathcal{R}^{bb}_{(0)}-V^{a}\,\mathcal{F}^{a}_{(0)}-M^{a}\,\mathcal{R}^{a}_{(0)}+2\Omega^{ab}\,\mathcal{R}^{ab}_{(1)}-\dfrac{2}{3}\Omega^{aa}\,\mathcal{R}^{bb}_{(1)}-W^{a}\,\mathcal{F}^{a}_{(1)}-S^{a}\,\mathcal{R}^{a}_{(1)}\bigg{)}$
$\displaystyle+\nu_{1}\left(W^{a}\,\mathcal{R}^{a}_{(1)}+V^{a}\,\mathcal{R}^{a}_{(0)}\right)+\frac{\mu_{0}}{\ell^{2}}\left(2E^{ab}\,\mathcal{R}^{ab}_{(1)}-\dfrac{2}{3}E^{aa}\,\mathcal{R}^{bb}_{(1)}-V^{a}\,\mathcal{F}^{a}_{(1)}-M^{a}\,\mathcal{R}^{a}_{(1)}\right)+\frac{\nu_{0}}{\ell^{2}}V^{a}\,\mathcal{R}^{a}_{(1)},$

where, as it happens for (A.2), we do not need to evaluate all the curvatures
in (3.43), but only

$\displaystyle\mathcal{R}^{ab}_{(0)}$
$\displaystyle=\mathrm{d}\Omega^{ab}+\epsilon^{c(a}\left(\frac{1}{2}W^{b)}\,\omega^{c}-\Omega^{b)c}\,w+\frac{1}{2\ell^{2}}V^{b)}\,e^{c}-\frac{1}{\ell^{2}}E^{b)c}\,\tau\right)-\delta^{ab}\epsilon_{cd}\left(\omega^{c}\,W^{d}+\frac{1}{\ell^{2}}e^{c}\,V^{d}\right),$
(A.5) $\displaystyle\mathcal{R}^{a}_{(0)}$
$\displaystyle=\mathrm{d}W^{a}+\epsilon^{ab}W^{b}\,w+\frac{1}{\ell^{2}}\epsilon^{ab}V^{b}\,\tau,$
$\displaystyle\mathcal{F}^{a}_{(0)}$
$\displaystyle=\mathrm{d}S^{a}+\epsilon^{ab}\left(S^{b}\,w+W^{b}\,s\right)-2\epsilon_{bc}\Omega^{ab}\,\omega^{c}$
$\displaystyle+\frac{1}{\ell^{2}}\epsilon^{ab}\left(M^{b}\,\tau+V^{b}\,m\right)-\frac{2}{\ell^{2}}\epsilon_{bc}E^{ab}\,e^{c},$
$\displaystyle\mathcal{R}^{ab}_{(1)}$
$\displaystyle=\mathrm{d}E^{ab}+\epsilon^{c(a}\left(\frac{1}{2}W^{b)}\,e^{c}+V^{b)}\,\omega^{c}-\Omega^{b)c}\,\tau-E^{b)c}\,\alpha_{(0)}\right)-\delta^{ab}\epsilon_{cd}\left(\omega^{c}\,+e^{c}\,W^{d}\right),$
$\displaystyle\mathcal{R}^{a}_{(1)}$
$\displaystyle=\mathrm{d}V^{a}+\epsilon^{ab}\left(W^{b}\,\tau+V^{b}\,w\right),$
$\displaystyle\mathcal{F}^{a}_{(1)}$
$\displaystyle=\mathrm{d}M^{a}+\epsilon^{ab}\left(S^{b}\,\tau+M^{b}\,w+W^{b}\,m+V^{b}\,s\right)-2\epsilon_{bc}\left(\Omega^{ab}\,e^{c}+E^{ab}\,\omega^{c}\right),$

These expressions also follow from using the values (A.1) for the pairs
$(i,j)$ to quickly find the terms appearing in each component of the curvature
two-form. One can similarly evaluate the spin-$3$ extension (3.46)
corresponding to the expanded algebra $\mathfrak{hs_{3}nh2_{2+1}}$, which is
given by

$\displaystyle\mathcal{L}_{\mathfrak{hs_{3}nh2_{2+1}}}=\mathcal{L}_{S^{(1)}_{M}\times\mathfrak{nw}}+\mu_{0}\left(2W^{ab}\,\tilde{\mathcal{F}}^{ab}_{(0)}+2S^{ab}\,\tilde{\mathcal{R}}^{ab}_{(0)}-\dfrac{2}{3}W^{aa}\,\tilde{\mathcal{F}}^{bb}_{(0)}-\dfrac{2}{3}S^{aa}\,\tilde{\mathcal{R}}^{bb}_{(0)}-\Omega^{a}\,\tilde{\mathcal{R}}^{a}_{(0)}\right)$
(A.6)
$\displaystyle+\mu_{1}\bigg{(}+2V^{ab}\,\tilde{\mathcal{F}}^{ab}_{(0)}+2M^{ab}\,\tilde{\mathcal{R}}^{ab}_{(0)}-\dfrac{2}{3}V^{aa}\,\tilde{\mathcal{F}}^{bb}_{(0)}-\dfrac{2}{3}M^{aa}\,\tilde{\mathcal{R}}^{bb}_{(0)}-E^{a}\,\tilde{\mathcal{R}}^{a}_{(0)}$
$\displaystyle+2W^{ab}\,\tilde{\mathcal{F}}^{ab}_{(1)}+2S^{ab}\,\tilde{\mathcal{R}}^{ab}_{(1)}-\dfrac{2}{3}W^{aa}\,\tilde{\mathcal{F}}^{bb}_{(1)}-\dfrac{2}{3}S^{aa}\,\tilde{\mathcal{R}}^{bb}_{(1)}-\Omega^{a}\,\tilde{\mathcal{R}}^{a}_{(1)}\bigg{)}$
$\displaystyle-2\nu_{0}\;W^{ab}\,\tilde{\mathcal{R}}^{ab}_{(0)}+\rho_{0}\;W^{aa}\,\tilde{\mathcal{R}}^{bb}_{(0)}-2\nu_{1}\left(W^{ab}\,\tilde{\mathcal{R}}^{ab}_{(1)}+V^{ab}\,\tilde{\mathcal{R}}^{ab}_{(0)}\right)+\rho_{1}\left(W^{aa}\,\tilde{\mathcal{R}}^{bb}_{(1)}+V^{aa}\,\tilde{\mathcal{R}}^{bb}_{(0)}\right)$
$\displaystyle+\frac{\mu_{0}}{\ell^{2}}\left(2V^{ab}\,\tilde{\mathcal{F}}^{ab}_{(1)}+2M^{ab}\,\tilde{\mathcal{R}}^{ab}_{(1)}-\dfrac{2}{3}V^{aa}\,\tilde{\mathcal{F}}^{bb}_{(1)}-\dfrac{2}{3}M^{aa}\,\tilde{\mathcal{R}}^{bb}_{(1)}-E^{a}\,\tilde{\mathcal{R}}^{a}_{(1)}\right)-\frac{2\nu_{0}}{\ell^{2}}V^{ab}\,\tilde{\mathcal{R}}^{ab}_{(1)}+\frac{\rho_{0}}{\ell^{2}}V^{aa}\,\tilde{\mathcal{R}}^{bb}_{(1)}.$

In this expression, the relevant curvature components follow from evaluating
(3.44) for $S^{(1)}_{M}$. One then finds

