# Bicrossproduct vs. twist quantum symmetries in noncommutative geometries:
the case of $\varrho$-Minkowski

Giuseppe Fabiano<EMAIL_ADDRESS>Dipartimento di Fisica “Ettore
Pancini”, Università di Napoli Federico II, Napoli, Italy INFN, Sezione di
Napoli, Italy Giulia Gubitosi<EMAIL_ADDRESS><EMAIL_ADDRESS>Dipartimento di Fisica “Ettore Pancini”, Università
di Napoli Federico II, Napoli, Italy INFN, Sezione di Napoli, Italy Fedele
Lizzi<EMAIL_ADDRESS><EMAIL_ADDRESS>Dipartimento di Fisica
“Ettore Pancini”, Università di Napoli Federico II, Napoli, Italy INFN,
Sezione di Napoli, Italy Departament de Física Quàntica i Astrofísica and
Institut de Cíencies del Cosmos (ICCUB), Universitat de Barcelona, Barcelona,
Spain Luca Scala<EMAIL_ADDRESS><EMAIL_ADDRESS>University of
Wrocław, Faculty of Physics and Astronomy, Maksa Borna 9, 50-204 Wrocław,
Poland Patrizia Vitale<EMAIL_ADDRESS><EMAIL_ADDRESS>Dipartimento di Fisica “Ettore Pancini”, Università di Napoli Federico II,
Napoli, Italy INFN, Sezione di Napoli, Italy

###### Abstract

We discuss the quantum Poincaré symmetries of the $\varrho$-Minkowski
spacetime, a space characterised by an angular form of noncommutativity. We
show that it is possible to give them both a bicrossproduct and a Drinfel’d
twist structure. We also obtain a new noncommutative $\star$-product, which is
cyclic with respect to the standard integral measure.

###### Contents

  1. 1 Introduction
  2. 2 The $\varrho$-Minkowski spacetime
  3. 3 The twisted Hopf algebra $U_{\varrho}(\mathfrak{p})$
  4. 4 The bicrossproduct construction
    1. 4.1 The bicrossproduct structure of the algebra $\mathcal{C}_{\varrho}(P)$
    2. 4.2 The bicrossproduct structure of the algebra $\mathcal{U}_{\varrho}(\mathfrak{p})$
    3. 4.3 The algebra isomorphism between $\mathcal{U}_{\varrho}(\mathfrak{p})$ and ${U}_{\varrho}(\mathfrak{p})$
      1. 4.3.1 The $\varrho$-Minkowski spacetime re-derived
  5. 5 Plane waves and $\star$-product in $\varrho$-Minkowski
    1. 5.1 Time-to-the-right ordering and bicrossproduct basis
      1. 5.1.1 The $\star$-product
    2. 5.2 Time-symmetric ordering and twist basis
      1. 5.2.1 Equivalence class of star products
  6. 6 Conclusions
  7. A The twist
  8. B The bicrossproduct structure
    1. B.1 The classical $r$-matrix deformation method
    2. B.2 The Quantum Group $\mathcal{C}_{\kappa}(P)$
  9. C Proof of Proposition 4.1
  10. D Some explicit calculations of the commutation relations for $\mathcal{U}_{\varrho}(\mathfrak{p})$
  11. E Plane waves in $\kappa$-Minkowski

##  1 Introduction

Quantization of gravity is likely to require some form of quantum spacetime,
which may be effectively described by a noncommutative geometry. This, in
turn, can be defined by noncommutativity between the spacetime coordinates,
similar to what happens with the coordinates of phase space in quantum
mechanics. Such nontrivial commutation relations are generally not covariant
under the action of classical relativity groups. Consequently, quantum
spacetimes require the introduction of quantum symmetries, which are realized
by quantum groups of isometries. In the present work we shall refer in
particular to Poincaré symmetries.

The latter can be approached either in terms of the noncommutative algebra of
continuous functions over the Poincaré group, the quantum Hopf algebra
$\mathcal{C}_{\star}(P)$, or in terms of the quantum Hopf algebra
$U_{\star}(\mathfrak{p})$ obtained from the universal enveloping algebra of
the Poincaré Lie algebra (the subscript indicating the specific deformation
under analysis). From the general theory of Hopf algebras, it is possible to
show that these two structures are Hopf-dual. The first one, owing to its
group-like properties, may be regarded as a “quantum symmetry group”, while
the second one can be regarded as a “quantum symmetry algebra”.

Quantum Hopf algebras can be built in different ways. Two of the most commonly
used structures are the _bicrossproduct_ and _twist_. The bicrossproduct
structure [1] can be regarded as a quantum generalization of the usual
semidirect product of classical groups. According to [1] its mathematical
structure is that of a Hopf algebra extension. It may concern both the algebra
of functions on the group $\mathcal{C}_{\star}(P)$ and the universal
enveloping of the Lie algebra $U_{\star}(\mathfrak{p})$. Therefore, as in the
classical case, it is possible to derive the associated noncommutative
spacetime in two ways.

One approach involves taking the quotient of $\mathcal{C}_{\star}(P)$ with
respect to the Lorentz Hopf algebra $\mathcal{C}(SO(1,3))$ (which we shall see
to be undeformed), so to obtain the (deformed) translations sector that is
isomorphic to the spacetime, as in the classical case. The second method
entails identifying the space on which $U_{\star}(\mathfrak{p})$ acts
covariantly, as the spacetime.

The twist approach [2] relies on an operator-valued map defined in terms of
the generators of some Lie algebra of symmetries. It may deform any bilinear
map of a given theory which carries a representation of the Lie algebra. It
affects both the algebra and the coalgebra structure of the associated Hopf
algebra of symmetries and, when applied to the product of functions on the
spacetime, it generates noncommutativity.

The bicrossproduct and the twist structures are usually considered as
alternative options for characterizing the quantum symmetries of
noncommutative spacetimes. Available examples generally deal exclusively with
one of these two approaches.111See however [3] where the light-like
$\kappa$-Minkowski is associated with two quantum groups of symmetries, one is
a bicrossproduct, the other is obtained with a twist. It is nevertheless
interesting to look for quantum isometries in both frameworks, as associated
with the same noncommutative spacetime.

The $\kappa$-Poincaré quantum group in the so-called “Majid-Ruegg basis”
(sometimes called “bicrossproduct basis”) is the most famous quantum group
with a bicrossproduct structure [1]. It describes the symmetries of the
$\kappa$-Minkowski spacetime algebra [4]. While the $\kappa$-Poincaré algebra
was first obtained in [5, 6] by contraction [7], it was also obtained later
within a twist approach [8, 9, 10, 11, 12, 13, 14]. However this requires to
enlarge the symmetry algebra, so to include at least the generator of Weyl
transformations.

On the other hand, the Moyal algebra is an example of a noncommutative
spacetime with deformed Poincaré symmetries which can only be obtained by a
twist. Indeed, it is possible to check that it is not the quotient of a
bicrossproduct structure [15, 16]. Its quantum symmetry group, the so-called
$\theta$-Poincaré, was first discussed in [17, 18, 19, 20].

In this paper we explore the relation between the twist and bicrossproduct
constructions. We provide an example of a noncommutative spacetime, dubbed
$\varrho$-Minkowski, whose quantum symmetries can be defined in terms of both
structures. While this spacetime was originally proposed in the context of
twist deformations of the Poincaré group [16], we show that its symmetries can
be naturally described also in terms of a bicrossproduct quantum group. The
deformed Poincaré symmetries obtained following the two approaches are
isomorphic quantum groups related one to the other by a nonlinear change of
generators. We shall indicate them with $U_{\varrho}(\mathfrak{p})$ and
$\mathcal{U}_{\varrho}(\mathfrak{p})$. They give rise to different
$\star$-products, which are both cyclic with respect to the standard
integration measure, an important property towards formulating gauge invariant
theories. This particular model can therefore serve as a bridge between the
two approaches, facilitating a comparison of the relevant structures and their
physical interpretation.

The paper is organised as follows. In Section 2 we introduce the
$\varrho$-Minkowski spacetime and shortly describe the $\kappa$-Minkowski one,
which will serve as a guiding example all over the paper. In Section 3 we
review the quantum enveloping algebra $U_{\varrho}(\mathfrak{p})$ [21]
describing the twisted quantum symmetries of the $\varrho$-Minkowski
spacetime, and recall the related $\star$-product.

Sections 4 and 5 contain the main original results of the paper. In Section
4.1 we describe $\mathcal{C}_{\varrho}(P)$, previously introduced in [22, 23]
via a classical $r$-matrix, and we show by direct calculation that it has a
standard bicrossproduct structure. We thus analyse the Hopf algebras
$U_{\varrho}(\mathfrak{p})$ and $\mathcal{U}_{\varrho}(\mathfrak{p})$ which
are both Hopf dual to $\mathcal{C}_{\varrho}(P)$. In Section 4.2 we prove that
$U_{\varrho}(\mathfrak{p})$ is not of the bicrossproduct kind. We thus
introduce a suitable non-linear change of generators of the Hopf algebra to
obtain the quantum enveloping algebra $\mathcal{U}(\mathfrak{p})$, which has a
bicrossproduct structure. We apply both a constructive method, as well as a
more abstract one, proposed by Majid and Ruegg in the $\kappa$-Poincaré
context in [1]. The quantum enveloping algebras $U_{\varrho}(\mathfrak{p})$
and $\mathcal{U}_{\varrho}(\mathfrak{p})$ are shown in Sec. 4.3 to be
isomorphic. They are both Hopf dual to $\mathcal{C}_{\varrho}(P)$, but only
the second one is bicrossproduct dual. In analogy with the classical and
$\kappa$-Minkowski spacetime, we finally re-derive the $\varrho$-Minkowski
spacetime as an appropriate quotient of the dual
$\mathcal{U}^{*}_{\varrho}(\mathfrak{p})$ with respect to the action of the
Lorentz sub-algebra.

In Sec. 5 we realise the algebra of noncommuting coordinate functions as
finite-dimensional operators and construct operator-valued plane waves in
terms of which we define a $\star$-product. Depending on the ordering
prescription (time-to-the-right or time-symmetric) we obtain compatible
coproducts which are either the bicrossproduct or the twisted one. The time-
to-the-right ordering is related with the bicrossproduct structure, in analogy
with the $\kappa$-Minkowski case. The time-symmetric ordering, related with
the twist, does not have a direct analogue in the $\kappa$-Poincaré case,
where the same ordering yields the ’standard’ $\kappa$-Poincaré basis, not
descending from a twist.

Finally, we retrieve the twisted star product already found in [24] as the one
associated with the time-symmetric ordering, and find a novel $\star$-product,
related with the bicrossproduct basis. We show that, similarly to the twisted
one, it is cyclic. However, differently from the latter, it is not closed.
This fact could have interesting consequences to be compared with the results
of [24, 21] where the twisted product has been employed in scalar field theory
models.

A section dedicated to summary and discussion and five appendices containing
technical details and review material conclude the paper.

##  2 The $\varrho$-Minkowski spacetime

In this section we introduce the $\varrho$-Minkowski spacetime, after
reviewing the more popular and well studied $\kappa$-Minkowski one, which we
shall use as a guiding example all over the paper. We work here with a
Lorentzian metric $g_{\mu\nu}$ with signature $(+,-,-,-)$.

### The $\kappa$-Minkowski spacetime

The $\kappa$-Minkowski spacetime is defined by the commutators

$[x^{\mu},x^{\nu}]=i\ell(v^{\mu}x^{\nu}-x^{\mu}v^{\nu}),$ (2.1)

where we use a length (rather than energy) deformation parameter
$\ell=1/\kappa$, for homogeneity with the notation used for
$\varrho$-Minkowski. The 4-vector $v^{\mu}$ characterizes physically different
models, based on whether it is timelike, null or spacelike with respect to the
metric $g_{\mu\nu}$ [25]. The timelike $\kappa$-Minkowski with commutation
relations

$[x^{0},x^{i}]=i\ell x^{i},\qquad[x^{i},x^{j}]=0,$ (2.2)

has been under the spotlight of theoretical investigations for several decades
(see for example [26, 27, 28, 29, 30, 31, 32, 33]), given its relevance in
phenomenological approaches to the quantum gravity problem [34]. The symmetry
group of the noncommutative spacetime (2.2) is known as $\kappa$-Poincaré [5,
6, 35] and was studied in several works regarding quantum deformations of the
Poincaré algebra (see for example [36, 37, 38]). As we will show, in the
context of the bicrossproduct structure of the deformed Poincaré symmetries
the commutation relations (2.2) were derived as the dual algebra with respect
to the deformed translations generators [1]. This accounts for one of the most
interesting properties of the bicrossproduct construction: the possibility of
defining a procedure similar to that of the classical case, in which the
Minkowski spacetime can be viewed as the dual space to the translation sector
of the Poincaré algebra.

### The $\varrho$-Minkowski spacetime

The $\varrho$-Minkowski spacetime is defined by the commutation relations

$\displaystyle[x^{1},x^{0}]$ $\displaystyle=i\varrho x^{2}\,,$
$\displaystyle[x^{2},x^{0}]$ $\displaystyle=-i\varrho x^{1}\,,$ (2.3)

all other commutators being zero.

This is a sort of angular noncommutativity, in that the time variable acts as
the generator of rotations in the $(x_{1},x_{2})$ plane. From the measurement
point of view, when the coordinates are represented as operators on a Hilbert
space, its spectrum will therefore be discrete [22].

The noncommutative spacetime (2.3), which has the structure of the Euclidean
algebra in 2+1 dimensions (the coordinate $x^{3}$ being central), was first
considered in [39] (also see [40]). It has been analyzed within the twist
approach in [16]. This kind of noncommutative spacetime might be physically
interesting, as argued in [41, 42, 43, 21] and phenomenologically relevant in
relation with relative locality [44]. Noncommutative field models with such
underlying spacetime have been studied in [24, 45] and, in the semiclassical
approach, by [46] in the context of Poisson gauge theory. The properties of
observers and localization problems in this spacetime were studied in [23].

A similar type of angular noncommutativity, when time is a commutative
variable while $x^{3}$ plays the role of a proper spatial rotation, is
considered in [47] in the context of a double quantization of both spacetime
and phase space. In order to distinguish it from the previous one, the latter
has been dubbed $\lambda$-Minkowski spacetime and is characterized by
commutation relations analogous to (2.3), with the roles of $x^{3}$ and
$x^{0}$ exchanged. Keeping this difference in mind, the analysis carried out
for $\varrho$-Minkowski in the subsequent sections can be repeated also for
the case of $\lambda$-Minkowski, with minor changes.

##  3 The twisted Hopf algebra $U_{\varrho}(\mathfrak{p})$

Given a Lie algebra, $\mathfrak{g}$, its universal enveloping algebra,
$U(\mathfrak{g})$, may be deformed to a quantum Hopf algebra if a twist
operator is available. This is an invertible map $\mathcal{F}\in
U(\mathfrak{g})\otimes U(\mathfrak{g})$ (called Drinfel’d twist), which
satisfies cocycle and normalization conditions. We shortly review in App. A
some of its properties. When an admissible twist is available, it is possible
to define both a quantum spacetime and its quantum symmetry group.