$\displaystyle\tilde{\mathcal{R}}^{a}_{(0)}$
$\displaystyle=\mathrm{d}\Omega^{a}+\epsilon^{ab}\Omega^{b}\,w-2\epsilon_{bc}W^{ab}\,\omega^{c}+\frac{1}{\ell^{2}}\left(\epsilon^{ab}E^{b}\,\tau-2\epsilon_{bc}V^{ab}\,e^{c}\right),$
(A.7) $\displaystyle\tilde{\mathcal{R}}^{ab}_{(0)}$
$\displaystyle=\mathrm{d}W^{ab}-\epsilon^{c(a}W^{b)c}\,w-\frac{1}{\ell^{2}}\epsilon^{c(a}V^{b)c}\,\tau,$
$\displaystyle\tilde{\mathcal{F}}^{ab}_{(0)}$
$\displaystyle=\mathrm{d}S^{ab}+\epsilon^{c(a}\left(\frac{1}{2}\Omega^{b)}\,\omega^{c}-W^{b)c}\,s-S^{b)c}\,w\right)-\delta^{ab}\epsilon_{cd}\omega^{c}\,\Omega^{d}$
$\displaystyle+\frac{1}{\ell^{2}}\epsilon^{c(a}\left(\frac{1}{2}E^{b)}\,e^{c}-V^{b)c}\,m-M^{b)c}\,\tau\right)-\frac{1}{\ell^{2}}\delta^{ab}\epsilon_{cd}e^{c}\,E^{d},$
$\displaystyle\tilde{\mathcal{R}}^{a}_{(1)}$
$\displaystyle=\mathrm{d}E^{a}+\epsilon^{ab}\left(\Omega^{b}\,\tau+E^{b}\,w\right)-2\epsilon_{bc}\left(W^{ab}\,e^{c}+V^{ab}\,\omega^{c}\right),$
$\displaystyle\tilde{\mathcal{R}}^{ab}_{(1)}$
$\displaystyle=\mathrm{d}V^{ab}-\epsilon^{c(a}\left(W^{b)c}\,\tau+V^{b)c}\,w\right),$
$\displaystyle\tilde{\mathcal{F}}^{ab}_{(1)}$
$\displaystyle=\mathrm{d}M^{ab}+\epsilon^{c(a}\left(\frac{1}{2}\Omega^{b)}\,e^{c}-W^{b)c}\,m-S^{b)c}\,\tau+\frac{1}{2}E^{b)}\,\omega^{c}-V^{b)c}\,s-M^{b)c}\,w\right)$
$\displaystyle-\delta^{ab}\epsilon_{cd}\left(\omega^{c}\,E^{d}+e^{c}\,\Omega^{d}\right).$

Due to the relation (3.60), it is clear that in the flat limit
$\ell\rightarrow\infty$ the forms (A.4) and (A.6) reduce to two spin-$3$
Bargmann gravity Lagrangians

$\mathcal{L}_{\mathfrak{hs_{3}nh1_{2+1}}}\xrightarrow[\ell\rightarrow\infty]{\,}\mathcal{L}_{\mathfrak{hs_{3}barg1_{2+1}}},\qquad\mathcal{L}_{\mathfrak{hs_{3}nh2_{2+1}}}\xrightarrow[\ell\rightarrow\infty]{\,}\mathcal{L}_{\mathfrak{hs_{3}barg2_{2+1}}},$
(A.8)

which could have been independently obtained by using the $S^{(1)}_{E}$
product law (3.54)

$\displaystyle i\diamond j=0\quad\rightarrow$
$\displaystyle(i,j)=\\{(0,0)\\},$ (A.9) $\displaystyle i\diamond
j=1\quad\rightarrow$ $\displaystyle(i,j)=\\{(0,1),(1,0)\\},$

when evaluating the expanded CS actions (3.45) and (3.46). It comes without
saying that (A.8) can be extended to include the relativistic case, in analogy
with (4.56) and (4.57), which requires to introduce CS actions invariant under
the relativistic algebra (4.51), such as the one considered in [10], together
with the CS forms associated to the extensions (4.53).

As stated before, the CS Lagrangians (A.4) and (A.6) could also be constructed
by replacing the expanded connections (3.41) and curvatures (3.42) in (3.29),
and using the invariant bilinear forms (3.25)-(3.26) for $S=S^{(1)}_{M}$.
However, here we have chosen the alternative path of evaluating the non-
vanishing combinations (A.1) directly in (3.45) and (3.46).

###  A.2 Non-relativistic AdS-Lorentz and Maxwell cases

Following the same procedure adopted in the previous example, we derive the
$S^{(2)}_{M}$-expanded CS forms. In this case we use the semigroup product law
(3.55) for $n=2$, for which the relevant pairs of semigroup indices are

$\displaystyle i\diamond j=0\quad\rightarrow$
$\displaystyle(i,j)=\\{(0,0)\\},$ (A.10) $\displaystyle i\diamond
j=1\quad\rightarrow$ $\displaystyle(i,j)=\\{(0,1),(1,0),(1,2),(2,1)\\},$
$\displaystyle i\diamond j=2\quad\rightarrow$
$\displaystyle(i,j)=\\{(0,2),(2,0),(1,1),(2,2)\\}.$

Using the gauge field definitions (4.75)-(4.77) and (4.78)-(4.80), together
with the rescaling (4.82), one can evaluate the CS forms (3.45) and (3.46),
which define higher-spin gravity theories invariant under the gauge symmetries
$\mathfrak{hs_{3}nr\textnormal{-}adslor1_{2+1}}$ and
$\mathfrak{hs_{3}nr\textnormal{-}adslor2_{2+1}}$. We first look at the
Lagrangian (3.35), which define NR AdS-Lorentz CS gravity [90, 41]

$\displaystyle\mathcal{L}_{S^{(2)}_{M}\times\mathfrak{nw}}=\mu_{0}\left(\omega^{a}\,\bar{R}^{a}_{(0)}-2w\,\mathrm{d}s\right)+\nu_{0}\,w\,\mathrm{d}w+\mu_{1}\left(\omega^{a}\,\bar{R}^{a}_{(1)}+e^{a}\,\bar{R}^{a}_{(0)}-2w\,\mathrm{d}m-2\tau\,\mathrm{d}s\right)$
(A.11)
$\displaystyle+2\nu_{1}\,w\,\mathrm{d}\tau+\mu_{2}\left(\omega^{a}\,\bar{R}^{a}_{(2)}+e^{a}\,\bar{R}^{a}_{(1)}+k^{a}\,\bar{R}^{a}_{(0)}-2w\,\mathrm{d}t-2\tau\,\mathrm{d}m-2y\,\mathrm{d}s\right)+\nu_{2}\left(2w\,\mathrm{d}y+\tau\,\mathrm{d}\tau\right)$
$\displaystyle+\frac{\mu_{1}}{\ell^{2}}\left(e^{a}\,\bar{R}^{a}_{(2)}+k^{a}\,\bar{R}^{a}_{(1)}-2\tau\,\mathrm{d}t-2y\,\mathrm{d}m\right)+\frac{2\nu_{1}}{\ell^{2}}\,\tau\,\mathrm{d}y+\frac{\mu_{2}}{\ell^{2}}\left(k^{a}\,\bar{R}^{a}_{(2)}-2y\,\mathrm{d}t\right)+\frac{\nu_{2}}{\ell^{2}}\,y\,\mathrm{d}y.$

Here, the curvature components (3.34) take the form

$\displaystyle\bar{R}^{a}_{(0)}$
$\displaystyle=\mathrm{d}\omega^{a}+\epsilon^{ab}\omega^{b}\,w,$ (A.12)
$\displaystyle\bar{R}^{a}_{(1)}$
$\displaystyle=\mathrm{d}e^{a}+\epsilon^{ab}\left(\omega^{b}\,\tau+e^{b}\,w\,\right)+\frac{1}{\ell^{2}}\epsilon^{ab}\left(e^{b}\,y+k^{b}\,\tau\right),$
$\displaystyle\bar{R}^{a}_{(2)}$
$\displaystyle=\mathrm{d}k^{a}+\epsilon^{ab}\left(\omega^{b}\,y+k^{b}\,w+e^{b}\,\tau\right)+\frac{1}{\ell^{2}}\epsilon^{ab}\;k^{b}\,y.$

In the limit $\ell\rightarrow\infty$, the last line in Eq. (A.2) disappears.
The first two lines correspond to the CS form
$\mathcal{L}_{S^{(2)}_{E}\times\mathfrak{nw}}$, which is precisely the
(exotic) Maxwellian NR gravity theory first found in [62]. The first higher-
spin extension (3.45) takes the form