The Drinfel’d twist for the $\varrho$-Minkowski case was first introduced in
[16]:

$\mathcal{F}=e^{\frac{i\varrho}{2}[P_{0}\wedge M_{12}]}\,,$ (3.1)

with $P_{\mu}$, $M_{\mu\nu}$ translations and Lorentz generators of the
Poincaré algebra $\mathfrak{p}$, respectively. The corresponding quantum Hopf
algebra, $U(\mathfrak{p})$, was first introduced in [42]. The twisted
coproducts of the Lie algebra generators, which we shall need in the
forthcoming sections, read

$\begin{array}[]{r@{}l}\Delta_{\mathcal{F}}P_{0}=&P_{0}\otimes 1+1\otimes
P_{0}\,,\\\
\Delta_{\mathcal{F}}P_{1}=&P_{1}\otimes\cos\left(\frac{\varrho}{2}P_{0}\right)+\cos\left(\frac{\varrho}{2}P_{0}\right)\otimes
P_{1}+P_{2}\otimes\sin\left(\frac{\varrho}{2}P_{0}\right)-\sin\left(\frac{\varrho}{2}P_{0}\right)\otimes
P_{2}\,,\\\
\Delta_{\mathcal{F}}P_{2}=&P_{2}\otimes\cos\left(\frac{\varrho}{2}P_{0}\right)+\cos\left(\frac{\varrho}{2}P_{0}\right)\otimes
P_{2}-P_{1}\otimes\sin\left(\frac{\varrho}{2}P_{0}\right)+\sin\left(\frac{\varrho}{2}P_{0}\right)\otimes
P_{1}\,,\\\ \Delta_{\mathcal{F}}P_{3}=&P_{3}\otimes 1+1\otimes P_{3}\,,\\\
\Delta_{\mathcal{F}}M_{12}=&M_{12}\otimes 1+1\otimes M_{12}\,,\\\
\Delta_{\mathcal{F}}M_{13}=&M_{13}\otimes\cos\left(\frac{\varrho}{2}P_{0}\right)+\cos\left(\frac{\varrho}{2}P_{0}\right)\otimes
M_{13}-M_{32}\otimes\sin\left(\frac{\varrho}{2}P_{0}\right)+\sin\left(\frac{\varrho}{2}P_{0}\right)\otimes
M_{32}\,,\\\
\Delta_{\mathcal{F}}M_{32}=&M_{32}\otimes\cos\left(\frac{\varrho}{2}P_{0}\right)+\cos\left(\frac{\varrho}{2}P_{0}\right)\otimes
M_{32}+M_{13}\otimes\sin\left(\frac{\varrho}{2}P_{0}\right)-\sin\left(\frac{\varrho}{2}P_{0}\right)\otimes
M_{13}\,,\\\
\Delta_{\mathcal{F}}M_{10}=&M_{10}\otimes\cos\left(\frac{\varrho}{2}P_{0}\right)+\cos\left(\frac{\varrho}{2}P_{0}\right)\otimes
M_{10}+M_{20}\otimes\sin\left(\frac{\varrho}{2}P_{0}\right)-\sin\left(\frac{\varrho}{2}P_{0}\right)\otimes
M_{20}\\\
&+P_{1}\otimes\frac{\varrho}{2}M_{12}\cos\left(\frac{\varrho}{2}P_{0}\right)-\frac{\varrho}{2}M_{12}\cos\left(\frac{\varrho}{2}P_{0}\right)\otimes
P_{1}\\\
&+P_{2}\otimes\frac{\varrho}{2}M_{12}\sin\left(\frac{\varrho}{2}P_{0}\right)+\frac{\varrho}{2}M_{12}\sin\left(\frac{\varrho}{2}P_{0}\right)\otimes
P_{2}\,,\\\
\Delta_{\mathcal{F}}M_{20}=&M_{20}\otimes\cos\left(\frac{\varrho}{2}P_{0}\right)+\cos\left(\frac{\varrho}{2}P_{0}\right)\otimes
M_{20}-M_{10}\otimes\sin\left(\frac{\varrho}{2}P_{0}\right)+\sin\left(\frac{\varrho}{2}P_{0}\right)\otimes
M_{10}\\\
&+P_{2}\otimes\frac{\varrho}{2}M_{12}\cos\left(\frac{\varrho}{2}P_{0}\right)-\frac{\varrho}{2}M_{12}\cos\left(\frac{\varrho}{2}P_{0}\right)\otimes
P_{2}\\\
&-P_{1}\otimes\frac{\varrho}{2}M_{12}\sin\left(\frac{\varrho}{2}P_{0}\right)-\frac{\varrho}{2}M_{12}\sin\left(\frac{\varrho}{2}P_{0}\right)\otimes
P_{1}\,,\\\ \Delta_{\mathcal{F}}M_{30}=&M_{30}\otimes 1+1\otimes
M_{30}+\frac{\varrho}{2}P_{3}\otimes M_{12}-\frac{\varrho}{2}M_{12}\otimes
P_{3}\,,\end{array}$ (3.2)

whereas the Lie algebra sector, the antipode and the counit are left
undeformed.

The homogeneous space on which the deformed symmetries act is characterised by
a noncommutative $\star$-product constructed with the inverse twist element
$\mathcal{F}^{-1}$. Following [42], we can express the twist in coordinate
representation

by realising $P_{\mu}$ as $-i\frac{\partial}{\partial x^{\mu}}$, and
$M_{\mu\nu}$ as $i(x_{\mu}\partial_{\nu}-x_{\nu}\partial_{\mu})$, to obtain

$\mathcal{F}=\exp\left\\{-\frac{i\varrho}{2}\left[\partial_{0}\otimes(x^{1}\partial_{2}-x^{2}\partial_{1})-(x^{1}\partial_{2}-x^{2}\partial_{1})\otimes\partial_{0}\right]\right\\}.$
(3.3)

The $\star$-product between two functions of the coordinates is defined as

$(f\star g)(x)\coloneqq\mu_{0}\circ\mathcal{F}^{-1}(f\otimes g)(x)\,,$ (3.4)

where $\mu_{0}$ is the ordinary product map. From the twisted $\star$-product,
the star-commutators between coordinate functions are immediately obtained,

$\displaystyle[x^{1},x^{0}]_{\star}=i\varrho x^{2}\,,$ (3.5)
$\displaystyle[x^{2},x^{0}]_{\star}=-i\varrho x^{1}\,,$

all the others being zero. They reproduce, as expected, the commutators (2.3).
We will derive analogous commutation relations in the context of the
bicrossproduct framework of quantum groups in the next section.

##  4 The bicrossproduct construction

Given two Hopf algebras $\mathcal{X},\mathcal{A}$, a bicrossproduct Hopf
algebra, $\mathcal{X}\vartriangleright\\!\blacktriangleleft\mathcal{A}$ [48],
is the tensor product $\mathcal{X}\otimes\mathcal{A}$ with Hopf algebra
operations, endowed with two structure maps, a covariant right action of
$\mathcal{X}$ on $\mathcal{A}$ and a covariant left coaction of $\mathcal{A}$
on $\mathcal{X}$, respectively

$\displaystyle\triangleleft$
$\displaystyle:\mathcal{A}\times\mathcal{X}\rightarrow\mathcal{A},$ (4.1a)
$\displaystyle\beta$
$\displaystyle:\mathcal{X}\rightarrow\mathcal{A}\otimes\mathcal{X},$ (4.1b)

which have to be compatible with the Hopf algebra structure. We review in App.
B the main definitions and properties. The meaning of the symbol
$\vartriangleright\\!\blacktriangleleft$ is that the first algebra acts on the
second from the right, while the second coacts back on the first from the
left, like a generalization of a semidirect product.

For the case under analysis, the bicross-structure of
$\mathcal{C}_{\varrho}(P)$ is obtained by identifying the algebra
$\mathcal{X}$ with a deformation of the Minkowski spacetime, while
$\mathcal{A}$ will be the algebra of functions on the Lorentz group. Dually,
the bicross-structure of the deformed universal enveloping $U(\mathfrak{p})$
is obtained by identifying $\mathcal{X}$ with $U(\mathfrak{so}(1,3))$ and
$\mathcal{A}$ with the deformed Lie algebra of translations.

###  4.1 The bicrossproduct structure of the algebra
$\mathcal{C}_{\varrho}(P)$

The quantum Hopf algebra $\mathcal{C}_{\varrho}(P)$ is obtained deforming the
commutative algebra of functions on the group manifold [22, 23]. It is
possible to obtain the noncommutative structure by quantizing the Poisson
bracket of coordinate functions associated with the classical $r$-matrix
(details are reported in App. B.1), which reads222This can be quickly verified
by expanding the twist (3.1) up to the first order in the deformation
parameter.

$r=i\varrho(P_{0}\wedge M_{12}).$ (4.2)

For comparison, an analogous analysis is summarized in App. B.2 for the
$\kappa$ case $\mathcal{C}_{\kappa}(P)$. Notice that the $\varrho$-Poincaré
$r$-matrix (4.2) satisfies the classical Yang-Baxter equation, differently
from the $r$-matrix of the $\kappa$ case, which is known to satisfy a modified
Yang-Baxter equation.

The noncommutative algebra of functions on the group manifold is obtained in
[23] by quantizing the Poisson brackets (B.11). The latter entails the
$r$-matrix (4.2), besides the left- and right-invariant vector fields of the
Poincaré group which are spelled out in Eqs. (B.12) for convenience. Similarly
to the $\kappa$-Minkowski case, the coproduct, antipode and counit of the
Lorentz sector are undeformed, while they are deformed for the translations
parameters. This result can be derived by imposing covariance of the
commutation relations (3.5) (analogously (2.2) for the $\kappa$ case) under
left coaction of the quantum group (see [23] for details). One finally obtains
the following quantum group structure:

$\displaystyle\left[a^{\mu},a^{\nu}\right]$
$\displaystyle=i\varrho[{\delta^{\nu}}_{0}(a^{2}{\delta^{\mu}}_{1}-a^{1}{\delta^{\mu}}_{2})-{\delta^{\mu}}_{0}(a^{2}{\delta^{\nu}}_{1}-a^{1}{\delta^{\nu}}_{2})]\,,$
(4.3a)
$\displaystyle\left[{\Lambda^{\mu}}_{\nu},{\Lambda^{\varrho}}_{\sigma}\right]$
$\displaystyle=0\,,$ (4.3b)
$\displaystyle\left[{\Lambda^{\mu}}_{\nu},a^{\rho}\right]$
$\displaystyle=i\varrho\left[{\Lambda^{\varrho}}_{0}({\Lambda^{\mu}}_{1}g_{2\nu}-{\Lambda^{\mu}}_{2}g_{1\nu})-{\delta^{\rho}}_{0}(\Lambda_{2\nu}{\delta^{\mu}}_{1}-\Lambda_{1\nu}{\delta^{\mu}}_{2})\right]\,,$
(4.3c) $\displaystyle\Delta({\Lambda^{\mu}}_{\nu})$
$\displaystyle={\Lambda^{\mu}}_{\alpha}\otimes{\Lambda^{\alpha}}_{\nu}\,,$
(4.3d) $\displaystyle S({\Lambda^{\mu}}_{\nu})$
$\displaystyle={(\Lambda^{-1})^{\mu}}_{\nu}\,,$ (4.3e)
$\displaystyle\varepsilon({\Lambda^{\mu}}_{\nu})$
$\displaystyle={\delta^{\mu}}_{\nu}\,,$ (4.3f) $\displaystyle\Delta(a^{\mu})$
$\displaystyle={\Lambda^{\mu}}_{\nu}\otimes a^{\nu}+a^{\mu}\otimes 1\,,$
(4.3g) $\displaystyle S(a^{\mu})$
$\displaystyle=-a^{\nu}{(\Lambda^{-1})^{\mu}}_{\nu}\,,$ (4.3h)
$\displaystyle\varepsilon(a^{\mu})$ $\displaystyle=0\,.$ (4.3i)

The $\varrho$-Minkowski spacetime in this approach is assigned a priori. It
determines its quantum group of symmetries, by requiring that it be the
noncommutative algebra of continuous functions on the spacetime, which is
covariant under the left coaction of the group: $x^{\mu}\rightarrow
x^{\mu^{\prime}}=\Lambda^{\mu}{}_{\nu}\otimes x^{\nu}+a^{\mu}\otimes 1$. We
shall see in Sec. 4.3.1 a dual picture, where the same spacetime emerges as an
appropriate quotient of the quantum universal enveloping algebra of the
Poincaré algebra $\mathfrak{p}$.

###### Proposition 4.1

The quantum Hopf algebra $\mathcal{C}_{\varrho}(P)$ has a bicrossproduct
structure, namely

$\mathcal{C}_{\varrho}(P)=\mathcal{T}_{\varrho}^{*}\vartriangleright\\!\blacktriangleleft\mathcal{C}(SO(1,3)),$
(4.4)

where $\mathcal{C}(SO(1,3))$ is the classical undeformed (Hopf) algebra of
continuous functions over the Lorentz group and $\mathcal{T}_{\varrho}^{*}$ is
the noncommutative (Hopf) algebra of functions of $\varrho$-Minkowski
spacetime, given by:

$\displaystyle[x^{\mu},x^{\nu}]$
$\displaystyle=i\varrho[{\delta^{\nu}}_{0}(x^{2}{\delta^{\mu}}_{1}-x^{1}{\delta^{\mu}}_{2})-{\delta^{\mu}}_{0}(x^{2}{\delta^{\nu}}_{1}-x^{1}{\delta^{\nu}}_{2})]\,,$
(4.5a) $\displaystyle\Delta(x^{\mu})$ $\displaystyle=x^{\mu}\otimes 1+1\otimes
x^{\mu}\,,$ (4.5b) $\displaystyle S(x^{\mu})$ $\displaystyle=-x^{\mu}\,,$
(4.5c) $\displaystyle\varepsilon(x^{\mu})$ $\displaystyle=0\,.$ (4.5d)

To prove this statement, we apply the construction described in App. B. In
order to define the right action of $\mathcal{T}^{*}_{\varrho}$ on
$\mathcal{C}(SO(1,3))$, we first extend the elements
$x^{\mu}\in\mathcal{T}_{\varrho}^{*}$ to $x^{\mu}\otimes 1\equiv
a^{\mu}\in\mathcal{T}_{\varrho}^{*}\otimes\mathcal{C}(SO(1,3))$ and the
elements ${\Lambda^{\alpha}}_{\beta}\in\mathcal{C}(SO(1,3))$ to
$1\otimes{\Lambda^{\alpha}}_{\beta}\equiv{\Lambda^{\alpha}}_{\beta}\in\mathcal{T}_{\varrho}^{*}\otimes\mathcal{C}(SO(1,3))$;
then, on assuming a bicrossproduct structure on the tensor product quantum
group, we compute by means of the Hopf algebra product (B.3a) and the
coproduct (4.5b)

$[x^{\rho}\otimes
1,1\otimes{\Lambda^{\mu}}_{\nu}]=x^{\rho}\otimes{\Lambda^{\mu}}_{\nu}-x^{\rho}\otimes{\Lambda^{\mu}}_{\nu}-1\otimes({\Lambda^{\mu}}_{\nu}\triangleleft
x^{\rho}),$ (4.6)

but $[x^{\rho}\otimes 1,1\otimes{\Lambda^{\mu}}_{\nu}]$ is (minus) the
commutator (4.3c). Therefore we obtain the right action in the form

${\Lambda^{\mu}}_{\nu}\triangleleft
x^{\rho}=i\varrho\left[{\Lambda^{\rho}}_{0}({\Lambda^{\mu}}_{1}g_{2\nu}-{\Lambda^{\mu}}_{2}g_{1\nu})-{\delta^{\rho}}_{0}(\Lambda_{2\nu}{\delta^{\mu}}_{1}-\Lambda_{1\nu}{\delta^{\mu}}_{2})\right].$
(4.7)

Note that, from (B.5), ${\Lambda^{\mu}}_{\nu}\triangleleft
1={\Lambda^{\mu}}_{\nu}$ and $1\triangleleft 1=1$, but $1\triangleleft
x^{\rho}=0$.