$\displaystyle\mathcal{L}_{\mathfrak{hs_{3}nr\textnormal{-}adslor1_{2+1}}}=\mathcal{L}_{S^{(1)}_{M}\times\mathfrak{nw}}+\mu_{0}\bigg{(}2\Omega^{ab}\,\mathcal{R}^{ab}_{(0)}-\dfrac{2}{3}\Omega^{aa}\,\mathcal{R}^{bb}_{(0)}-W^{a}\,\mathcal{F}^{a}_{(0)}-S^{a}\,\mathcal{R}^{a}_{(0)}\bigg{)}$
(A.13)
$\displaystyle+\mu_{1}\bigg{(}2\Omega^{ab}\,\mathcal{R}^{ab}_{(1)}+2E^{ab}\,\mathcal{R}^{ab}_{(0)}-\dfrac{2}{3}\Omega^{aa}\,\mathcal{R}^{bb}_{(1)}-\dfrac{2}{3}E^{aa}\,\mathcal{R}^{bb}_{(0)}-W^{a}\,\mathcal{F}^{a}_{(1)}-V^{a}\,\mathcal{F}^{a}_{(0)}-S^{a}\,\mathcal{R}^{a}_{(1)}-M^{a}\,\mathcal{R}^{a}_{(0)}\bigg{)}$
$\displaystyle+\mu_{2}\bigg{(}2\Omega^{ab}\,\mathcal{R}^{ab}_{(2)}+2K^{ab}\,\mathcal{R}^{ab}_{(0)}+2E^{ab}\,\mathcal{R}^{ab}_{(1)}-\dfrac{2}{3}\Omega^{aa}\,\mathcal{R}^{bb}_{(2)}-\dfrac{2}{3}K^{aa}\,\mathcal{R}^{bb}_{(0)}-\dfrac{2}{3}E^{aa}\,\mathcal{R}^{bb}_{(1)}$
$\displaystyle-W^{a}\,\mathcal{F}^{a}_{(2)}-Y^{a}\,\mathcal{F}^{a}_{(0)}-V^{a}\,\mathcal{F}^{a}_{(1)}-S^{a}\,\mathcal{R}^{a}_{(2)}-T^{a}\,\mathcal{R}^{a}_{(0)}-M^{a}\,\mathcal{R}^{a}_{(1)}\bigg{)}$
$\displaystyle+\nu_{0}\;W^{a}\,\mathcal{R}^{a}_{(0)}+\nu_{1}\left(W^{a}\,\mathcal{R}^{a}_{(1)}+V^{a}\,\mathcal{R}^{a}_{(0)}\right)+\nu_{2}\left(W^{a}\,\mathcal{R}^{a}_{(2)}+Y^{a}\,\mathcal{R}^{a}_{(0)}+V^{a}\,\mathcal{R}^{a}_{(1)}\right)$
$\displaystyle+\frac{\mu_{1}}{\ell^{2}}\bigg{(}2E^{ab}\,\mathcal{R}^{ab}_{(2)}+2K^{ab}\,\mathcal{R}^{ab}_{(1)}-\dfrac{2}{3}E^{aa}\,\mathcal{R}^{bb}_{(2)}-\dfrac{2}{3}K^{aa}\,\mathcal{R}^{bb}_{(1)}-V^{a}\,\mathcal{F}^{a}_{(2)}-A^{a}_{(2)}\,\mathcal{F}^{a}_{(1)}-M^{a}\,\mathcal{R}^{a}_{(2)}-T^{a}\,\mathcal{R}^{a}_{(1)}\bigg{)}$
$\displaystyle+\frac{\nu_{1}}{\ell^{2}}\left(V^{a}\,\mathcal{R}^{a}_{(2)}+Y^{a}\,\mathcal{R}^{a}_{(1)}\right)+\frac{\mu_{2}}{\ell^{2}}\bigg{(}2K^{ab}\,\mathcal{R}^{ab}_{(2)}-\dfrac{2}{3}K^{aa}\,\mathcal{R}^{bb}_{(2)}-Y^{a}\,\mathcal{F}^{a}_{(2)}-T^{a}\,\mathcal{R}^{a}_{(2)}\bigg{)}+\frac{\nu_{2}}{\ell^{2}}Y^{a}\,\mathcal{R}^{a}_{(2)},$

where the higher-spin curvatures appearing in the action can be found by
evaluating Eq. (3.43) and are given by

$\displaystyle\mathcal{R}^{ab}_{(0)}$
$\displaystyle=\mathrm{d}\Omega^{ab}+\epsilon^{c(a}\left(\frac{1}{2}W^{b)}\,\omega^{c}-\Omega^{b)c}\,w\right)-\delta^{ab}\epsilon_{cd}\omega^{c}\,W^{d},$
(A.14) $\displaystyle\mathcal{R}^{a}_{(0)}$
$\displaystyle=\mathrm{d}W^{a}+\epsilon^{ab}W^{b}\,w,$
$\displaystyle\mathcal{F}^{a}_{(0)}$
$\displaystyle=\mathrm{d}S^{a}+\epsilon^{ab}\left(S^{b}\,w+W^{b}\,s\right)-2\epsilon_{bc}\Omega^{ab}\,\omega^{c},$
$\displaystyle\mathcal{R}^{ab}_{(1)}$
$\displaystyle=\mathrm{d}E^{ab}+\epsilon^{c(a}\left(\frac{1}{2}W^{b)}\,e^{c}+\frac{1}{2}V^{b)}\,\omega^{c}-\Omega^{b)c}\,\tau-E^{b)c}\,w\right)-\delta^{ab}\epsilon_{cd}\left(\omega^{c}\,V^{d}+e^{c}\,W^{d}\right)$
$\displaystyle+\frac{1}{\ell^{2}}\epsilon^{c(a}\left(\frac{1}{2}V^{b)}\,k^{c}+\frac{1}{2}Y^{b)}\,e^{c}-E^{b)c}\,y-K^{b)c}\,\tau\right)-\frac{1}{\ell^{2}}\delta^{ab}\epsilon_{cd}\left(e^{c}\,Y^{d}+k^{c}\,V^{d}\right),$
$\displaystyle\mathcal{R}^{a}_{(1)}$
$\displaystyle=\mathrm{d}V^{a}+\epsilon^{ab}W^{b}\left(\,\tau+V^{b}\,w\right)+\frac{1}{\ell^{2}}\epsilon^{ab}\left(V^{b}\,\alpha_{(2)}+Y^{b}\,\tau\right),$
$\displaystyle\mathcal{F}^{a}_{(1)}$
$\displaystyle=\mathrm{d}M^{a}+\epsilon^{ab}\left(S^{b}\,\tau+M^{b}\,w+W^{b}\,m+V^{b}\,s\right)-2\epsilon_{bc}\left(\Omega^{ab}\,e^{c}+E^{ab}\,\omega^{c}\right)$
$\displaystyle+\frac{1}{\ell^{2}}\epsilon^{ab}\left(M^{b}\,y+T^{b}\,\tau+V^{b}\,t+Y^{b}\,m\right)-\frac{2}{\ell^{2}}\epsilon_{bc}\left(E^{ab}\,k^{c}+K^{ab}\,e^{c}\right),$
$\displaystyle\mathcal{R}^{ab}_{(2)}$
$\displaystyle=\mathrm{d}K^{ab}+\frac{1}{2}\epsilon^{c(a}\left(W^{b)}\,k^{c}+Y^{b)}\,\omega^{c}+V^{b)}\,e^{c}\right)-\epsilon^{c(a}\left(\Omega^{b)c}\,y+K^{b)c}\,w+E^{b)c}\,\tau\right)$
$\displaystyle-\delta^{ab}\epsilon_{cd}\left(\omega^{c}\,Y^{d}+k^{c}\,W^{d}+e^{c}\,V^{d}\right)+\frac{1}{\ell^{2}}\epsilon^{c(a}\left(\frac{1}{2}Y^{b)}\,k^{c}-K^{b)c}\,y\right)-\frac{1}{\ell^{2}}\delta^{ab}\epsilon_{cd}k^{c}\,Y^{d},$
$\displaystyle\mathcal{R}^{a}_{(2)}$
$\displaystyle=\mathrm{d}Y^{a}+\epsilon^{ab}\left(W^{b}\,y+Y^{b}\,w+V^{b}\,\tau\right)+\frac{1}{\ell^{2}}\epsilon^{ab}Y^{b}\,y,$
$\displaystyle\mathcal{F}^{a}_{(2)}$
$\displaystyle=\mathrm{d}T^{a}+\epsilon^{ab}\left(S^{b}\,y+T^{b}\,w+M^{b}\,\tau+W^{b}\,t+Y^{b}\,s+V^{b}\,m\right)$
$\displaystyle-2\epsilon_{bc}\left(\Omega^{ab}\,k^{c}+K^{ab}\,\omega^{c}+E^{ab}\,e^{c}\right)+\frac{1}{\ell^{2}}\epsilon^{ab}\left(T^{b}\,y+Y^{b}\,t\right)-\frac{2}{\ell^{2}}\epsilon_{bc}K^{ab}\,k^{c}.$