To find the left coaction of $\mathcal{C}(SO(1,3))$ on
$\mathcal{T}^{*}_{\varrho}$, we compute $\Delta(x^{\mu}\otimes 1)$ from (B.3c)
and the coproduct (4.5b):

$\Delta(x^{\mu}\otimes 1)=x^{\mu}\otimes 1\otimes 1\otimes
1+1\otimes{x^{\mu}}^{(\bar{1})}\otimes{x^{\mu}}^{(\bar{2})}\otimes 1\,.$ (4.8)

Thus, comparing with (4.3g), we find that
${x^{\mu}}^{(\bar{1})}={\Lambda^{\mu}}_{\nu}$ and
${x^{\mu}}^{(\bar{2})}=x^{\nu}$, therefore:

$\beta(x^{\mu})={\Lambda^{\mu}}_{\nu}\otimes x^{\nu}.\\\ $ (4.9)

Note that $\beta(1)=1\otimes 1$.

Once the right action and left coaction are obtained, we have to verify that
the compatibility conditions (B.4a)-(B.4d) hold. This is proven in App. C in
order not to burden the text.

It remains to check that the quantum group relations (4.3) are reproduced by
means of the bicrossproduct structure given by Eqs. (B.3a)-(B.3e).

From (B.3a) we have

$\begin{array}[]{r@{}l}(1\otimes{\Lambda^{\mu}}_{\nu})\cdot(1\otimes{\Lambda^{\alpha}}_{\beta})-(1\otimes{\Lambda^{\alpha}}_{\beta})\cdot(1\otimes{\Lambda^{\mu}}_{\nu})&=1\otimes({\Lambda^{\mu}}_{\nu}{\Lambda^{\alpha}}_{\beta}-{\Lambda^{\alpha}}_{\beta}{\Lambda^{\mu}}_{\nu}),\\\
(x^{\mu}\otimes 1)\cdot(x^{\nu}\otimes 1)-(x^{\nu}\otimes
1)\cdot(x^{\mu}\otimes 1)&=(x^{\mu}x^{\nu}\otimes
1+x^{\mu}\otimes(1\triangleleft x^{\nu}))-(\mu\leftrightarrow\nu)\\\
&=x^{\mu}x^{\nu}\otimes 1-x^{\nu}x^{\mu}\otimes 1,\\\ (x^{\alpha}\otimes
1)\cdot(1\otimes{\Lambda^{\mu}}_{\nu})&=x^{\alpha}\otimes{\Lambda^{\mu}}_{\nu},\\\
(1\otimes{\Lambda^{\mu}}_{\nu})\cdot(x^{\alpha}\otimes
1)&=x^{\alpha}\otimes{\Lambda^{\mu}}_{\nu}+1\otimes({\Lambda^{\mu}}_{\nu}\triangleleft
x^{\alpha});\end{array}$ (4.10)

from which one finds the commutation rules (4.3a)-(4.3c).

From (B.3c) one finds

$\begin{array}[]{r@{}l}\Delta(x^{\mu}\otimes 1)&=x^{\mu}\otimes 1\otimes
1\otimes 1+1\otimes{\Lambda^{\mu}}_{\lambda}\otimes x^{\lambda}\otimes 1,\\\
\Delta(1\otimes{\Lambda^{\alpha}}_{\beta})&=1\otimes{\Lambda^{\alpha}}_{\gamma}\otimes
1\otimes{\Lambda^{\gamma}}_{\beta},\\\
\Delta(x^{\alpha}\otimes{\Lambda^{\mu}}_{\nu})&=x^{\alpha}\otimes{\Lambda^{\mu}}_{\beta}\otimes
1\otimes{\Lambda^{\beta}}_{\nu}+1\otimes{\Lambda^{\alpha}}_{\beta}{\Lambda^{\mu}}_{\gamma}\otimes
x^{\beta}\otimes{\Lambda^{\gamma}}_{\nu},\end{array}$ (4.11)

and Eqs. (4.3d), (4.3g) are recovered. From (B.3d):

$\varepsilon(x^{\mu}\otimes 1)=\varepsilon(x^{\mu})\varepsilon(1),$ (4.12)

but from (4.5d) $\varepsilon(x^{\mu})=0$, so that we obtain (4.3i) by
identifying the LHS with $\varepsilon(a)$. Moreover

$\varepsilon(1\otimes{\Lambda^{\alpha}}_{\beta})=\varepsilon(1)\varepsilon({\Lambda^{\alpha}}_{\beta}),$
(4.13)

but $\varepsilon({\Lambda^{\alpha}}_{\beta})={\delta^{\alpha}}_{\beta}$ and we
find (4.3f). Finally

$\varepsilon(x^{\mu}\otimes{\Lambda^{\alpha}}_{\beta})=\varepsilon(x^{\mu})\varepsilon({\Lambda^{\alpha}}_{\beta}),$
(4.14)

which is the expected result from the homomorphism property of the counit.

The last condition to analyse is (B.3e). It yields

$\displaystyle S(x^{\mu}\otimes 1)$ $\displaystyle=(1\otimes
S({\Lambda^{\mu}}_{\nu})\cdot(S(x^{\mu})\otimes
1)=-x^{\nu}\otimes{(\Lambda^{-1})^{\mu}}_{\nu},$ (4.15a) $\displaystyle
S(1\otimes{\Lambda^{\mu}}_{\nu})$
$\displaystyle=(1\otimes{\Lambda^{\mu}}_{\nu})\cdot(1\otimes
1)=1\otimes{(\Lambda^{-1})^{\mu}}_{\nu},$ (4.15b) $\displaystyle
S(x^{\alpha}\otimes{\Lambda^{\mu}}_{\nu})$ $\displaystyle=S(x^{\beta})\otimes
S({\Lambda^{\alpha}}_{\beta}{\Lambda^{\mu}}_{\nu})=-x^{\beta}\otimes{(\Lambda^{-1})^{\alpha}}_{\beta}{(\Lambda^{-1})^{\mu}}_{\nu},$
(4.15c)

leading to (4.3e) and (4.3h). This completes the proof of the bicrossproduct
structure (4.4).

###  4.2 The bicrossproduct structure of the algebra
$\mathcal{U}_{\varrho}(\mathfrak{p})$

In this section we build a noncommutative Hopf algebra which is dual to
$\mathcal{C}_{\varrho}(P)$ and possesses a bicrossproduct structure. In order
to distinguish it from $U_{\varrho}(\mathfrak{p})$, the one obtained from the
twist in Sec. 3, we will indicate it with a calligraphic notation,
$\mathcal{U}_{\varrho}(\mathfrak{p})$. To this, we will perform an invertible
nonlinear change of generators relating the two algebras, which results to be
a Hopf algebra homomorphism as shown in the next section.

The algebra $\mathcal{U}_{\varrho}(\mathfrak{p})$ is the analogue, for
$\varrho$-deformation, of the famous quantum Hopf algebra
$\mathcal{U}_{\kappa}(\mathfrak{p})$, introduced by Majid and Ruegg in [1],
that is, the bicrossproduct dual of the $\kappa$-Poincaré
$\mathcal{C}_{\kappa}(P)$ described in App. B.2. In Sec. 4.3 we show that
their construction applies to our case and reproduces our results.

Primarily, let us show that the twist-related Hopf algebra
$U_{\varrho}(\mathfrak{p})$ defined by (3.2) does not have a bicrossproduct
structure, namely it is not of the form
$U_{\varrho}(\mathfrak{p})=U(\mathfrak{so}(1,3))\vartriangleright\\!\blacktriangleleft
T_{\varrho},$ where $U(\mathfrak{so}(1,3))$ is the undeformed universal
enveloping algebra of the Lorentz sector, and $T_{\varrho}$ is the
$\varrho$-deformed (Hopf) algebra of translations. To do this, let us denote
the generators of the Lorentz and translations sector of the tensor product
above respectively with $m_{\mu\nu}$ and $p_{\mu}$ and let us extend them to
the whole tensor product algebra, by posing

$P_{\lambda}=1\otimes p_{\lambda}\in U(\mathfrak{so}(1,3))\otimes
T_{\varrho},\;\;\;\;\;M_{\mu\nu}=m_{\mu\nu}\otimes 1\in
U(\mathfrak{so}(1,3))\otimes T_{\varrho}.$ (4.16)

Given the coproducts for translation generators in (3.2) we deduce the
relevant Hopf algebraic structures of $T_{\varrho}$, naturally identified as a
subalgebra of $U_{\varrho}(\mathfrak{p})$, so to have

$\displaystyle[p_{\mu},p_{\nu}]=$ $\displaystyle 0,$ (4.17a)
$\displaystyle\Delta_{\mathcal{F}}p_{0}=$ $\displaystyle p_{0}\otimes
1+1\otimes p_{0},$ (4.17b) $\displaystyle\Delta_{\mathcal{F}}p_{1}=$
$\displaystyle
p_{1}\otimes\cos\left(\frac{\varrho}{2}p_{0}\right)+\cos\left(\frac{\varrho}{2}p_{0}\right)\otimes
p_{1}+p_{2}\otimes\sin\left(\frac{\varrho}{2}p_{0}\right)-\sin\left(\frac{\varrho}{2}p_{0}\right)\otimes
p_{2},$ (4.17c) $\displaystyle\Delta_{\mathcal{F}}p_{2}=$ $\displaystyle
p_{2}\otimes\cos\left(\frac{\varrho}{2}p_{0}\right)+\cos\left(\frac{\varrho}{2}p_{0}\right)\otimes
p_{2}-p_{1}\otimes\sin\left(\frac{\varrho}{2}p_{0}\right)+\sin\left(\frac{\varrho}{2}p_{0}\right)\otimes
p_{1},$ (4.17d) $\displaystyle\Delta_{\mathcal{F}}p_{3}=$ $\displaystyle
p_{3}\otimes 1+1\otimes p_{3},$ (4.17e) $\displaystyle S(p_{\mu})=$
$\displaystyle-p_{\mu},$ (4.17f) $\displaystyle\varepsilon(p_{\mu})=$
$\displaystyle 0\,,$ (4.17g)

while the Hopf algebra $U(\mathfrak{so}(1,3))$ remains undeformed.

We want to show that the factorization
$U_{\varrho}(\mathfrak{p})=U(\mathfrak{so}(1,3))\otimes T_{\varrho}$ generated
by the elements (4.16) is not compatible with the bicrossproduct structure.
Indeed, while it is possible to read the right action of
$U(\mathfrak{so}(1,3))$ on $T_{\varrho}$, the left coaction of $T_{\varrho}$
is ill-defined. For the right action we apply Eq. (B.5),

$\left[M_{\mu\nu},P_{\lambda}\right]=m_{\mu\nu}\otimes
p_{\lambda}-\bigl{(}m_{\mu\nu}\otimes(p_{\lambda}\triangleleft
1)+1\otimes(p_{\lambda}\triangleleft m_{\mu\nu})\bigr{)}\,,$ (4.18)

where we used the undeformed coproduct $\Delta m_{\mu\nu}=m_{\mu\nu}\otimes
1+1\otimes m_{\mu\nu}$. On the other hand, the Lie algebra sector of
$U_{\varrho}(\mathfrak{p})$ is undeformed, so that

$\left[M_{\mu\nu},P_{\lambda}\right]=i(g_{\nu\lambda}P_{\mu}-g_{\mu\lambda}P_{\nu}).$
(4.19)

Comparing the two, we finally obtain

$p_{\lambda}\triangleleft
m_{\mu\nu}=-i(g_{\nu\lambda}p_{\mu}-g_{\mu\lambda}p_{\nu}).$ (4.20)

For the left coaction, however, we encounter a problem. Let us rewrite the
last of Eqs. (3.2) by expanding the tensor product, as in (4.16). We have:

$\Delta(m_{30}\otimes 1)=m_{30}\otimes 1\otimes 1\otimes 1+1\otimes 1\otimes
m_{30}\otimes 1-\frac{\varrho}{2}1\otimes p_{3}\otimes m_{12}\otimes
1+\frac{\varrho}{2}m_{12}\otimes 1\otimes 1\otimes p_{3}.$ (4.21)

Comparing the latter with Eq. (B.3c), we obtain

$\Delta(m_{30}\otimes
1)=(m_{30(1)}\otimes{m_{30(2)}}^{(\bar{1})})\otimes({m_{30(2)}}^{(\bar{2})}\otimes
1),$ (4.22)

where we used the notation $\beta(x)=x^{(\bar{1})}\otimes x^{(\bar{2})}$, with
$x,x^{(\bar{2})}\in{U}(\mathfrak{so}(1,3))$, $x^{(\bar{1})}\in{T}_{\varrho},$
and $\beta$ the sought left coaction. It is easy to see that there is no left
coaction of ${T}_{\varrho}$ on $U(\mathfrak{so}(1,3)$, since the last term in
Eq. (4.21) has $p_{3}$ as fourth component of the tensor product, while all
terms in (4.22) have the identity as fourth component.

This means that, although $\mathcal{C}_{\varrho}(P)$ and
$U_{\varrho}(\mathfrak{p})$ are dual in the Hopf-algebraic sense, they are not
bicrossproduct-dual.

In what follows we show that it is possible to obtain, through a nonlinear
change of generators of $U_{\varrho}(\mathfrak{p})$, a new quantum group that
is not only Hopf-dual to $\mathcal{C}_{\varrho}(P)$, but also bicrossproduct-
dual, having a bicossproduct structure itself. As already mentioned, this is
analogous to the $\kappa$-Poincaré case, where the quantum group
$\mathcal{C}_{\kappa}(P)$ is Hopf-dual to both the quantum groups of the
universal enveloping algebra in the standard and Majid-Ruegg bases, but
bicrossproduct-dual only to that of the Majid-Ruegg basis.

To do this, let us rewrite the Poincaré generators of
$U_{\varrho}(\mathfrak{p})$ in a non-covariant way by defining
$R_{i}=\frac{1}{2}\epsilon_{ijk}M_{jk}$, $N_{i}=M_{i0}$, so that the algebra
reads:

$\begin{matrix}[P_{\mu},P_{\nu}]=0,&[N_{i},P_{0}]=iP_{i},&[N_{i},P_{j}]=i\delta_{ij}P_{0},&[N_{i},N_{j}]=-i\epsilon_{ijk}R_{k},\\\
[R_{i},P_{0}]=0,&[R_{i},P_{j}]=i\epsilon_{ijk}P_{k},&[R_{i},N_{j}]=i\epsilon_{ijk}N_{k},&[R_{i},R_{j}]=i\epsilon_{ijk}R_{k}.\end{matrix}$
(4.23)

The twisted coproducts in (3.2) become then

$\begin{array}[]{r@{}l}\Delta_{\mathcal{F}}P_{0}=&P_{0}\otimes 1+1\otimes
P_{0},\\\
\Delta_{\mathcal{F}}P_{1}=&P_{1}\otimes\cos\left(\frac{\varrho}{2}P_{0}\right)+\cos\left(\frac{\varrho}{2}P_{0}\right)\otimes
P_{1}+P_{2}\otimes\sin\left(\frac{\varrho}{2}P_{0}\right)-\sin\left(\frac{\varrho}{2}P_{0}\right)\otimes
P_{2},\\\
\Delta_{\mathcal{F}}P_{2}=&P_{2}\otimes\cos\left(\frac{\varrho}{2}P_{0}\right)+\cos\left(\frac{\varrho}{2}P_{0}\right)\otimes
P_{2}-P_{1}\otimes\sin\left(\frac{\varrho}{2}P_{0}\right)+\sin\left(\frac{\varrho}{2}P_{0}\right)\otimes
P_{1},\\\ \Delta_{\mathcal{F}}P_{3}=&P_{3}\otimes 1+1\otimes P_{3},\\\
\Delta_{\mathcal{F}}R_{1}=&R_{1}\otimes\cos\left(\frac{\varrho}{2}P_{0}\right)+\cos\left(\frac{\varrho}{2}P_{0}\right)\otimes
R_{1}+R_{2}\otimes\sin\left(\frac{\varrho}{2}P_{0}\right)-\sin\left(\frac{\varrho}{2}P_{0}\right)\otimes
R_{2},\\\
\Delta_{\mathcal{F}}R_{2}=&R_{2}\otimes\cos\left(\frac{\varrho}{2}P_{0}\right)+\cos\left(\frac{\varrho}{2}P_{0}\right)\otimes
R_{2}-R_{1}\otimes\sin\left(\frac{\varrho}{2}P_{0}\right)+\sin\left(\frac{\varrho}{2}P_{0}\right)\otimes
R_{1},\\\ \Delta_{\mathcal{F}}R_{3}=&R_{3}\otimes 1+1\otimes R_{3},\\\
\Delta_{\mathcal{F}}N_{1}=&N_{1}\otimes\cos\left(\frac{\varrho}{2}P_{0}\right)+\cos\left(\frac{\varrho}{2}P_{0}\right)\otimes
N_{1}+N_{2}\otimes\sin\left(\frac{\varrho}{2}P_{0}\right)-\sin\left(\frac{\varrho}{2}P_{0}\right)\otimes
N_{2}\\\
&+P_{1}\otimes\frac{\varrho}{2}R_{3}\cos\left(\frac{\varrho}{2}P_{0}\right)-\frac{\varrho}{2}R_{3}\cos\left(\frac{\varrho}{2}P_{0}\right)\otimes
P_{1}+P_{2}\otimes\frac{\varrho}{2}R_{3}\sin\left(\frac{\varrho}{2}P_{0}\right)\\\
&+\frac{\varrho}{2}R_{3}\sin\left(\frac{\varrho}{2}P_{0}\right)\otimes
P_{2},\\\
\Delta_{\mathcal{F}}N_{2}=&N_{2}\otimes\cos\left(\frac{\varrho}{2}P_{0}\right)+\cos\left(\frac{\varrho}{2}P_{0}\right)\otimes
N_{2}-N_{1}\otimes\sin\left(\frac{\varrho}{2}P_{0}\right)+\sin\left(\frac{\varrho}{2}P_{0}\right)\otimes
N_{1}\\\
&+P_{2}\otimes\frac{\varrho}{2}R_{3}\cos\left(\frac{\varrho}{2}P_{0}\right)-\frac{\varrho}{2}R_{3}\cos\left(\frac{\varrho}{2}P_{0}\right)\otimes
P_{2}-P_{1}\otimes\frac{\varrho}{2}R_{3}\sin\left(\frac{\varrho}{2}P_{0}\right)\\\
&-\frac{\varrho}{2}R_{3}\sin\left(\frac{\varrho}{2}P_{0}\right)\otimes
P_{1},\\\ \Delta_{\mathcal{F}}N_{3}=&N_{3}\otimes 1+1\otimes
N_{3}+\frac{\varrho}{2}P_{3}\otimes R_{3}-\frac{\varrho}{2}R_{3}\otimes
P_{3}.\end{array}$ (4.24)

Inspecting Eq. (4.22) one infers that, in order to have a well defined left
coaction, a new basis is needed such that the new coproduct has all the
translation generators on one side of the tensor product, while the Lorentz
ones are on the other side. To do this, we perform the following non-linear
transformation of the generators:

$\begin{array}[]{r@{}l}\widetilde{P}_{0}&=P_{0},\\\
\widetilde{P}_{1}&=P_{1}\cos(\frac{\varrho}{2}P_{0})-P_{2}\sin(\frac{\varrho}{2}P_{0}),\\\
\widetilde{P}_{2}&=P_{2}\cos(\frac{\varrho}{2}P_{0})+P_{1}\sin(\frac{\varrho}{2}P_{0}),\\\
\widetilde{P}_{3}&=P_{3},\\\
\widetilde{R}_{1}&=R_{1}\cos(\frac{\varrho}{2}P_{0})-R_{2}\sin(\frac{\varrho}{2}P_{0}),\\\
\widetilde{R}_{2}&=R_{2}\cos(\frac{\varrho}{2}P_{0})+R_{1}\sin(\frac{\varrho}{2}P_{0}),\\\
\widetilde{R}_{3}&=R_{3},\\\
\widetilde{N}_{1}&={N}_{1}\cos(\frac{\varrho}{2}P_{0})-{N}_{2}\sin(\frac{\varrho}{2}P_{0})+\frac{\varrho}{2}R_{3}\widetilde{P}_{1},\\\
\widetilde{N}_{2}&={N}_{2}\cos(\frac{\varrho}{2}P_{0})+{N}_{1}\sin(\frac{\varrho}{2}P_{0})+\frac{\varrho}{2}R_{3}\widetilde{P}_{2},\\\
\widetilde{N}_{3}&={N}_{3}+\frac{\varrho}{2}R_{3}P_{3},\end{array}$ (4.25)

which is such that the Lie algebra relations continue to be undeformed.
Details of the computation of the commutators are reported in appendix D. Let
us prove that we have the desired property for the coproducts.

Exploiting the fact that

$\displaystyle\Delta\cos(x)=\cos(x)\otimes\cos(x)-\sin(x)\otimes\sin(x),$
(4.26)
$\displaystyle\Delta\sin(x)=\cos(x)\otimes\sin(x)+\cos(x)\otimes\sin(x),$
(4.27)

for $x=\varrho P_{0}/2$, we can write the coproducts of the new generators as

$\begin{array}[]{r@{}l}&\Delta\widetilde{P}_{0}=\widetilde{P}_{0}\otimes
1+1\otimes\widetilde{P}_{0},\\\
&\Delta\widetilde{P}_{1}=\widetilde{P}_{1}\otimes
1+\cos(\varrho\widetilde{P}_{0})\otimes\widetilde{P}_{1}-\sin(\varrho\widetilde{P}_{0})\otimes\widetilde{P}_{2},\\\
&\Delta\widetilde{P}_{2}=\widetilde{P}_{2}\otimes
1+\cos(\varrho\widetilde{P}_{0})\otimes\widetilde{P}_{2}+\sin(\varrho\widetilde{P}_{0})\otimes\widetilde{P}_{1},\\\
&\Delta\widetilde{P}_{3}=\widetilde{P}_{3}\otimes
1+1\otimes\widetilde{P}_{3},\\\
&\Delta\widetilde{R}_{1}=\widetilde{R}_{1}\otimes
1+\cos(\varrho\widetilde{P}_{0})\otimes\widetilde{R}_{1}-\sin(\varrho\widetilde{P}_{0})\otimes\widetilde{R}_{2},\\\
&\Delta\widetilde{R}_{2}=\widetilde{R}_{2}\otimes
1+\cos(\varrho\widetilde{P}_{0})\otimes\widetilde{R}_{2}+\sin(\varrho\widetilde{P}_{0})\otimes\widetilde{R}_{1},\\\
&\Delta\widetilde{R}_{3}=\widetilde{R}_{3}\otimes
1+1\otimes\widetilde{R}_{3},\\\
&\Delta\widetilde{N}_{1}=\widetilde{N}_{1}\otimes
1+\cos(\varrho\widetilde{P}_{0})\otimes\widetilde{N}_{1}-\sin(\varrho\widetilde{P}_{0})\otimes\widetilde{N}_{2}+\varrho\widetilde{P}_{1}\otimes\widetilde{R}_{3},\\\
&\Delta\widetilde{N}_{2}=\widetilde{N}_{2}\otimes
1+\cos(\varrho\widetilde{P}_{0})\otimes\widetilde{N}_{2}+\sin(\varrho\widetilde{P}_{0})\otimes\widetilde{N}_{1}+\varrho\widetilde{P}_{2}\otimes\widetilde{R}_{3},\\\
&\Delta\widetilde{N}_{3}=\widetilde{N}_{3}\otimes
1+1\otimes\widetilde{N}_{3}+\varrho\widetilde{P}_{3}\otimes\widetilde{R}_{3}.\end{array}$
(4.28)

Since the counits for the old generators are all zero, it is easy to see that
the same result holds for the new generators:

$\varepsilon(\widetilde{P}_{\mu})=\varepsilon(\widetilde{N}_{i})=\varepsilon(\widetilde{R}_{i})=0.$
(4.29)

For the antipodes, by means of direct calculation, one finds:

$\begin{array}[]{r@{}l}&S(\widetilde{P}_{0})=-\widetilde{P}_{0},\\\
&S(\widetilde{P}_{1})=-\widetilde{P}_{1}\cos(\varrho\widetilde{P}_{0})-\widetilde{P}_{2}\sin(\varrho\widetilde{P}_{0}),\\\
&S(\widetilde{P}_{2})=-\widetilde{P}_{2}\cos(\varrho\widetilde{P}_{0})+\widetilde{P}_{1}\sin(\varrho\widetilde{P}_{0}),\\\
&S(\widetilde{P}_{3})=-\widetilde{P}_{3},\\\
&S(\widetilde{R}_{1})=-\widetilde{R}_{1}\cos(\varrho\widetilde{P}_{0})-\widetilde{R}_{2}\sin(\varrho\widetilde{P}_{0}),\\\
&S(\widetilde{R}_{2})=-\widetilde{R}_{2}\cos(\varrho\widetilde{P}_{0})+\widetilde{R}_{1}\sin(\varrho\widetilde{P}_{0}),\\\
&S(\widetilde{R}_{3})=-\widetilde{R}_{3},\\\
&S(\widetilde{N}_{1})=-\cos(\varrho\widetilde{P}_{0})\widetilde{N}_{1}-\sin(\varrho\widetilde{P}_{0})\widetilde{N}_{2}+\varrho\cos(\varrho\widetilde{P}_{0})\widetilde{P}_{1}\widetilde{R}_{3}+\varrho\sin(\varrho\widetilde{P}_{0})\widetilde{P}_{2}\widetilde{R}_{3},\\\
&S(\widetilde{N}_{2})=-\cos(\varrho\widetilde{P}_{0})\widetilde{N}_{2}+\sin(\varrho\widetilde{P}_{0})\widetilde{N}_{1}+\varrho\cos(\varrho\widetilde{P}_{0})\widetilde{P}_{2}\widetilde{R}_{3}-\varrho\sin(\varrho\widetilde{P}_{0})\widetilde{P}_{1}\widetilde{R}_{3},\\\
&S(\widetilde{N}_{3})=-\widetilde{N}_{3}+\varrho\widetilde{R}_{3}\widetilde{P}_{3}.\end{array}$
(4.30)

Therefore we have obtained a new Hopf algebra, which we call
$\mathcal{U}_{\varrho}(\mathfrak{p})$, to distinguish it from the twisted one,
$U_{\varrho}(\mathfrak{p})$.

###### Proposition 4.2

The quantum Hopf algebra $\mathcal{U}_{\varrho}(\mathfrak{p})$ admits the
bicrossproduct decomposition

$\mathcal{U}_{\varrho}(\mathfrak{p})=U(\mathfrak{so}(1,3))\vartriangleright\\!\blacktriangleleft\mathcal{T}_{\varrho},$
(4.31)

with $\mathcal{T}_{\varrho}$ the new deformed translations sector.

The proof of the statement proceeds in two steps. One first has to find the
right action of $U(\mathfrak{so}(1,3))$ on the translations and the left
coaction of $\mathcal{T}_{\varrho}$ on the Lorentz sector. Then one has to
verify the compatibility conditions of these maps, Eqs. (B.4a)-(B.4d), having
checked the Hopf algebra structure given by Eqs. (B.3a)-(B.3e).