Following the same steps to find a second spin-$3$ extension of the NR AdS-
Lorentz CS form leads to

$\displaystyle\mathcal{L}_{\mathfrak{hs_{3}nr\textnormal{-}adslor2_{2+1}}}=\mathcal{L}_{S^{(1)}_{M}\times\mathfrak{nw}}+\mu_{0}\left(2W^{ab}\,\tilde{\mathcal{F}}^{ab}_{(0)}+2S^{ab}\,\tilde{\mathcal{R}}^{ab}_{(0)}-\dfrac{2}{3}W^{aa}\,\tilde{\mathcal{F}}^{bb}_{(0)}-\dfrac{2}{3}S^{aa}\,\tilde{\mathcal{R}}^{bb}_{(0)}-\Omega^{a}\,\tilde{\mathcal{R}}^{a}_{(0)}\right)$
(A.15)
$\displaystyle+\mu_{1}\bigg{(}2W^{ab}\,\tilde{\mathcal{F}}^{ab}_{(1)}+2V^{ab}\,\tilde{\mathcal{F}}^{ab}_{(0)}+2S^{ab}\,\tilde{\mathcal{R}}^{ab}_{(1)}+2M^{ab}\,\tilde{\mathcal{R}}^{ab}_{(0)}-\dfrac{2}{3}W^{aa}\,\tilde{\mathcal{F}}^{bb}_{(1)}$
$\displaystyle-\dfrac{2}{3}V^{aa}\,\tilde{\mathcal{F}}^{bb}_{(0)}-\dfrac{2}{3}S^{aa}\,\tilde{\mathcal{R}}^{bb}_{(1)}-\dfrac{2}{3}M^{aa}\,\tilde{\mathcal{R}}^{bb}_{(0)}-\Omega^{a}\,\tilde{\mathcal{R}}^{a}_{(1)}-E^{a}\,\tilde{\mathcal{R}}^{a}_{(0)}\bigg{)}$
$\displaystyle+\mu_{2}\bigg{(}2W^{ab}\,\tilde{\mathcal{F}}^{ab}_{(2)}+2Y^{ab}\,\tilde{\mathcal{F}}^{ab}_{(0)}+2V^{ab}\,\tilde{\mathcal{F}}^{ab}_{(1)}+2S^{ab}\,\tilde{\mathcal{R}}^{ab}_{(2)}+2T^{ab}\,\tilde{\mathcal{R}}^{ab}_{(0)}+2M^{ab}\,\tilde{\mathcal{R}}^{ab}_{(1)}-\dfrac{2}{3}W^{aa}\,\tilde{\mathcal{F}}^{bb}_{(2)}$
$\displaystyle-\dfrac{2}{3}Y^{aa}\,\tilde{\mathcal{F}}^{bb}_{(0)}-\dfrac{2}{3}V^{aa}\,\tilde{\mathcal{F}}^{bb}_{(1)}-\dfrac{2}{3}S^{aa}\,\tilde{\mathcal{R}}^{bb}_{(2)}-\dfrac{2}{3}T^{aa}\,\tilde{\mathcal{R}}^{bb}_{(0)}-\dfrac{2}{3}M^{aa}\,\tilde{\mathcal{R}}^{bb}_{(1)}-\Omega^{a}\,\tilde{\mathcal{R}}^{a}_{(2)}-K^{a}\,\tilde{\mathcal{R}}^{a}_{(0)}-E^{a}\,\tilde{\mathcal{R}}^{a}_{(1)}\bigg{)}$
$\displaystyle-2\nu_{0}\;W^{ab}\,\tilde{\mathcal{R}}^{ab}_{(0)}-2\nu_{1}\left(W^{ab}\,\tilde{\mathcal{R}}^{ab}_{(1)}+V^{ab}\,\tilde{\mathcal{R}}^{ab}_{(0)}\right)-2\nu_{2}\left(W^{ab}\,\tilde{\mathcal{R}}^{ab}_{(2)}+Y^{ab}\,\tilde{\mathcal{R}}^{ab}_{(0)}+V^{ab}\,\tilde{\mathcal{R}}^{ab}_{(1)}\right)$
$\displaystyle+\rho_{0}\;W^{aa}\,\tilde{\mathcal{R}}^{bb}_{(0)}+\rho_{1}\left(W^{aa}\,\tilde{\mathcal{R}}^{bb}_{(1)}+V^{aa}\,\tilde{\mathcal{R}}^{bb}_{(0)}\right)+\rho_{2}\left(W^{aa}\,\tilde{\mathcal{R}}^{bb}_{(2)}+A^{aa}_{(2)}\,\tilde{\mathcal{R}}^{bb}_{(0)}+V^{aa}\,\tilde{\mathcal{R}}^{bb}_{(1)}\right)$
$\displaystyle+\frac{\mu_{1}}{\ell^{2}}\bigg{(}2V^{ab}\,\tilde{\mathcal{F}}^{ab}_{(2)}+2Y^{ab}\,\tilde{\mathcal{F}}^{ab}_{(1)}+2M^{ab}\,\tilde{\mathcal{R}}^{ab}_{(2)}+2T^{ab}\,\tilde{\mathcal{R}}^{ab}_{(1)}-\dfrac{2}{3}V^{aa}\,\tilde{\mathcal{F}}^{bb}_{(2)}$
$\displaystyle-\dfrac{2}{3}Y^{aa}\,\tilde{\mathcal{F}}^{bb}_{(1)}-\dfrac{2}{3}M^{aa}\,\tilde{\mathcal{R}}^{bb}_{(2)}-\dfrac{2}{3}T^{aa}\,\tilde{\mathcal{R}}^{bb}_{(1)}-E^{a}\,\tilde{\mathcal{R}}^{a}_{(2)}-K^{a}\,\tilde{\mathcal{R}}^{a}_{(1)}\bigg{)}$