Let us denote the standard Poincaré rotation and boost generators of
$U(\mathfrak{so}(1,3))$ by $m_{k},n_{k}$, respectively and the generators of
$\mathcal{T}_{\varrho}$ by $p_{\mu}$. We then identify $\mathcal{X}\equiv
U(\mathfrak{so}(1,3))$, $\mathcal{A}\equiv\mathcal{T}_{\varrho}$

and we extend the generators of the two Hopf algebras to the whole tensor
product by posing

$\widetilde{M}_{\mu\nu}=m_{\mu\nu}\otimes
1,\qquad\widetilde{N}_{i}=n_{i}\otimes 1,\qquad\widetilde{R}_{i}=m_{i}\otimes
1,\qquad\widetilde{P}_{\mu}=1\otimes p_{\mu}.$ (4.32)

Repeating the same steps as in Eqs. (4.18), (4.19) and (4.20), we apply (B.5)
to find

$\displaystyle\left[\widetilde{R}_{i},\widetilde{P}_{\mu}\right]$
$\displaystyle=$ $\displaystyle m_{i}\otimes
p_{\mu}-\left(m_{i}\otimes(p_{\mu}\triangleleft
1)+1\otimes(p_{\mu}\triangleleft m_{i})\right)\,,$ (4.33)
$\displaystyle\left[\widetilde{N}_{i},\widetilde{P}_{\mu}\right]$
$\displaystyle=$ $\displaystyle n_{i}\otimes
p_{\mu}-\left(n_{i}\otimes(p_{\mu}\triangleleft
1)+1\otimes(p_{\mu}\triangleleft n_{i})\right)\,,$ (4.34)

where we used the undeformed coproducts $\Delta m_{i}=m_{i}\otimes 1+1\otimes
m_{i}$, $\Delta n_{i}=n_{i}\otimes 1+1\otimes n_{i}$. On the other hand, the
Lie algebra sector of $\mathcal{U}_{\varrho}(\mathfrak{p})$ is undeformed,
namely

$\displaystyle\left[\widetilde{R}_{i},\widetilde{P}_{\mu}\right]$
$\displaystyle=$ $\displaystyle i\,\delta_{\mu
j}\;\varepsilon_{ijk}\widetilde{P}_{k}\,,$ (4.35)
$\displaystyle\left[\widetilde{N}_{i},\widetilde{P}_{\mu}\right]$
$\displaystyle=$ $\displaystyle
i\,\delta_{i\mu}\;\widetilde{P}_{0}+i\,\delta_{\mu 0}\;\widetilde{P}_{i}.$
(4.36)

Comparing the two sets of equations, one finally obtains the right action of
the Lorentz generators on $\mathcal{T}_{\varrho}$

$p_{\mu}\triangleleft m_{i}=-i\,\delta_{\mu j}\;\varepsilon_{ijk}p_{k},\qquad
p_{\mu}\triangleleft n_{i}=-i\,(\delta_{\mu i}\;p_{0}+\delta_{\mu 0}\;p_{i}).$
(4.37)

In order to find the left coaction of $\mathcal{T}_{\varrho}$ on
$U(\mathfrak{so}(1,3))$ we expand the Lorentz coproducts in (4.28)

$\begin{array}[]{r@{}l}\Delta(m_{A}\otimes 1)=&m_{A}\otimes 1\otimes 1\otimes
1+1\otimes\cos(\varrho p_{0})\otimes m_{a}\otimes 1-1\otimes\sin(\varrho
p_{0})\otimes\varepsilon_{AB}m_{B}\otimes 1\,,\\\ \Delta(m_{3}\otimes
1)=&m_{3}\otimes 1\otimes 1\otimes 1+1\otimes 1\otimes m_{3}\otimes 1\,,\\\
\Delta(n_{A}\otimes 1)=&n_{A}\otimes 1\otimes 1\otimes 1+1\otimes\cos(\varrho
p_{0})\otimes n_{a}\otimes 1-1\otimes\sin(\varrho
p_{0})\otimes\varepsilon_{AB}n_{B}\otimes 1\\\ &+\varrho(1\otimes p_{A}\otimes
m_{3}\otimes 1)\,,\\\ \Delta(n_{3}\otimes 1)=&n_{3}\otimes 1\otimes 1\otimes
1+1\otimes 1\otimes n_{3}\otimes 1+\varrho(1\otimes p_{3}\otimes m_{3}\otimes
1)\,,\end{array}$ (4.38)

where $A,B=1,2$. We thus compare with (B.3c), which now reads

$\begin{array}[]{r@{}l}\Delta(m_{i}\otimes
1)&=(m_{i(1)}\otimes{m_{i(2)}}^{(\bar{1})})\otimes({m_{i(2)}}^{(\bar{2})}\otimes
1)\,,\\\ \Delta(n_{i}\otimes
1)&=(n_{i(1)}\otimes{n_{i(2)}}^{(\bar{1})})\otimes({n_{i(2)}}^{(\bar{2})}\otimes
1)\,,\end{array}$ (4.39) $\displaystyle\beta(m_{A})$
$\displaystyle=\cos(\varrho p_{0})\otimes m_{A}-\varepsilon_{AB}\sin(\varrho
p_{0})\otimes m_{B},$ (4.40) $\displaystyle\beta(m_{3})$
$\displaystyle=1\otimes m_{3},$ $\displaystyle\beta(n_{A})$
$\displaystyle=\cos(\varrho p_{0})\otimes n_{A}-\varepsilon_{AB}\sin(\varrho
p_{0})\otimes n_{B}+\varrho p_{i}\otimes m_{3},$ $\displaystyle\beta(n_{3})$
$\displaystyle=1\otimes n_{3}+\varrho p_{3}\otimes m_{3}.$

The Hopf algebra structure of the tensor product algebra
$U(\mathfrak{so}(1,3))\otimes\mathcal{T}_{\varrho}$, with the right action and
left coaction found above, is assured by the validity of Eqs. (B.3a)-(B.3e).
Indeed, Eqs. (B.3a), (B.3c) are certainly satisfied since we have employed
them to obtain the right action and left coaction. Eq.(B.3b) is trivially
satisfied, as well as Eq. (B.3d), since the counits are all zero in both the
factor Hopf algebras as well as in the bicrossproduct algebra. The remaining
conditions (B.3e) are easily verified by direct calculations.

To complete the proof that that $\mathcal{U}_{\varrho}(\mathfrak{p})$ has the
proper bicrossproduct structure, it remains to check the compatibility
conditions Eqs. (B.4a)-(B.4d). Eq. (B.4a) is trivially satisfied since the
counits of all the generators are zero, and the right action gives always
terms proportional to generators. Eq (B.4b) can be verified by straightforward
computations applying the coproduct to the actions (4.37):

$\Delta(p_{\lambda}\triangleleft
m_{\mu\nu})=-i\left(g_{\nu\lambda}\Delta(p_{\mu})-g_{\mu\lambda}\Delta(p_{\nu})\right).$
(4.41)

To verify Eq. (B.4c) one has to apply the coaction $\beta$ to the commutators
of Lorentz generators:

$\beta([m_{\mu\nu},m_{\lambda\sigma}])=i[g_{\mu\sigma}\beta(m_{\nu\lambda})-g_{\nu\sigma}\beta(m_{\mu\lambda})+g_{\nu\lambda}\beta(m_{\mu\sigma})-g_{\mu\lambda}\beta(m_{\nu\sigma})].$
(4.42)

Then, a straightforward evaluation of all possible Lorentz commutators shows
that (B.4c) holds. Eq.(B.4d) is trivially satisfied since the starting
coproducts of $U(so(1,3))$ are cocommutative and the terms $a\triangleleft
x_{(2)}$ in the LHS, being proportional to $p$’s, always commute with
${x_{(1)}}^{(\bar{1})}$.

This concludes the proof of Prop. 4.2.

###  4.3 The algebra isomorphism between $\mathcal{U}_{\varrho}(\mathfrak{p})$
and ${U}_{\varrho}(\mathfrak{p})$

In this subsection we will follow the steps described in [1] to show that the
map between $\mathcal{U}_{\varrho}(\mathfrak{p})$ and
${U}_{\varrho}(\mathfrak{p})$ is an algebra homomorphism (indeed an
isomorphism, it being invertible). Moreover, we shall obtain the
$\varrho$-Minkowski spacetime as $\mathcal{T}_{\varrho}^{*}$, dual to the
enveloping algebra of translations, $\mathcal{T}_{\varrho}$, which is also an
algebra, on which $\mathcal{T}$ and the whole
$\mathcal{U}_{\varrho}(\mathfrak{p})$ act covariantly.

At first we consider the classical Poincaré algebra, which has the structure
of a semidirect sum, $\mathfrak{g}=\mathfrak{h}\oplus_{S}\mathfrak{f}$, with
$\mathfrak{h}$ the algebra of translations and $\mathfrak{f}$ the Lorentz
algebra (notice that here we follow the notation used by mathematicians with
the abelian algebra, which is acted upon by the Lorentz algebra, on the left).
The mathematical structure of the semidirect sum is that of a ‘split
extension’ namely a short exact sequence

$\mathfrak{h}\hookrightarrow\mathfrak{g}\rightarrow\mathfrak{f},$ (4.43)

with the two maps, an inclusion $i:\mathfrak{h}\rightarrow\mathfrak{g}$ and a
projection $\pi:\mathfrak{g}\rightarrow\mathfrak{f}$, such that $\mathfrak{h}$
is an invariant subalgebra of $\mathfrak{g}$ and $\mathfrak{f}$ is the
quotient of $\mathfrak{g}$ by $\mathfrak{h}$ . The same holds for Lie groups.
A Lie group $G$ is a semidirect product of $H$ and $F$, namely $G=H\rtimes F$,
iff

$H\hookrightarrow G\rightarrow F,$ (4.44)

is a split short exact sequence.

In [48, 49, 50] a generalization is provided for quantum groups. A
bicrossproduct Hopf algebra is there constructed as a split extension of Hopf
algerbas. The approach is applied in [1] to obtain the $\kappa$-Poincaré
quantum group $\mathcal{U}_{\kappa}(\mathfrak{p})$ (with the the so-called
Majid-Ruegg basis) and a homomorphism is established with the
$\kappa$-Poincaré ${U}_{\kappa}(\mathfrak{p})$ (with the so called standard
basis). In the following we adapt the procedure to the $\varrho$-Poincaré
case.

The starting observation is that $\mathcal{U}_{\varrho}(\mathfrak{p})$
contains $\mathcal{T}_{\varrho}$ as a sub-Hopf Algebra, which can be included
in $\mathcal{U}_{\varrho}(\mathfrak{p})$ via an inclusion map $i$. In a
complementary way, $\mathcal{U}_{\varrho}(\mathfrak{p})$ projects onto the
classical $U(\mathfrak{so}(1,3))$ with a Hopf Algebra map $\pi$:

$\mathcal{T}_{\varrho}\xhookrightarrow{i}\mathcal{U}_{\varrho}(\mathfrak{p})\xrightarrow{\pi}U(\mathfrak{so}(1,3)).$
(4.45)

From an operative point of view, $i$ is a standard immersion map while $\pi$
is such that

$\pi(\widetilde{N}_{i})=n_{i},\qquad\pi(\widetilde{R}_{i})=m_{i},\qquad\pi(\widetilde{P}_{\mu})=0,$
(4.46)

where
$\widetilde{N}_{i},\widetilde{R}_{i},\widetilde{P}_{\mu}\in\mathcal{U}_{\varrho}(\mathfrak{p})$
and $n_{i},m_{i}\in U(\mathfrak{so}(1,3))$, as in Sec. 4.2. Namely, $\pi$
projects the generators of the deformed enveloping algebra into their
classical Lorentz counterparts.

To implement the bicrossproduct construction it is also necessary to define a
Hopf Algebra homomorphism $j$ and a linear map $p$ that play an inverse role
with respect to $i,\pi$. Namely

$\mathcal{T}_{\varrho}\xleftarrow{p}\mathcal{U}_{\varrho}(\mathfrak{p})\xhookleftarrow{j}U(\mathfrak{so}(1,3)),$
(4.47)

with

$\qquad\pi\circ j=id,\hskip 14.22636ptp\circ i=id.$ (4.48)

The linear map $p$ has to to be a co-algebra homomorphism, meaning that

$(p\otimes p)\circ\Delta=\Delta\circ p,\qquad\varepsilon\circ p=p.$ (4.49)

Furthermore, if $j,p$ satisfy the properties

$\displaystyle(id\otimes j)\circ\Delta=(\pi\otimes id)\circ\Delta\circ j,$
(4.50) $\displaystyle p(u)t=p(u\,i(t)),\qquad
u\in\mathcal{U}_{\varrho}(\mathfrak{p}),\qquad t\in\mathcal{T}_{\varrho},$
(4.51)

then $\mathcal{U}_{\varrho}(\mathfrak{p})$ is a bicrossproduct extension of
$U(\mathfrak{so}(1,3))$ by $\mathcal{T}_{\varrho}$[48, 49, 50]. Let us analyse
the construction in some details. By defining

$\begin{array}[]{r@{}l}\widetilde{R}_{A}&\coloneqq
j(m_{A})=R_{A}\cos(\frac{\varrho}{2}P_{0})-\epsilon_{AB}R_{B}\sin(\frac{\varrho}{2}P_{0})\,,\\\
\widetilde{R}_{3}&\coloneqq j(m_{3})=R_{3}\,,\\\ \widetilde{N}_{A}&\coloneqq
j(n_{A})=N_{A}\cos(\frac{\varrho}{2}P_{0})-\epsilon_{AB}N_{B}\sin(\frac{\varrho}{2}P_{0})+\frac{\varrho}{2}R_{3}\left(P_{A}\cos(\frac{\varrho}{2}P_{0})-\epsilon_{AB}P_{B}\sin(\frac{\varrho}{2}P_{0})\right)\,,\\\
\widetilde{N}_{3}&\coloneqq
j(n_{3})=N_{3}+\frac{\varrho}{2}R_{3}P_{3}\,,\end{array}$ (4.52)

where, as in the previous section, $\epsilon_{AB}\,,A,B=1,2$ is the Levi-
Civita pseudotensor in $2$ dimensions, one can verify that Eq. (4.50) is
satisfied. Moreover, Eq. (4.51) holds trivially. Let us recall that in Eqs.
(4.52) the lowercase generators are the ones of the classical
$U(\mathfrak{so}(1,3))$, with undeformed commutators and coproducts, the
uppercase generators with tilde, on the LHS, are the generators of the quantum
Hopf algebra $\mathcal{U}_{\varrho}(\mathfrak{p})$ and the uppercase
generators on the RHS are related to ${U}_{\varrho}(\mathfrak{p})$. In other
words, the map $j$ defined by Eqs. (4.52) is nothing but the nonlinear change
of basis that we have performed in the previous section from
${U}_{\varrho}(\mathfrak{p})$ to $\mathcal{U}_{\varrho}(\mathfrak{p})$.

As for the right action and left coaction, they are given in this approach by
the following definitions

$\displaystyle t\triangleleft h$ $\displaystyle=$ $\displaystyle
j(Sh_{(1)})tj(h_{(2)}),\qquad t\in\mathcal{T}_{\varrho},\,\ h\in
U(\mathfrak{so}(1,3)),$ (4.53) $\displaystyle\beta(\pi(u))$ $\displaystyle=$
$\displaystyle p(u_{(1)})Sp(u_{(3)})\otimes\pi(u_{(2)}),\qquad
u\in\mathcal{U}_{\varrho}(\mathfrak{p}),$ (4.54)

where the notation $a_{(3)}$ comes from the square of the coproduct
$\Delta^{2}a=a_{(1)}\otimes a_{(2)}\otimes a_{(3)}$.

Equipped with these maps, the set
$(\mathcal{T}_{\varrho},U(\mathfrak{so}(1,3)),\triangleleft,\beta)$ completely
determines the Hopf algebra structure of
$\mathcal{U}_{\varrho}(\mathfrak{p})$. In fact, the commutators and coproducts
are determined by the cross relations

$\displaystyle i(t)j(h)$ $\displaystyle=j(h_{(1)})i(t\triangleleft h_{(2)}),$
(4.55) $\displaystyle\Delta(i(t))$ $\displaystyle=i(t_{(1)})\otimes
i(t_{(2)}),$ $\displaystyle\Delta(j(h))$ $\displaystyle=j(h_{(1)})(i\otimes
j)\circ\beta(h_{(2)}),$

where $h\in U(\mathfrak{so}(1,3))$, $t\in\mathcal{T}_{\varrho}$. By direct
checking, it is easily verified that Eqs. (4.53), (4.54), and (4.55) reproduce
the algebra and coalgebra sectors of $\mathcal{U}_{\varrho}(\mathfrak{p})$
presented in previous section.

Therefore we can conclude that

  * •

The quantum Hopf algebra $\mathcal{U}_{\varrho}(\mathfrak{p})$ derived in the
previous section by a non-linear change of basis for the twisted Hopf algebra
${U}_{\varrho}(\mathfrak{p})$ is a bicrossproduct algebra complying with the
definitions in [48, 49, 50];

  * •

It is the same Hopf algebra which one would obtain by rigorous mathematical
construction via an Hopf algebra extension of the undeformed
$U(\mathfrak{so}(1,3))$ through the deformed Hopf algebra of translations
$\mathcal{T}_{\varrho}$;

  * •

Because of that, and by virtue of Eqs.(4.52), the bicrossproduct algebra
$\mathcal{U}_{\varrho}(\mathfrak{p})$ and the twisted Hopf algebra
${U}_{\varrho}(\mathfrak{p})$ are homomorphic as Hopf algebras (indeed
isomorphic, being the non-linear map invertible).