$\displaystyle+\frac{\mu_{2}}{\ell^{2}}\bigg{(}2Y^{ab}\,\tilde{\mathcal{F}}^{ab}_{(2)}+2T^{ab}\,\tilde{\mathcal{R}}^{ab}_{(2)}-\dfrac{2}{3}Y^{aa}\,\tilde{\mathcal{F}}^{bb}_{(2)}-\dfrac{2}{3}T^{aa}\,\tilde{\mathcal{R}}^{bb}_{(2)}-K^{a}\,\tilde{\mathcal{R}}^{a}_{(2)}\bigg{)}$
$\displaystyle-\frac{2\nu_{1}}{\ell^{2}}\left(V^{ab}\,\tilde{\mathcal{R}}^{ab}_{(2)}+Y^{ab}\,\tilde{\mathcal{R}}^{ab}_{(1)}\right)-\frac{2\nu_{2}}{\ell^{2}}Y^{ab}\,\tilde{\mathcal{R}}^{ab}_{(2)}+\frac{\rho_{1}}{\ell^{2}}\left(V^{aa}\,\tilde{\mathcal{R}}^{bb}_{(2)}+Y^{aa}\,\tilde{\mathcal{R}}^{bb}_{(1)}\right)+\frac{\rho_{2}}{\ell^{2}}Y^{aa}\,\tilde{\mathcal{R}}^{bb}_{(2)},$

where the curvature components relevant here follow from (3.44) and read

$\displaystyle\tilde{\mathcal{R}}^{a}_{(0)}$
$\displaystyle=\mathrm{d}\Omega^{a}+\epsilon^{ab}\Omega^{b}\,w-2\epsilon_{bc}W^{ab}\,\omega^{c}$
(A.16) $\displaystyle\tilde{\mathcal{R}}^{ab}_{(0)}$
$\displaystyle=\mathrm{d}W^{ab}-\epsilon^{c(a}W^{b)c}\,w$
$\displaystyle\tilde{\mathcal{F}}^{ab}_{(0)}$
$\displaystyle=\mathrm{d}S^{ab}+\epsilon^{c(a}\left(\frac{1}{2}\Omega^{b)}\,\omega^{c}-W^{b)c}\,s-S^{b)c}\,w\right)-\delta^{ab}\epsilon_{cd}\omega^{c}\,\Omega^{d}$
$\displaystyle\tilde{\mathcal{R}}^{a}_{(1)}$
$\displaystyle=\mathrm{d}E^{a}+\epsilon^{ab}\left(\Omega^{b}\,\tau+E^{b}\,w\right)-2\epsilon_{bc}\left(W^{ab}\,e^{c}+V^{ab}\,\omega^{c}\right)$
$\displaystyle+\frac{1}{\ell^{2}}\epsilon^{ab}\left(E^{b}\,y+K^{b}\,\tau\right)-\frac{2}{\ell^{2}}\epsilon_{bc}\left(V^{ab}\,k^{c}+Y^{ab}\,e^{c}\right)$
$\displaystyle\tilde{\mathcal{R}}^{ab}_{(1)}$
$\displaystyle=\mathrm{d}V^{ab}-\epsilon^{c(a}\left(W^{b)c}\,\tau+V^{b)c}\,w\right)-\frac{1}{\ell^{2}}\epsilon^{c(a}\left(V^{b)c}\,y+Y^{b)c}\,\tau\right)$
$\displaystyle\tilde{\mathcal{F}}^{ab}_{(1)}$
$\displaystyle=\mathrm{d}M^{ab}+\epsilon^{c(a}\left(\frac{1}{2}\Omega^{b)}\,e^{c}+\frac{1}{2}E^{b)}\,\omega^{c}-W^{b)c}\,m-V^{b)c}\,s-S^{b)c}\,\tau-M^{b)c}\,w\right)$
$\displaystyle-\delta^{ab}\epsilon_{cd}\left(\omega^{c}\,E^{d}+e^{c}\,\Omega^{d}\right)-\frac{1}{\ell^{2}}\delta^{ab}\epsilon_{cd}\left(e^{c}\,K^{d}+k^{c}\,E^{d}\right)$
$\displaystyle+\frac{1}{\ell^{2}}\epsilon^{c(a}\left(\frac{1}{2}E^{b)}\,k^{c}+\frac{1}{2}K^{b)}\,e^{c}-V^{b)c}\,t-Y^{b)c}\,m-M^{b)c}\,y-T^{b)c}\,\tau\right)$
$\displaystyle\tilde{\mathcal{R}}^{a}_{(2)}$
$\displaystyle=\mathrm{d}K^{a}+\epsilon^{ab}\left(\Omega^{b}\,\alpha_{(2)}+K^{b}\,w+E^{b}\,\tau\right)-2\epsilon_{bc}\left(W^{ab}\,k^{c}+Y^{ab}\,\omega^{c}+V^{ab}\,e^{c}\right)$
$\displaystyle+\frac{1}{\ell^{2}}\epsilon^{ab}K^{b}\,y-\frac{2}{\ell^{2}}\epsilon_{bc}Y^{ab}\,k^{c},$
$\displaystyle\tilde{\mathcal{R}}^{ab}_{(2)}$
$\displaystyle=\mathrm{d}Y^{ab}-\epsilon^{c(a}\left(W^{b)c}\,y+Y^{b)c}\,w+V^{b)c}\,\tau\right)-\frac{1}{\ell^{2}}\epsilon^{c(a}Y^{b)c}\,y$
$\displaystyle\tilde{\mathcal{F}}^{ab}_{(2)}$
$\displaystyle=\mathrm{d}T^{ab}-\delta^{ab}\epsilon_{cd}\left(\omega^{c}\,K^{d}+k^{c}\,\Omega^{d}+e^{c}\,E^{d}\right)$
$\displaystyle+\epsilon^{c(a}\left(\frac{1}{2}\Omega^{b)}\,k^{c}-W^{b)c}\,t-S^{b)c}\,y+\frac{1}{2}K^{b)}\,\omega^{c}-Y^{b)c}\,s-T^{b)c}\,w+\frac{1}{2}E^{b)}\,e^{c}-V^{b)c}\,m-M^{b)c}\,\tau\right)$
$\displaystyle+\frac{1}{\ell^{2}}\epsilon^{c(a}\left(\frac{1}{2}K^{b)}\,k^{c}-Y^{b)c}\,t-T^{b)c}\,y\right)-\frac{1}{\ell^{2}}\delta^{ab}\epsilon_{cd}\;k^{c}\,K^{d}.$

Using (3.60), one finds that applying the flat limit $\ell\rightarrow\infty$
to (A.13) and (A.15) yields the CS for the corresponding spin-$3$ extensions
of the NR Maxwell algebra (4.23)

$\mathcal{L}_{\mathfrak{hs_{3}nr\textnormal{-}adslor1_{2+1}}}\xrightarrow[\ell\rightarrow\infty]{\,}\mathcal{L}_{\mathfrak{hs_{3}nr\textnormal{-}max1_{2+1}}},\qquad\mathcal{L}_{\mathfrak{hs_{3}nr\textnormal{-}adslor2_{2+1}}}\xrightarrow[\ell\rightarrow\infty]{\,}\mathcal{L}_{\mathfrak{hs_{3}nr\textnormal{-}max2_{2+1}}}.$
(A.17)

The Maxwellian action can be independently constructed by repeating the same
computation with the semigroup $S^{(2)}_{E}$ as starting point. This requires
to replace (A.10) by

$\displaystyle i\diamond j=0\quad\rightarrow$
$\displaystyle(i,j)=\\{(0,0)\\},$ (A.18) $\displaystyle i\diamond
j=1\quad\rightarrow$ $\displaystyle(i,j)=\\{(0,1),(1,0)\\},$ $\displaystyle
i\diamond j=2\quad\rightarrow$
$\displaystyle(i,j)=\\{(0,2),(2,0),(1,1),(2,2)\\}.$

One could easily extend (A.17) to include underlying relativistic gravity
theories in analogy with the diagrams (4.73) and (4.74) at the level of the
gauge symmetries. The relativistic CS forms are obtained by extending the AdS-
Lorentz higher-spin gravity action invariant under (4.69) [49] using the CS
forms associated to the extensions (4.70). As before, the CS theories defined
by (A.13) and (A.15) could be alternatively obtained by evaluating the
connections (3.41) and the curvatures (3.42) for $S=S^{(2)}_{M}$ in (3.29),
and using the explicit form of the invariant bilinear forms (3.25)-(3.26).

## References

  * [1] X. Bekaert, N. Boulanger, and P. Sundell, “How higher-spin gravity surpasses the spin two barrier: no-go theorems versus yes-go examples,” Rev. Mod. Phys. 84 (2012) 987–1009, arXiv:1007.0435 [hep-th].
  * [2] M. A. Vasiliev, “Consistent equation for interacting gauge fields of all spins in (3+1)-dimensions,” Phys. Lett. B 243 (1990) 378–382.
  * [3] M. A. Vasiliev, “Nonlinear equations for symmetric massless higher spin fields in (A)dS(d),” Phys. Lett. B 567 (2003) 139–151, arXiv:hep-th/0304049.
  * [4] M. P. Blencowe, “A Consistent Interacting Massless Higher Spin Field Theory in $D$ = (2+1),” Class. Quant. Grav. 6 (1989) 443.
  * [5] E. Bergshoeff, M. P. Blencowe, and K. S. Stelle, “Area Preserving Diffeomorphisms and Higher Spin Algebra,” Commun. Math. Phys. 128 (1990) 213.
  * [6] M. A. Vasiliev, “Higher spin gauge theories in four-dimensions, three-dimensions, and two-dimensions,” Int. J. Mod. Phys. D 05 (1996) 763–797, arXiv:hep-th/9611024.
  * [7] C. Bunster, M. Henneaux, A. Perez, D. Tempo, and R. Troncoso, “Generalized Black Holes in Three-dimensional Spacetime,” JHEP 05 (2014) 031, arXiv:1404.3305 [hep-th].
  * [8] A. Castro, “Lectures on Higher Spin Black Holes in AdS3 Gravity,” Acta Phys. Polon. B 47 (2016) 2479–2508.
  * [9] M. Henneaux and S.-J. Rey, “Nonlinear $W_{infinity}$ as Asymptotic Symmetry of Three-Dimensional Higher Spin Anti-de Sitter Gravity,” JHEP 12 (2010) 007, arXiv:1008.4579 [hep-th].
  * [10] A. Campoleoni, S. Fredenhagen, S. Pfenninger, and S. Theisen, “Asymptotic symmetries of three-dimensional gravity coupled to higher-spin fields,” JHEP 11 (2010) 007, arXiv:1008.4744 [hep-th].
  * [11] H. Afshar, A. Bagchi, R. Fareghbal, D. Grumiller, and J. Rosseel, “Spin-3 Gravity in Three-Dimensional Flat Space,” Phys. Rev. Lett. 111 no. 12, (2013) 121603, arXiv:1307.4768 [hep-th].
  * [12] H. A. Gonzalez, J. Matulich, M. Pino, and R. Troncoso, “Asymptotically flat spacetimes in three-dimensional higher spin gravity,” JHEP 09 (2013) 016, arXiv:1307.5651 [hep-th].
  * [13] E. Bergshoeff, D. Grumiller, S. Prohazka, and J. Rosseel, “Three-dimensional Spin-3 Theories Based on General Kinematical Algebras,” JHEP 01 (2017) 114, arXiv:1612.02277 [hep-th].
  * [14] D. Chernyavsky and D. Sorokin, “Three-dimensional (higher-spin) gravities with extended Schrödinger and $l$-conformal Galilean symmetries,” arXiv:1905.13154 [hep-th].
  * [15] A. Cappelli, C. A. Trugenberger, and G. R. Zemba, “Infinite symmetry in the quantum Hall effect,” Nucl. Phys. B 396 (1993) 465–490, arXiv:hep-th/9206027.
  * [16] S. Iso, D. Karabali, and B. Sakita, “Fermions in the lowest Landau level: Bosonization, W infinity algebra, droplets, chiral bosons,” Phys. Lett. B 296 (1992) 143–150, arXiv:hep-th/9209003.
  * [17] A. Cappelli and L. Maffi, “W-infinity Symmetry in the Quantum Hall Effect Beyond the Edge,” JHEP 05 (2021) 120, arXiv:2103.04163 [hep-th].
  * [18] S. M. Girvin, A. H. MacDonald, and P. M. Platzman, “Magneto-roton theory of collective excitations in the fractional quantum Hall effect,” Phys. Rev. B 33 (1986) 2481–2494.
  * [19] S. Golkar, D. X. Nguyen, M. M. Roberts, and D. T. Son, “Higher-Spin Theory of the Magnetorotons,” Phys. Rev. Lett. 117 no. 21, (2016) 216403, arXiv:1602.08499 [cond-mat.mes-hall].
  * [20] A. Gromov and D. T. Son, “Bimetric Theory of Fractional Quantum Hall States,” Phys. Rev. X 7 no. 4, (2017) 041032, arXiv:1705.06739 [cond-mat.str-el]. [Addendum: Phys.Rev.X 8, 019901 (2018)].
  * [21] R. M. Nandkishore and M. Hermele, “Fractons,” Ann. Rev. Condensed Matter Phys. 10 (2019) 295–313, arXiv:1803.11196 [cond-mat.str-el].
  * [22] K. T. Grosvenor, C. Hoyos, F. Peña Benitez, and P. Surówka, “Space-Dependent Symmetries and Fractons,” Front. in Phys. 9 (2022) 792621, arXiv:2112.00531 [hep-th].
  * [23] M. Taylor, “Non-relativistic holography,” arXiv:0812.0530 [hep-th].
  * [24] K. Balasubramanian and J. McGreevy, “Gravity duals for non-relativistic CFTs,” Phys. Rev. Lett. 101 (2008) 061601, arXiv:0804.4053 [hep-th].
  * [25] C. P. Herzog, M. Rangamani, and S. F. Ross, “Heating up Galilean holography,” JHEP 11 (2008) 080, arXiv:0807.1099 [hep-th].
  * [26] A. Bagchi and R. Gopakumar, “Galilean Conformal Algebras and AdS/CFT,” JHEP 07 (2009) 037, arXiv:0902.1385 [hep-th].
  * [27] C. Duval and P. A. Horvathy, “Exotic Galilean symmetry in the noncommutative plane, and the Hall effect,” J. Phys. A34 (2001) 10097–10108, arXiv:hep-th/0106089 [hep-th].
  * [28] D. T. Son, “Toward an AdS/cold atoms correspondence: A Geometric realization of the Schrodinger symmetry,” Phys. Rev. D 78 (2008) 046003, arXiv:0804.3972 [hep-th].
  * [29] M. Geracie, D. T. Son, C. Wu, and S.-F. Wu, “Spacetime Symmetries of the Quantum Hall Effect,” Phys. Rev. D91 (2015) 045030, arXiv:1407.1252 [cond-mat.mes-hall].
  * [30] A. Gromov and A. G. Abanov, “Thermal Hall Effect and Geometry with Torsion,” Phys. Rev. Lett. 114 (2015) 016802, arXiv:1407.2908 [cond-mat.str-el].
  * [31] A. Campoleoni and S. Pekar, “Carrollian and Galilean conformal higher-spin algebras in any dimensions,” JHEP 02 (2022) 150, arXiv:2110.07794 [hep-th].
  * [32] M. Hatsuda and M. Sakaguchi, “Wess-Zumino term for the AdS superstring and generalized Inonu-Wigner contraction,” Prog. Theor. Phys. 109 (2003) 853–867, arXiv:hep-th/0106114 [hep-th].
  * [33] J. A. de Azcarraga, J. M. Izquierdo, M. Picon, and O. Varela, “Generating Lie and gauge free differential (super)algebras by expanding Maurer-Cartan forms and Chern-Simons supergravity,” Nucl. Phys. B662 (2003) 185–219, arXiv:hep-th/0212347 [hep-th].
  * [34] N. Boulanger, M. Henneaux, and P. van Nieuwenhuizen, “Conformal (super)gravities with several gravitons,” JHEP 01 (2002) 035, arXiv:hep-th/0201023.
  * [35] F. Izaurieta, E. Rodriguez, and P. Salgado, “Expanding Lie (super)algebras through Abelian semigroups,” J. Math. Phys. 47 (2006) 123512, arXiv:hep-th/0606215 [hep-th].
  * [36] J. A. de Azcarraga, J. M. Izquierdo, M. Picon, and O. Varela, “Expansions of algebras and superalgebras and some applications,” Int. J. Theor. Phys. 46 (2007) 2738–2752, arXiv:hep-th/0703017 [HEP-TH].
  * [37] D. M. Peñafiel and L. Ravera, “Infinite S-Expansion with Ideal Subtraction and Some Applications,” J. Math. Phys. 58 no. 8, (2017) 081701, arXiv:1611.05812 [hep-th].
  * [38] E. Bergshoeff, J. M. Izquierdo, T. Ortín, and L. Romano, “Lie Algebra Expansions and Actions for Non-Relativistic Gravity,” JHEP 08 (2019) 048, arXiv:1904.08304 [hep-th].
  * [39] J. A. de Azcárraga, D. Gútiez, and J. M. Izquierdo, “Extended $D=3$ Bargmann supergravity from a Lie algebra expansion,” Nucl. Phys. B 946 (2019) 114706, arXiv:1904.12786 [hep-th].
  * [40] D. Hansen, J. Hartong, and N. A. Obers, “Gravity between Newton and Einstein,” Int. J. Mod. Phys. D 28 no. 14, (2019) 1944010, arXiv:1904.05706 [gr-qc].
  * [41] D. M. Peñafiel and P. Salgado-Rebolledo, “Non-relativistic symmetries in three space-time dimensions and the Nappi-Witten algebra,” Phys. Lett. B 798 (2019) 135005, arXiv:1906.02161 [hep-th].
  * [42] J. Gomis, A. Kleinschmidt, J. Palmkvist, and P. Salgado-Rebolledo, “Newton-Hooke/Carrollian expansions of (A)dS and Chern-Simons gravity,” JHEP 02 (2020) 009, arXiv:1912.07564 [hep-th].
  * [43] O. Kasikci, N. Ozdemir, M. Ozkan, and U. Zorba, “Three-dimensional higher-order Schrödinger algebras and Lie algebra expansions,” JHEP 04 (2020) 067, arXiv:2002.03558 [hep-th].
  * [44] P. Concha, M. Ipinza, L. Ravera, and E. Rodríguez, “Non-relativistic three-dimensional supergravity theories and semigroup expansion method,” JHEP 02 (2021) 094, arXiv:2010.01216 [hep-th].
  * [45] O. Kasikci and M. Ozkan, “Lie algebra expansions, non-relativistic matter multiplets and actions,” JHEP 01 (2022) 081, arXiv:2111.14568 [hep-th].
  * [46] J. Gomis and A. Kleinschmidt, “Infinite-Dimensional Algebras as Extensions of Kinematic Algebras,” Front. in Phys. 10 (2022) 892812, arXiv:2202.05026 [hep-th].
  * [47] E. Ekiz, O. Kasikci, M. Ozkan, C. B. Senisik, and U. Zorba, “Non-relativistic and ultra-relativistic scaling limits of multimetric gravity,” JHEP 10 (2022) 151, arXiv:2207.07882 [hep-th].
  * [48] N. González, G. Rubio, P. Salgado, and S. Salgado, “Generalized Galilean algebras and Newtonian gravity,” Phys. Lett. B755 (2016) 433–438, arXiv:1604.06313 [hep-th].