####  4.3.1 The $\varrho$-Minkowski spacetime re-derived

Further building upon the results from [1], we construct the noncommutative
spacetime upon which the Hopf algebra $\mathcal{U}_{\varrho}(\mathfrak{p})$
acts in a covariant way.

In order to appreciate the different approaches, let us recall that, within
the twist framework, the covariant spacetime and its symmetries are defined at
once by twisting both structures, see Sect. 3. On the other hand, in terms of
the quantum group of functions over the group manifold,
$\mathcal{C}_{\varrho}(P)$, the quantum spacetime is the primary object and
the quantum symmetries are defined by requiring covariance of the former (or,
viceversa, one can start from $\mathcal{C}_{\varrho}(P)$ and check the
covariance of the quantum space), see Sect. 4.1.

In this section we follow a dual approach to the latter, which exploits the
bicrossproduct structure of $\mathcal{U}_{\varrho}(\mathfrak{p})$ and its
duality with $\mathcal{C}_{\varrho}(P)$. We will show that the algebra of
noncommutative coordinates can be obtained as the dual of the algebra of
translations $\mathcal{T}_{\varrho}$, which is precisely
$\mathcal{T}_{\varrho}^{*}$, the Hopf subalgebra of $\mathcal{C}_{\varrho}(P)$
that we have already encountered in the bicrossproduct factorization of
$\mathcal{C}_{\varrho}(P)$, see Prop. 4.1.

To this, let us apply the definitions of Hopf algebra duality

$\displaystyle\langle t,xy\rangle=\langle t_{(1)},x\rangle\langle
t_{(2)},y\rangle,$ (4.56) $\displaystyle\langle ts,x\rangle=\langle
t,x_{(1)}\rangle\langle s,x_{(2)}\rangle,\qquad\forall
t,s\in\mathcal{T}_{\varrho},\,\ \forall x,y\in\mathcal{T}_{\varrho}^{*}.$

Upon introducing the dual pairing between $\mathcal{T}_{\varrho}$ and
$\mathcal{T}_{\varrho}^{*}$, with $t_{\mu}=i\widetilde{P}_{\mu}$,

$\langle\widetilde{P}_{\mu},x^{\nu}\rangle=-i\delta^{\nu}_{\mu},$ (4.57)

and using the first of Eqs. (4.56), we can read off the commutators between
coordinates

$[x^{1},x^{0}]=i\varrho x^{2},\qquad[x^{2},x^{0}]=-i\varrho
x^{1},\qquad[x^{3},x^{\mu}]=0.$ (4.58)

After pplying the second of Eqs. (4.56), the coproducts turn out to be

$\Delta x^{\mu}=x^{\mu}\otimes 1+1\otimes x^{\mu}.$ (4.59)

The $\varrho$-Minkowski spacetime is thus immediately recovered. Furthermore,
within this framework it is possible to define a natural action of the
$\varrho$-Poincaré generators on the coordinates $x^{\mu}$, still exploiting
the Hopf algebra duality relations. The canonical action of
$\mathcal{T}_{\varrho}$ on $\mathcal{T}_{\varrho}^{*}$ is given by

$t\triangleright x=\langle t,x_{(1)}\rangle x_{(2)}\,,$ (4.60)

which in our case simply reads

$\widetilde{P}_{\mu}\triangleright x^{\nu}=-i\delta^{\mu}_{\nu}.$ (4.61)

The action of $U(\mathfrak{so}(1,3))$ on $\mathcal{T}_{\varrho}^{*}$ can be
obtained by dualizing its action on $\mathcal{T}_{\varrho}$, namely:

$\langle t,h\triangleright x\rangle=\langle t\triangleleft h,x\rangle,$ (4.62)

with $t\in\mathcal{T}_{\varrho}$, $x\in\mathcal{T}_{\varrho}^{*}$ and $h\in
U(\mathfrak{so}(1,3))$. Upon using Eq. (4.20), such action can be read off
immediately from the definition above:

$\widetilde{N}_{i}\triangleright
x^{0}=ix^{i},\qquad\tilde{N_{i}}\triangleright
x^{j}=-i\delta^{j}_{i}x^{0},\qquad\widetilde{R}_{i}\triangleright
x^{0}=0,\qquad\widetilde{R}_{i}\triangleright x^{j}=\epsilon_{ijk}x^{k}.$
(4.63)

With these building blocks, the action on the product of two algebra elements,
say, $a$ and $b$, is given by the canonical Hopf Algebra action

$h\triangleright ab=(h_{(1)}\triangleright a)(h_{(2)}\triangleright b).$
(4.64)

for any $h\in\mathcal{U}_{\varrho}(\mathfrak{p})$, as required for an action
to be covariant.

As an example, we prove the covariance of the commutator
$[x^{1},x^{0}]=i\varrho x^{2}$ under the boost $\widetilde{N}_{2}$. Using
(4.63) and recalling the coproducts (4.28), it is easy to see that

$\widetilde{N}_{2}\triangleright(x^{1}x^{0})=0,\qquad\widetilde{N}_{2}\triangleright(x^{0}x^{1})=-\varrho
x^{0},\qquad\widetilde{N}_{2}\triangleright(i\varrho x^{2})=\varrho x^{0},$
(4.65)

so that the covariance condition

$\widetilde{N}_{2}\triangleright[x^{1},x^{0}]=i\varrho\widetilde{N}_{2}\triangleright
x^{2}$ (4.66)

is verified. Covariance of the commutation relations under the action of the
other Lorentz generators can be verified in a similar fashion.

For the translation generators the covariance of the commutation relations

$\widetilde{P}_{\sigma}\triangleright[x^{\mu},x^{\nu}]=i\,\varrho\left(\delta^{\mu}_{j}\delta^{\nu}_{0}-\delta^{\mu}_{0}\delta^{\nu}_{j}\right){\epsilon^{3j}}_{k}\widetilde{P}_{\sigma}\triangleright
x^{k},$ (4.67)

can be verified by computing the left-hand side with the coproduct (4.28) and
subsequently using the action of translations on the dual, (4.60). As an
example, we show this for $t_{2}=i\widetilde{P}_{2}$.

We have

$t_{2}\triangleright(x^{\mu}x^{\nu})=i(\widetilde{P}_{2(1)}\triangleright
x^{\mu})(\widetilde{P}_{2(2)}\triangleright x^{\nu}),$ (4.68)

with $\widetilde{P}_{2(1)}\otimes\widetilde{P}_{2(2)}=\widetilde{P}_{2}\otimes
1+\cos(\varrho\widetilde{P}_{0})\otimes\widetilde{P}_{2}+\sin(\varrho
P_{0})\otimes\widetilde{P}_{1}$. We thus apply Eq. (4.60) to each term of the
product, to find

$t_{2}\triangleright(x^{\mu}x^{\nu})=-\left(\delta^{\mu}_{2}x^{\nu}+\delta^{\nu}_{2}x^{\mu}+\varrho\delta_{0}^{\mu}\delta^{\nu}_{1}\right),$
(4.69)

namely

$t_{2}\triangleright(x^{\mu}x^{\nu}-x^{\nu}x^{\mu})=-\varrho\left(\delta_{0}^{\mu}\delta^{\nu}_{1}-\delta_{0}^{\nu}\delta^{\mu}_{1}\right),$
(4.70)

which is the same result that we get by computing the right hand side of
(4.67)

$RHS=\,\varrho\left(\delta^{\mu}_{j}\delta^{\nu}_{0}-\delta^{\mu}_{0}\delta^{\nu}_{j}\right){\epsilon^{3j}}_{k}\delta^{k}_{2}=\varrho\left(\delta_{1}^{\mu}\delta^{\nu}_{0}-\delta_{1}^{\nu}\delta^{\mu}_{0}\right).$
(4.71)

The proof can be repeated in a similar way for the other generators of
translations. Therefore (4.67) holds true.

##  5 Plane waves and $\star$-product in $\varrho$-Minkowski

In the previous sections we have seen that the $\varrho$-Minkowski spacetime
can be interpreted in two different ways. From the perspective of the twist
construction, the commutation rules of spacetime coordinates are computed by
means of the $\star$-product corresponding to the twist that generates the
algebra of symmetries discussed in section 3. From the perspective of the
bicrossproduct construction, the $\varrho$-Minkowski commutation relations are
deduced by exploiting the duality between the translation generators and
spacetime coordinates.

On the other hand, starting from the commutation rules of spacetime one can
represent the algebra of noncommutative functions as an algebra of operators
by choosing a specific ordering for the basis of noncommutative plane waves
$\hat{\phi}(p)=:e^{ip_{\mu}\hat{x}^{\mu}}:$ [51]. The operator product of
plane waves $\hat{\phi}(p)$ defines a noncommutative $\star$-product for
functions on spacetime, according to

$(f\star
g)(x)=\int\differential^{4}p\,\differential^{4}k\,\tilde{f}(p)\tilde{g}(k)\;\langle\hat{\phi}(p)\hat{\phi}(k)\rangle\,,$
(5.1)

where the tilde indicates the standard Fourier transform. Different ordering
prescriptions will define different products. In App. E a short review for the
analogous case of $\kappa$-Minkowski is given, in order to highlight the
similarities. The notation $\langle\hat{\phi}(p)\hat{\phi}(k)\rangle$ defines
the noncommuting function, $\phi(p\oplus k)=\exp\left[i(p\oplus k)\cdot
x\right]$, which corresponds to the operator between brackets once the
operator product is computed, $\oplus$ representing a deformed sum. In terms
of noncommutative exponentials $e^{ip_{\mu}x^{\mu}}$ the $\star$-product
between functions reads, then,333We ignore here convergence issues. The
product is well defined for a fairly large class of functions, which includes
the Schwarzian ones.

$(f\star
g)(x)=\int\differential^{4}p\,\differential^{4}k\,\tilde{f}(p)\tilde{g}(k)\;\rm{e}^{ip_{\mu}x^{\mu}}\star\rm{e}^{ik_{\nu}x^{\nu}}\,.$
(5.2)

with

$e^{ip\cdot x}\star e^{ik\cdot x}=e^{i(p\oplus k)\cdot x}.$ (5.3)

Moreover, the composition rule of plane waves defines a coproduct for the
translation generators. This will be derived by requiring consistency of the
action of translations on both sides of (5.3), or, equivalently, by acting on
both sides of the operator version $\hat{\phi}(p)\hat{\phi}(k)=\hat{\phi}(q)$,
with $q=p\oplus k$. The operator $\hat{\phi}(q)$ and the related function will
be different for different ordering prescriptions. Therefore, the deformed sum
$\oplus$ defines different coproducts for the translation generators, as we
shall see in detail below.

The question arises as to whether the $\star$-product and coproduct derived
from plane wave multiplication correspond to any of the structures defined
either through the bicrossproduct or the twist constructions.

Different ordering prescriptions of the plane waves result in different
coproducts of the translation generators, and in different $\star$-products.
We shall show that in fact only a specific ordering choice, the so called
time-to-the-right prescription, results in a coproduct that is compatible with
translation generators associated with the bicrossproduct construction. This
ordering also defines a novel $\star$-product which appears to be different
from previously known ones [24, 45]. Other ordering choices correspond to
different coproducts, that is to non-linear redefinitions of the translation
generators. In particular, the so called time-symmetric prescription is
associated with the translation generators in the twisted basis, and it
coherently defines the $\star$-product which is obtained from the twist [24].

In order to define plane waves and their rule of multiplication, we follow
[52, 53] and start with (finite dimensional) representations of the spacetime
algebra. The commutation relations (2.3) can be represented by means of
$4\crossproduct 4$ matrices

$\displaystyle\hat{x}^{0}=\begin{pmatrix}0&i\varrho&0&0\\\ -i\varrho&0&0&0\\\
0&0&0&0\\\
0&0&0&0\end{pmatrix},\quad\hat{x}^{1}=\begin{pmatrix}0&0&0&i\varrho\\\
0&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\end{pmatrix},$ (5.4)
$\displaystyle\hat{x}^{2}=\begin{pmatrix}0&0&0&0\\\ 0&0&0&i\varrho\\\
0&0&0&0\\\ 0&0&0&0\end{pmatrix},\quad\hat{x}^{3}=\begin{pmatrix}0&0&0&0\\\
0&0&0&0\\\ 0&0&0&i\varrho\\\ 0&0&0&0\end{pmatrix},$

which formally close the Lie algebra of the centrally extended Euclidean group
in 3 dimensions, with one rotation ($\hat{x}^{0}$) and two spatial
translations ($\hat{x}^{1},\hat{x}^{2}$), while $\hat{x}^{3}$ is a central
generator.

###  5.1 Time-to-the-right ordering and bicrossproduct basis

Using the representation (5.4), we can write plane waves in matrix form.
Choosing the time-to-the-right ordering we define

$\hat{\phi}_{R}(p)=e^{ip_{k}\hat{x}^{k}}e^{ip_{0}\hat{x}^{0}}=\begin{pmatrix}\cos(\varrho
p_{0})&-\sin(\varrho p_{0})&0&-\varrho p_{1}\\\ \sin(\varrho
p_{0})&\cos(\varrho p_{0})&0&-\varrho p_{2}\\\ 0&0&1&-\varrho p_{3}\\\
0&0&0&1\end{pmatrix}.$ (5.5)

Multiplying two plane waves we get

$\displaystyle\hat{\phi}_{R}(p)\hat{\phi}_{R}(k)=\hat{\phi}_{R}(p\oplus_{R}k)=$
(5.6)
$\displaystyle=\begin{pmatrix}\cos(\varrho(k_{0}+p_{0}))&-\sin(\varrho(k_{0}+p_{0}))&0&-\varrho[p_{1}+k_{1}\cos(\varrho
p_{0})-k_{2}\sin(\varrho p_{0})]\\\
\sin(\varrho(k_{0}+p_{0}))&\cos(\varrho(k_{0}+p_{0}))&0&-\varrho[p_{2}+k_{2}\cos(\varrho
p_{0})+k_{1}\sin(\varrho p_{0})]\\\ 0&0&1&-\varrho(p_{3}+k_{3})\\\
0&0&0&1\end{pmatrix}.$

Comparing the last column of the single plane wave representation, Eq. (5.5),
to the last column of (LABEL:ttrsum), we find the law of addition of momenta,

$\begin{cases}(p\oplus_{R}k)_{0}=p_{0}+k_{0},\\\
(p\oplus_{R}k)_{1}=p_{1}+k_{1}\cos(\varrho p_{0})-k_{2}\sin(\varrho p_{0}),\\\
(p\oplus_{R}k)_{2}=p_{2}+k_{2}\cos(\varrho p_{0})+k_{1}\sin(\varrho p_{0}),\\\
(p\oplus_{R}k)_{3}=p_{3}+k_{3}.\end{cases}$ (5.7)

Therefore, the multiplication of plane waves is a new plane wave of momentum
$p\oplus_{R}k$. Namely, it is still an eigenfunction of translations. This can
in turn be related to the coproduct of translation generators $P_{\mu}^{R}$,
as we now demonstrate.