  * [49] R. Caroca, P. Concha, O. Fierro, E. Rodríguez, and P. Salgado-Rebolledo, “Generalized Chern–Simons higher-spin gravity theories in three dimensions,” Nucl. Phys. B 934 (2018) 240–264, arXiv:1712.09975 [hep-th].
  * [50] P. K. Concha, R. Durka, C. Inostroza, N. Merino, and E. K. Rodríguez, “Pure Lovelock gravity and Chern-Simons theory,” Phys. Rev. D94 no. 2, (2016) 024055, arXiv:1603.09424 [hep-th].
  * [51] F. Izaurieta, E. Rodriguez, P. Minning, P. Salgado, and A. Perez, “Standard General Relativity from Chern-Simons Gravity,” Phys. Lett. B 678 (2009) 213–217, arXiv:0905.2187 [hep-th].
  * [52] E. Bergshoeff, J. Figueroa-O’Farrill, and J. Gomis, “A non-lorentzian primer,” arXiv:2206.12177 [hep-th].
  * [53] E. Inönü and E. P. Wigner, “On the contraction of groups and their representations,” in The Collected Works of Eugene Paul Wigner, pp. 488–502.  Springer, 1993.
  * [54] P. Concha, C. Henríquez-Báez, and E. Rodríguez, “Non-relativistic and Ultra-relativistic Expansions of Three-dimensional Spin-3 Gravity Theories,” arXiv:2208.01013 [hep-th].
  * [55] E. J. Saletan, “Contraction of lie groups,” Journal of Mathematical physics 2 no. 1, (1961) 1–21.
  * [56] D. I. Olive, E. Rabinovici, and A. Schwimmer, “A Class of string backgrounds as a semiclassical limit of WZW models,” Phys. Lett. B 321 (1994) 361–364, arXiv:hep-th/9311081.
  * [57] K. Sfetsos, “Gauged WZW models and nonAbelian duality,” Phys. Rev. D 50 (1994) 2784–2798, arXiv:hep-th/9402031.
  * [58] C. R. Nappi and E. Witten, “A WZW model based on a nonsemisimple group,” Phys. Rev. Lett. 71 (1993) 3751–3753, arXiv:hep-th/9310112 [hep-th].
  * [59] J. M. Figueroa-O’Farrill and S. Stanciu, “More D-branes in the Nappi-Witten background,” JHEP 01 (2000) 024, arXiv:hep-th/9909164 [hep-th].
  * [60] G. Papageorgiou and B. J. Schroers, “A Chern-Simons approach to Galilean quantum gravity in 2+1 dimensions,” JHEP 11 (2009) 009, arXiv:0907.2880 [hep-th].
  * [61] P. D. Alvarez, J. Gomis, K. Kamimura, and M. S. Plyushchay, “(2+1)D Exotic Newton-Hooke Symmetry, Duality and Projective Phase,” Annals Phys. 322 (2007) 1556–1586, arXiv:hep-th/0702014.
  * [62] L. Avilés, E. Frodden, J. Gomis, D. Hidalgo, and J. Zanelli, “Non-Relativistic Maxwell Chern-Simons Gravity,” JHEP 05 (2018) 047, arXiv:1802.08453 [hep-th].
  * [63] E. A. Bergshoeff and J. Rosseel, “Three-Dimensional Extended Bargmann Supergravity,” Phys. Rev. Lett. 116 no. 25, (2016) 251601, arXiv:1604.08042 [hep-th].
  * [64] J. Hartong, Y. Lei, and N. A. Obers, “Nonrelativistic Chern-Simons theories and three-dimensional Horava-Lifshitz gravity,” Phys. Rev. D94 no. 6, (2016) 065027, arXiv:1604.08054 [hep-th].
  * [65] J. Matulich, S. Prohazka, and J. Salzer, “Limits of three-dimensional gravity and metric kinematical Lie algebras in any dimension,” JHEP 07 (2019) 118, arXiv:1903.09165 [hep-th].
  * [66] H. Bacry and J. Levy-Leblond, “Possible kinematics,” J. Math. Phys. 9 (1968) 1605–1614.
  * [67] Y. Brihaye, C. Gonera, S. Giller, and P. Kosinski, “Galilean invariance in (2+1)-dimensions,” arXiv:hep-th/9503046.
  * [68] C. Duval and P. A. Horvathy, “Spin and exotic Galilean symmetry,” Phys. Lett. B 547 (2002) 306–312, arXiv:hep-th/0209166. [Erratum: Phys.Lett.B 588, 228–228 (2004)].
  * [69] C. R. Hagen, “Second central extension in Galilean covariant field theory,” Phys. Lett. B 539 (2002) 168–171, arXiv:quant-ph/0203109.
  * [70] G. Barnich and B. Oblak, “Notes on the BMS group in three dimensions: I. Induced representations,” JHEP 06 (2014) 129, arXiv:1403.5803 [hep-th].
  * [71] A. Campoleoni, H. A. Gonzalez, B. Oblak, and M. Riegler, “BMS Modules in Three Dimensions,” Int. J. Mod. Phys. A 31 no. 12, (2016) 1650068, arXiv:1603.03812 [hep-th].
  * [72] H. Bacry, P. Combe, and J. L. Richard, “Group-theoretical analysis of elementary particles in an external electromagnetic field. 1. the relativistic particle in a constant and uniform field,” Nuovo Cim. A 67 (1970) 267–299.
  * [73] R. Schrader, “The maxwell group and the quantum theory of particles in classical homogeneous electromagnetic fields,” Fortsch. Phys. 20 (1972) 701–734.
  * [74] G. Rubio and P. Salgado, “Modified Newtonian dynamics and non-relativistic ChSAS gravity,” Phys. Lett. B787 (2018) 30–35, arXiv:1811.01717 [hep-th].
  * [75] J. Gomis, A. Kleinschmidt, and J. Palmkvist, “Galilean free Lie algebras,” JHEP 09 (2019) 109, arXiv:1907.00410 [hep-th].
  * [76] H. Bacry, P. Combe, and J. L. Richard, “Group-theoretical analysis of elementary particles in an external electromagnetic field. 2. the nonrelativistic particle in a constant and uniform field,” Nuovo Cim. A 70 (1970) 289–312.
  * [77] J. Beckers and V. Hussin, “Minimal electromagnetic coupling schemes. II. Relativistic and nonrelativistic Maxwell groups,” J. Math. Phys. 24 (1983) 1295–1298.
  * [78] J. Negro and M. A. del Olmo, “Local realizations of kinematical groups with a constant electromagnetic field. ii. the nonrelativistic case,” Journal of Mathematical Physics 31 no. 12, (1990) 2811–2821.
  * [79] S. Bansal and D. Sorokin, “Can Chern-Simons or Rarita-Schwinger be a Volkov-Akulov Goldstone?,” JHEP 07 (2018) 106, arXiv:1806.05945 [hep-th].
  * [80] D. Chernyavsky, N. S. Deger, and D. Sorokin, “Spontaneously broken $3d$ Hietarinta/Maxwell Chern–Simons theory and minimal massive gravity,” Eur. Phys. J. C 80 no. 6, (2020) 556, arXiv:2002.07592 [hep-th].
  * [81] P. Salgado-Rebolledo, “The Maxwell group in 2+1 dimensions and its infinite-dimensional enhancements,” JHEP 10 (2019) 039, arXiv:1905.09421 [hep-th].
  * [82] S. Fedoruk and J. Lukierski, “Maxwell group and HS field theory,” J. Phys. Conf. Ser. 474 (2013) 012016, arXiv:1309.6878 [hep-th].
  * [83] P. D. Alvarez, J. Gomis, K. Kamimura, and M. S. Plyushchay, “Anisotropic harmonic oscillator, non-commutative Landau problem and exotic Newton-Hooke symmetry,” Phys. Lett. B659 (2008) 906–912, arXiv:0711.2644 [hep-th].
  * [84] J. Hartong, Y. Lei, N. A. Obers, and G. Oling, “Zooming in on AdS3/CFT2 near a BPS bound,” JHEP 05 (2018) 016, arXiv:1712.05794 [hep-th].
  * [85] E. Joung and W. Li, “Nonrelativistic limits of colored gravity in three dimensions,” Phys. Rev. D97 no. 10, (2018) 105020, arXiv:1801.10143 [hep-th].
  * [86] D. V. Soroka and V. A. Soroka, “Semi-simple extension of the (super)Poincare algebra,” Adv. High Energy Phys. 2009 (2009) 234147, arXiv:hep-th/0605251 [hep-th].
  * [87] J. Gomis, K. Kamimura, and J. Lukierski, “Deformations of Maxwell algebra and their Dynamical Realizations,” JHEP 08 (2009) 039, arXiv:0906.4464 [hep-th].
  * [88] P. Concha, N. Merino, E. Rodríguez, P. Salgado-Rebolledo, and O. Valdivia, “Semi-simple enlargement of the $\mathfrak{bms}_{3}$ algebra from a $\mathfrak{so}(2,2)\oplus\mathfrak{so}(2,1)$ Chern-Simons theory,” JHEP 02 (2019) 002, arXiv:1810.12256 [hep-th].
  * [89] G. Papageorgiou and B. J. Schroers, “Galilean quantum gravity with cosmological constant and the extended $q$-Heisenberg algebra,” JHEP 11 (2010) 020, arXiv:1008.0279 [hep-th].
  * [90] P. Concha and E. Rodríguez, “Non-Relativistic Gravity Theory based on an Enlargement of the Extended Bargmann Algebra,” JHEP 07 (2019) 085, arXiv:1906.00086 [hep-th].
  * [91] P. Salgado-Rebolledo and G. Palumbo, “Extended Nappi-Witten Geometry for the Fractional Quantum Hall Effect,” Phys. Rev. D 103 (2021) 125006, arXiv:2102.03886 [cond-mat.mes-hall].
  * [92] P. Salgado-Rebolledo and G. Palumbo, “Nonrelativistic supergeometry in the Moore-Read fractional quantum Hall state,” Phys. Rev. D 106 no. 6, (2022) 065020, arXiv:2112.14339 [cond-mat.mes-hall].
  * [93] A. Cappelli and E. Randellini, “Multipole Expansion in the Quantum Hall Effect,” JHEP 03 (2016) 105, arXiv:1512.02147 [cond-mat.str-el].
  * [94] A. Gromov, E. J. Martinec, and S. Ryu, “Collective excitations at filling factor 5/2: The view from superspace,” Phys. Rev. Lett. 125 no. 7, (2020) 077601, arXiv:1909.06384 [cond-mat.str-el].
  * [95] O. Coussaert, M. Henneaux, and P. van Driel, “The Asymptotic dynamics of three-dimensional Einstein gravity with a negative cosmological constant,” Class. Quant. Grav. 12 (1995) 2961–2966, arXiv:gr-qc/9506019.