In the language of noncommutative plane wave operators
$\hat{\phi}_{R}=e^{ip_{k}\hat{x}^{k}}e^{ip_{0}\hat{x}^{0}}$, on which the
translation generators act as
$P_{\mu}^{R}\hat{\phi}_{R}(p)=p_{\mu}\hat{\phi}_{R}(p)$, we must have

$P^{R}_{\mu}\left(\hat{\phi}(p)\hat{\phi}(k)\right)=P^{R}_{\mu}\hat{\phi}(p\oplus_{R}k)=(p\oplus_{R}k)_{\mu}\hat{\phi}(p\oplus_{R}k).$
(5.8)

One can verify by direct calculation that, in order for this to hold, the
coproduct of $P_{\mu}^{R}$ should be:

$\begin{array}[]{r@{}l}&\Delta P_{0}^{R}=P_{0}^{R}\otimes 1+1\otimes
P_{0}^{R},\\\ &\Delta P_{1}^{R}=P_{1}^{R}\otimes 1+\cos(\varrho
P_{0}^{R})\otimes P_{1}^{R}-\sin(\varrho P_{0}^{R})\otimes P_{2}^{R},\\\
&\Delta P_{2}^{R}=P_{2}^{R}\otimes 1+\cos(\varrho P_{0}^{R})\otimes
P_{2}^{R}+\sin(\varrho P_{0}^{R})\otimes P_{1}^{R},\\\ &\Delta
P_{3}^{R}=P_{3}^{R}\otimes 1+1\otimes P_{3}^{R}\,.\end{array}$ (5.9)

Let us check it for $P_{2}^{R}$, as the others can be easily verified
analogously. On indicating with $\hat{\mu}$ the standard operator product, we
have, applying (5.9) to the product of plane waves

$\begin{array}[]{r@{}l}P^{R}_{\mu}\left(\hat{\phi}(p)\hat{\phi}(k)\right)&=\hat{\mu}\circ\Delta
P^{R}_{2}(\hat{\phi}(p)\otimes\hat{\phi}(k))\\\
&=\hat{\mu}\circ\Bigl{[}P^{R}_{2}\hat{\phi}(p)\otimes\hat{\phi}(k)+\cos(\varrho
P^{R}_{0})\hat{\phi}(p)\otimes\hat{\phi}(k)+\sin(\varrho
P^{R}_{0})\hat{\phi}(p)\otimes P^{R}_{1}\hat{\phi}(k)\Bigr{]}\\\
&=\left(p_{2}+\cos(\varrho p_{0})k_{2}+\sin(\varrho
p_{0})k_{1}\right)\hat{\phi}(p)\hat{\phi}(k)\,.\end{array}$ (5.10)

In terms of noncommutative functions $\phi(p)=e^{ip_{\nu}x^{\nu}}$ the
translation generators act as differential operators,
$P_{\mu}^{R}\phi(p)=-i\partial_{\mu}e^{ip_{\nu}x^{\nu}}=p_{\mu}\phi(p)$.
444Notice that the superscript ${\;}^{R}$ for the generator of translations is
redundant when acting on a single plane wave. We maintain it for book keeping.
Therefore we must have

$P^{R}_{\mu}\left(\phi(p)\star\phi(k)\right)=P^{R}_{\mu}\phi(p\oplus_{R}k)=-i\partial_{\mu}\exp(i(p\oplus_{R}k)_{\nu}x^{\nu})=(p\oplus_{R}k)_{\mu}\phi(p\oplus_{R}k).$
(5.11)

Again, this holds if the coproduct of $P_{\mu}^{R}$ takes the form (5.9). As
done before, we check it for $P_{2}^{R}$. On indicating with $\mu$ the $\star$
product between functions, and applying (5.9) to the product of plane waves,
we have

$\begin{array}[]{r@{}l}P^{R}_{\mu}\left(\phi(p)\star\phi(k)\right)&=\mu\circ\Delta
P^{R}_{2}(\phi(p)\otimes\phi(k))=\mu\circ\Bigl{[}P^{R}_{2}\phi(p)\otimes\phi(k)+\cos(\varrho
P^{R}_{0})\phi(p)\otimes\phi(k)\bigr{.}\\\ &+\bigl{.}\sin(\varrho
P^{R}_{0})\phi(p)\otimes P^{R}_{1}\phi(k)\Bigr{]}=\left(p_{2}+\cos(\varrho
p_{0})k_{2}+\sin(\varrho
p_{0})k_{1}\right)\left(\phi(p)\star\phi(k)\right).\end{array}$ (5.12)

Comparing with the results obtained in the previous sections, we find that the
$\varrho$-Minkowski coproducts for translation operators in the bicrossproduct
basis, given in (4.28), take exactly the form (5.9). We conclude that the
bicrossproduct basis is compatible with the time-to-the-right ordering for
plane-waves.

We briefly mention what happens upon choosing the time-to-the-left ordering.
With analogous calculations, it is possible to show that the structure of the
coproducts is similar to the one obtained with the time-to-the-right ordering,
namely the enveloping algebra is still a bicrossproduct, modulo an inversion
of the deformation parameter (i.e. $\varrho\rightarrow-\varrho$) and a swap
between the tensor product spaces, namely:

$\displaystyle\Delta P^{L}_{1}=1\otimes P^{L}_{1}+P^{L}_{1}\otimes\cos(\varrho
P^{L}_{0})+P^{L}_{2}\otimes\sin(\varrho P_{0}^{L}),$ (5.13)
$\displaystyle\Delta P^{L}_{2}=1\otimes P^{L}_{2}+P^{L}_{2}\otimes\cos(\varrho
P^{L}_{0})-P^{L}_{1}\otimes\sin(\varrho P_{0}^{L}),$ $\displaystyle\Delta
P^{L}_{0,3}=P^{L}_{0,3}\otimes 1+1\otimes P^{L}_{0,3}\;.$

An analogous result is also present when analysing the time-to-the-right and
time-to-the-left plane waves in $\kappa$-Minkowski.

####  5.1.1 The $\star$-product

While the $\star$-product can be defined for any ordering, in the following we
concentrate on the time-to-the-right one. From Eqs. (5.2), (5.3) we find

$\displaystyle(f\star g)(x)$ $\displaystyle=$
$\displaystyle\int\differential^{4}p\;\differential^{4}k\tilde{f}(p_{0}-k_{0},p_{A}-R_{AB}(-\varrho
p_{0}+\varrho k_{0})k_{B},p_{3}-k_{3})\tilde{g}(k)e^{ip_{\mu}x^{\mu}}$ (5.14)
$\displaystyle=$
$\displaystyle\int\differential^{4}p\;(\tilde{f}\circ\tilde{g})(p)e^{ip_{\mu}x^{\mu}}\,,$

with $R_{AB}(\theta),\;A,B=1,2$ the rotation matrix in the 1-2 plane of
argument $\theta$ and

$(\tilde{f}\circ\tilde{g})(p)=\int\differential^{4}k\;\tilde{f}(p_{0}-k_{0},p_{A}-R_{AB}(-\varrho
p_{0}+\varrho k_{0})k_{B},p_{3}-k_{3})\;\tilde{g}(k)$ (5.15)

the deformed convolution of Fourier transforms.

Let us show by direct calculation that this product is cyclic with respect to
the standard integration measure on $\mathbb{R}^{4}$ namely that

$\int\differential^{4}{x}\;f({x})\star
g({x})=\int\differential^{4}{x}\;g({x})\star f({x}).$ (5.16)

From the expression obtained in terms of plane waves (5.14) we compute

$\displaystyle\int\differential^{4}x\;(f\star g)(x)=$
$\displaystyle\int\differential^{4}x\;\differential^{4}p\;\differential^{4}k\;\tilde{f}(p_{0}-k_{0},p_{A}-R_{AB}(-\varrho
p_{0}+\varrho k_{0})k_{B},p_{3}-k_{3})\tilde{g}(k)e^{ip_{\mu}x^{\mu}}$ (5.17)
$\displaystyle=\int\differential^{4}p\;\differential^{4}k\tilde{f}(p_{0}-k_{0},p_{A}-R_{AB}(-\varrho
p_{0}+\varrho k_{0})k_{B},p_{3}-k_{3})\tilde{g}(k)\delta(p_{\mu})$
$\displaystyle=\int\differential^{4}k\;\tilde{f}(-k_{0},-R_{AB}(\varrho
k_{0}),-k_{3})\tilde{g}(k)$
$\displaystyle=\int\differential^{4}k\;\tilde{f}(k)\tilde{g}(-k_{0},-R_{AB}(\varrho
k_{0})k_{B},-k_{3})$
$\displaystyle=\int\differential^{4}p\;\differential^{4}k\tilde{f}(k)\tilde{g}(p_{0}-k_{0},p_{A}-R_{AB}(-\varrho
p_{0}+\varrho k_{0})k_{B},p_{3}-k_{3})\delta(p_{\mu})$
$\displaystyle=\int\differential^{4}x\;\differential^{4}p\;\differential^{4}k\;\tilde{f}(k)\tilde{g}(p_{0}-k_{0},p_{A}-R_{AB}(-\varrho
p_{0}+\varrho k_{0})k_{B},p_{3}-k_{3})e^{ip_{\mu}x^{\mu}}$
$\displaystyle=\int\differential^{4}x\;(g\star f)(x).$

We thus conclude that the $\star$-product derived from the time-to-the-right
ordering for plane waves is indeed cyclic. Furthermore, it is also
$\varrho$-Poincaré invariant, since the measure is undeformed.

Notice however that the stronger closure condition [54] $\int f\star g=\int
f\cdot g$ does not hold. It has been already noticed that the cyclicity
property is a fundamental property of the $\star$-product because it allows to
formulate field theories in terms of a $\star$-gauge invariant action (namely
such that the fields transform according to $\psi(x)\rightarrow
g(x)\star\psi(x)$, with $g$ a unitary element of the noncommutative algebra).
When the stronger closure condition is fulfilled one has the further
simplifying property that quadratic terms of the action, such as the kinetic
and the mass term, are undeformed, hence they produce the standard tree level
propagator. This is the case of the twisted $\varrho$-Minkowski product, for
which the closure property has been employed to investigate the behaviour of
scalar field theories at one loop [21]. We recall that an analogous situation
holds for Moyal and Wick-Voros products, which represent the same spacetime
noncommutativity, both being cyclic, but only Moyal being closed. Their
application to scalar field theory and a comparison between the results has
been analysed in [55]. It would be interesting to analyse the consequences of
the new product (5.14) within a similar setting. We plan to come back to
applications in a further publication.555Another $\star$-product for the
$\varrho$-Minkowski spacetime has been recently found [45], based on Weyl
quantization. It appears to be different from the ones presented here. The
latter is cyclic but not closed. It has the interesting property that $\int
f^{\dagger}\star g=\int\bar{f}\cdot g$ where the adjoint is defined w.r.t. a
suitable integration measure.

###  5.2 Time-symmetric ordering and twist basis

Consider the time-symmetric ordering for plane waves

$\displaystyle\hat{\phi}_{S}(p)=e^{i\frac{p_{0}\hat{x}^{0}}{2}}e^{ip_{k}\hat{x}^{k}}e^{\frac{p_{0}\hat{x}^{0}}{2}}=$
(5.18) $\displaystyle=\begin{pmatrix}\cos(\varrho p_{0})&-\sin(\varrho
p_{0})&0&-\varrho(p_{1}\cos(\varrho\frac{p_{0}}{2})-p_{2}\sin(\varrho\frac{p_{0}}{2}))\\\
\sin(\varrho p_{0})&\cos(\varrho
p_{0})&0&-\varrho(p_{2}\cos(\varrho\frac{p_{0}}{2})+p_{1}\sin(\varrho\frac{p_{0}}{2}))\\\
0&0&1&-\varrho p_{3}\\\ 0&0&0&1\end{pmatrix}.$

The product of symmetric ordered plane waves
$\hat{\phi}_{S}(p)\hat{\phi}_{S}(k)$ is given by

$\begin{pmatrix}\cos(\varrho(k_{0}+p_{0}))&-\sin(\varrho(k_{0}+p_{0}))&0&-\varrho
F(p_{0},k_{0},p_{1},k_{1},p_{2},k_{2})\\\
\sin(\varrho(k_{0}+p_{0}))&\cos(\varrho(k_{0}+p_{0}))&0&-\varrho
G(p_{0},k_{0},p_{1},k_{1},p_{2},k_{2})\\\ 0&0&1&-\varrho(p_{3}+k_{3})\\\
0&0&0&1\end{pmatrix},$ (5.19)

with

$\displaystyle F(p_{0},k_{0},p_{1},k_{1},p_{2},k_{2})=$ $\displaystyle
p_{1}\cos(\varrho\frac{p_{0}}{2})+k_{1}\cos(\frac{\varrho}{2}(k_{0}+2p_{0}))$
(5.20)
$\displaystyle+p_{2}\sin(\varrho\frac{p_{0}}{2})+k_{2}\sin(\frac{\varrho}{2}(k_{0}+2p_{0})),$
$\displaystyle G(p_{0},k_{0},p_{1},k_{1},p_{2},k_{2})=$ $\displaystyle
p_{2}\cos(\varrho\frac{p_{0}}{2})+k_{2}\cos(\frac{\varrho}{2}(k_{0}+2p_{0}))$
$\displaystyle+p_{1}\sin(\varrho\frac{p_{0}}{2})+k_{1}\sin(\frac{\varrho}{2}(k_{0}+2p_{0})).$

Using trigonometric identities, this can be put in the form
$\hat{\phi}_{S}(p\oplus_{S}k)$ where the deformed composition law $\oplus_{S}$
is defined as follows

$\begin{cases}(p\oplus_{S}k)_{0}=p_{0}+k_{0},\\\
(p\oplus_{S}k)_{1}=p_{1}\cos(\frac{\varrho}{2}k_{0})+\cos(\frac{\varrho}{2}p_{0})k_{1}+p_{2}\sin(\frac{\varrho}{2}k_{0})-\sin(\frac{\varrho}{2}p_{0})k_{2},\\\
(p\oplus_{S}k)_{2}=p_{2}\cos(\frac{\varrho}{2}k_{0})+\cos(\frac{\varrho}{2}p_{0})k_{2}-p_{1}\sin(\frac{\varrho}{2}k_{0})+\sin(\frac{\varrho}{2}p_{0})k_{1},\\\
(p\oplus_{S}k)_{3}=p_{3}+k_{3}.\end{cases}$ (5.21)

Let us notice that this is exactly the deformed sum obtained in [24] by
computing the twisted $\star$-product of plane waves. By introducing time-
symmetric translation generators, we want to find, as in the previous
subsection, the coproduct $\Delta P^{S}$ which is compatible with the request
that $\phi(p\oplus_{S}k)$ be an eigenfunction of translations
$P^{S}_{\mu}\phi(p\oplus_{S}k)=(p\oplus_{S}k)_{\mu}\phi(p\oplus_{S}k)$. It
turns out that this composition law can be checked to be compatible with the
twisted coproducts (3.2) of translation generators. Indeed, acting with the
twisted coproducts on the product of two symmetric ordered plane-waves, one
obtains a composition law of $p$ and $k$ that is the same as the one given by
the $\oplus_{S}$ operation. Consistently, one can verify that the
$\star$-product which is obtained by convolution of time-symmetric plane waves
along the same lines as in the previous section, matches the one that is
computed in [24] using the twist, Eq. (3.3). We have, therefore, obtained the
plane waves representation which corresponds to the twisted $\star$-product.