* [177] M. Byra, M. Wu, X. Zhang, H. Jang, Y.-J. Ma, E. Y. Chang, S. Shah, and J. Du, “Knee menisci segmentation and relaxometry of 3d ultrashort echo time cones mr imaging using attention u-net with transfer learning,” _Magnetic Resonance in Medicine_ , vol. 83, no. 3, pp. 1109–1122, 2020.
  * [178] C. Tan, F. Sun, T. Kong, W. Zhang, C. Yang, and C. Liu, “A survey on deep transfer learning,” in _International conference on artificial neural networks_. Springer, 2018, pp. 270–279.
  * [179] S. Mishra, B. L. Sturm, and S. Dixon, “Local interpretable model-agnostic explanations for music content analysis.” in _ISMIR_ , 2017, pp. 537–543.
  * [180] S. M. Lundberg and S.-I. Lee, “A unified approach to interpreting model predictions,” in _Advances in neural information processing systems_ , 2017, pp. 4765–4774.
  * [181] J. H. Friedman, “Greedy function approximation: a gradient boosting machine,” _Annals of statistics_ , pp. 1189–1232, 2001.
  * [182] M. T. Ribeiro, S. Singh, and C. Guestrin, “Anchors: High-precision model-agnostic explanations.” in _AAAI_ , vol. 18, 2018, pp. 1527–1535.

[]Narinder Singh Punn received his Bachelor’s degree in Computer Science and
Engineering from National Institute of Technology Hamirpur, India in 2015. He
is is presently pursuing his PhD from Indian Institute of Information
Technology Allahabad, India. His research areas are: machine learning, deep
learning and biomedical image analysis.

[]Sonali Agarwal is presently working as Associate Professor in department of
information technology at Indian Institute of Information Technology
Allahabad, India. Her research interests are in the areas of stream analytics,
big data, stream data mining, complex event processing system, deep learning,
support vector machines and software engineering.