In the previous sections, we have shown that the twist translation generators
and bicrossproduct translation generators are related by a nonlinear
transformation. We can verify that the same relation holds between the momenta
of the time-symmetric and the time-to-the-right ordered plane waves. To do
this, starting from the momenta in the time-symmetric plane waves, $p_{i}$, we
perform the following transformations

$\displaystyle\tilde{p}_{1}$
$\displaystyle=p_{1}\cos(\varrho\frac{p_{0}}{2})-p_{2}\sin(\varrho\frac{p_{0}}{2}),$
(5.22) $\displaystyle\tilde{p}_{2}$
$\displaystyle=p_{2}\cos(\varrho\frac{p_{0}}{2})+p_{1}\sin(\varrho\frac{p_{0}}{2}),$
$\displaystyle\tilde{p}_{0,3}$ $\displaystyle=p_{0,3}\;,$

which we apply to the time-symmetric plane waves (5.18). One can verify that
the transformed plane waves are eigenfunctions of the translation generators

$\displaystyle\widetilde{P}_{1}$
$\displaystyle=P^{S}_{1}\cos(\varrho\frac{P^{S}_{0}}{2})-P^{S}_{2}\sin(\varrho\frac{P^{S}_{0}}{2}),$
(5.23) $\displaystyle\widetilde{P}_{2}$
$\displaystyle=P^{S}_{2}\cos(\varrho\frac{P^{S}_{0}}{2})+P^{S}_{1}\sin(\varrho\frac{P^{S}_{0}}{2}),$
$\displaystyle\widetilde{P}_{0,3}$ $\displaystyle=P^{S}_{0,3}\;,$

which are precisely the generators obtained in Sec. 4.2 through the change of
basis (4.25) and one can identify $\widetilde{P}_{\mu}$ with $P_{\mu}^{R}$.

####  5.2.1 Equivalence class of star products

It is known that different $\star$-products associated with a given spacetime
noncommutativity are equivalent, in the sense that an invertible map $T$ can
always be found, such that

$T(f\star g)=Tf\star^{\prime}Tg.$ (5.24)

In the framework of phase-space quantization this redundance is well known and
related to ordering ambiguities. For quantum spacetime, it has been studied
for the class of translation invariant $\star$-products [56], of which the
Moyal product is the only closed representative. For rotation invariant
products the issue is discussed in [57].

Interestingly, for the $\varrho$-Minkowski case such a map is easily found in
terms of the invertible relations between the Poincaré generators related to
the various orderings that it is possible to choose for the $\star$-product of
plane waves. In particular, Eqs. (5.22) connecting the time-symmetric ordering
with the time-to-the right ordering allow to connect the twisted
$\star$-product found in [24], which is closed, with the bicrossproduct (5.14)
which is only cyclic. The map can be explicitly written in Fourier transform

$Tf:=\int\differential^{4}p\,\tilde{f}(p)\tilde{T}{\rm e}^{ip_{\mu}x^{\mu}}$
(5.25)

with $\tilde{T}{\rm e}^{ip_{\mu}x^{\mu}}:={\rm e}^{i\tilde{p}_{\mu}x^{\mu}}$
according to (5.22).

##  6 Conclusions

The noncommutative spacetime $\varrho$-Minkowski shows some interesting
features, which single it out among the variations of the original
$\kappa$-Minkowski. Similarly to the latter, it has an undeformed algebra of
Lorentz symmetries, $C(SO(3,1))$, as a subalgebra of the quantum Poincaré
group $C_{\varrho}(P)$, as well as un undeformed Lorentz Lie algebra sector
and a bicrossproduct construction which make it possible to read its quantum
relativity group as a natural extension of the semidirect product of Lie
groups to the Hopf algebraic framework. Moreover, the bicrossproduct structure
allows to obtain the spacetime (i.e. the noncommutative algebra
$\mathcal{T}^{*}_{\varrho}$) as a generalization of the concept of homogeneous
space which holds for the Minkowski spacetime in the classical setting.

Besides the bicrossproduct derivation, the $\varrho$-Minkowski symmetries may
be obtained by a twist. In this case the algebra is generated by an $r$-matrix
which is solution of the classical Yang-Baxter equation, whereas, up to our
knowledge, the twists related to the $\kappa$-Minkowski algebra are associated
with modified Yang-Baxter equations and, as such, they require an enlargement
of the Poincaré symmetry in order to be defined. The deformed Poincaré
symmetries, $\mathcal{U}_{\varrho}\mathfrak{p})$ and
$U_{\varrho}\mathfrak{p})$, obtained following the bicrossproduct and the
twist approaches, are isomorphic quantum groups related by a nonlinear change
of generators.

Using a finite-dimensional representation of the algebra of coordinates and
expanding the noncommutative functions on a basis of plane waves, it has been
possible to define different $\star$-products related to different ordering
prescriptions. We have found that the time-symmetric ordering singles out the
twisted star product already found in [21], whereas the time-to-the-right
ordering yields a novel $\star$-product, consistent with the bicrossproduct
construction. We have shown that it enjoys the cyclicity property with respect
to the standard measure in $\mathbb{R}^{4}$ and, unlike the twisted product,
it is not closed. Therefore, it would be interesting to to study gauge
theories within the two approaches, invariance being ensured by cyclicity, and
compare the results.

There are other interesting features, such as the discretization of the
spectrum of the time observable due to the non-commutativity of coordinates
[22], which could have phenomenological applications. Moreover, a
$2+1$-dimensional version of the $\varrho$-Minkowski framework was taken as a
starting point for quantum gravity phenomenology studies. In [58, 41], the
$\varrho$-deformations of the Poincaré symmetries predict the so-called dual-
lensing effect, which might open a new window on Planck-scale phenomenology.
The algebra of relativistic symmetries used in these studies is however
different from the ones we have discussed in this work. In particular, the
Lorentz sector is deformed. An analysis of the different implications for
these models will be matter of future work. Another aspect which deserves
further investigation is the relation with the $\lambda$-Minkowski spacetime
(where the role of $x^{0}$ and $x^{3}$ is exchanged), whose analysis is likely
to be very similar, although the physics it describes would be different.

## Acknowledgments

We acknowledge support from the INFN Iniziativa Specifica GeoSymQFT (F.L.,
P.V.) and Quagrap (G.F., G.G.). F.L. acknowledges financial support from the
State Agency for Research of the Spanish Ministry of Science and Innovation
through the “Unit of Excellence Maria de Maeztu 2020–2023” award to the
Institute of Cosmos Sciences (Grant No. CEX2019-000918-M) and from Grants No.
PID2019–105614 GB- C21 and No. 2017-SGR-929. The research of G.F, G.G, F.L.
and P.V. was carried out in the frame of Programme STAR Plus, financially
supported by UniNA and Compagnia di San Paolo. L.S. acknowledges financial
support from the doctoral school of the University of Wrocław and the SONATA
BIS grant 2021/42/E/ST2/00304 from the National Science Centre (NCN), Poland.

##  Appendix A The twist

In this appendix we give some defintions and show some properties of the twist
approach.

A Drinfel’d twist $\mathcal{F}$ is an invertible map $\mathcal{F}\in
U(\mathfrak{g})\otimes U(\mathfrak{g})$, with an action on the algebra of
functions on the group,

$\mathcal{F}:\mathcal{C}(G)\otimes\mathcal{C}(G)\rightarrow\mathcal{C}(G)\otimes\mathcal{C}(G),$
(A.1)

which satisfies the following cocycle and normalization conditions

$\displaystyle(\mathcal{F}\otimes 1)(\Delta\otimes id)\mathcal{F}$
$\displaystyle=(1\otimes\mathcal{F})(id\otimes\Delta)\mathcal{F},$ (A.2)
$\displaystyle(\varepsilon\otimes 1)\mathcal{F}$
$\displaystyle=(1\otimes\varepsilon)\mathcal{F}=1.$ (A.3)

Twist operators that do not satisfy the cocycle condition are called non
admissible twists.

In terms of this map it is possible to introduce a noncommutative
$\star$-product between functions defined on $G$ as follows

$f\star g=\mu_{\star}(f\otimes g)\doteq\mu\circ\mathcal{F}^{-1}(f\otimes
g),\hskip 14.22636ptf,g\in\mathcal{C}(G);$ (A.4)

where $\mu_{\star}$ is the noncommutative deformation of the classical
commutative product
$\mu:\mathcal{C}(G)\otimes\mathcal{C}(G)\rightarrow\mathcal{C}(G)$. The
cocycle and the normalization conditions imply that the $\star$-product is
associative and the existence of the neutral element $1$: $f\star 1=1\star
f=f$.

In our work we will deal with twists of a particular kind called abelian
twists (namely constructed in terms of commuting generators of the Lie
algebra), and for this case it is possible to show that

$\mathcal{F}^{-1}\approx 1\otimes 1+\frac{1}{2}r+\dots$ (A.5)

i.e., that the classical $r$-matrix is given by the first order in the
deformation parameter expansion of the twist operator.

Connected with this property, it is possible to show that if the classical
$r$-matrix does not solve a CYBE but a modified one, then the dual universal
enveloping algebra does not admit an abelian admissible twist. This is the
case of $\kappa$-Poincaré, and the reason we are forced to obtain its quantum
algebra via other procedures, while in the $\varrho$-Poincaré case, since the
$r$-matrix satisfies a CYBE, we can employ the twist procedure.

For a Lie algebra-type deformation that admits an admissible twist operator, a
simple way to obtain a quantum enveloping algebra of Poincaré is achieved
deforming the standard Poincaré Hopf algebra via the relation

$\Delta_{\mathcal{F}}=\mathcal{F}\Delta\mathcal{F}^{-1},$ (A.6)

leaving all the other maps undeformed.

The $\kappa$-Minkowski algebra has been described in terms of a
$\star$-product derived via many different twisting operators. These are the
so called Jordanian [8, 9, 10, 11] and Abelian twists [12, 13, 14]. In
general, they both need the enveloping algebra of symmetries to be enlarged,
including at least the Weyl generator, yielding for example the deformed
universal enveloping algebra of the Weyl group [59].

##  Appendix B The bicrossproduct structure

Following [60, 61] we present here a constructive definition of the
bicrossproduct structure.

Let $\mathcal{X},\mathcal{A}$ be two Hopf algebras, a bicrossproduct algebra
$\mathcal{X}\vartriangleright\\!\blacktriangleleft\mathcal{A}$ is the tensor
product $\mathcal{X}\otimes\mathcal{A}$ endowed with two additional structure
maps, a covariant right action of $\mathcal{X}$ on $\mathcal{A}$ and a
covariant left coaction of $\mathcal{A}$ on $\mathcal{X}$

$\displaystyle\triangleleft$
$\displaystyle:\mathcal{A}\times\mathcal{X}\rightarrow\mathcal{A},$ (B.1a)
$\displaystyle\beta$
$\displaystyle:\mathcal{X}\rightarrow\mathcal{A}\otimes\mathcal{X},$ (B.1b)

such that

$\displaystyle a\triangleleft(xy)$ $\displaystyle=(a\triangleleft
x)\triangleleft y,$ (B.2a) $\displaystyle 1\triangleleft x$
$\displaystyle=\varepsilon(x)1,$ (B.2b) $\displaystyle(a\cdot b)\triangleleft
x$ $\displaystyle=(a\triangleleft x_{(1)})(b\triangleleft x_{(2)}),$ (B.2c)
$\displaystyle(id\otimes\beta)\circ\beta$ $\displaystyle=(\Delta\otimes
id)\circ\beta,$ (B.2d) $\displaystyle(\varepsilon\otimes id)\circ\beta$
$\displaystyle=id,$ (B.2e) $\displaystyle\beta(ab)$
$\displaystyle=\beta(a)\beta(b),$ (B.2f) $\displaystyle\beta(1)$
$\displaystyle=1\otimes 1,$ (B.2g)

and with an Hopf algebra structure given by:

$\displaystyle\mu((x\otimes a),(y\otimes b))$ $\displaystyle=(x\otimes
a)\cdot(y\otimes b)=xy_{(1)}\otimes(a\triangleleft y_{(2)})b,$ (B.3a)
$\displaystyle
1_{\mathcal{X}\vartriangleright\\!\blacktriangleleft\mathcal{A}}$
$\displaystyle=1_{\mathcal{X}}\otimes 1_{\mathcal{A}},$ (B.3b)
$\displaystyle\Delta(x\otimes a)$
$\displaystyle=(x_{(1)}\otimes{x_{(2)}}^{(\bar{1})}a_{(1)})\otimes({x_{(2)}}^{(\bar{2})}\otimes
a_{(2)}),$ (B.3c) $\displaystyle\varepsilon(x\otimes a)$
$\displaystyle=\varepsilon(x)\varepsilon(a),$ (B.3d) $\displaystyle S(x\otimes
a)$ $\displaystyle=(1_{\mathcal{X}}\otimes
S(x^{(\bar{1})}a))\cdot(S(x^{(\bar{2})})\otimes 1_{\mathcal{A}}),$ (B.3e)

where $x,y\in\mathcal{X}$, $a,b\in\mathcal{A}$ and following the Sweedler
notation, we have defined
$\Delta(h)=\sum_{i}{h_{(1)}}_{i}\otimes{h_{(2)}}_{i}=h_{(1)}\otimes h_{(2)}$
and $\beta(x)=x^{(\bar{1})}\otimes x^{(\bar{2})}$, with
$x^{(\bar{1})}\in\mathcal{A}$ and $x^{(\bar{2})}\in\mathcal{X}$. The structure
maps must also satisfy the following compatibility conditions:

$\displaystyle\varepsilon(a\triangleleft x)$
$\displaystyle=\varepsilon(a)\varepsilon(x),$ (B.4a)
$\displaystyle\Delta(a\triangleleft x)$ $\displaystyle=(a_{(1)}\triangleleft
x_{(1)}){x_{(2)}}^{(\bar{1})}\otimes(a_{(2)}\triangleleft{x_{(2)}}^{(\bar{2})}),$
(B.4b) $\displaystyle\beta(xy)$ $\displaystyle=(x^{(\bar{1})}\triangleleft
y_{(1)}){y_{(2)}}^{(\bar{1})}\otimes x^{(\bar{2})}{y_{(2)}}^{(\bar{2})},$
(B.4c) $\displaystyle{x_{(1)}}^{(\bar{1})}(a\triangleleft
x_{(2)})\otimes{x_{(1)}}^{(\bar{2})}$ $\displaystyle=(a\triangleleft
x_{(1)}){x_{(2)}}^{(\bar{1})}\otimes{x_{(2)}}^{(\bar{2})}.$ (B.4d)

An important thing to note is that the bicrossproduct algebra
$\mathcal{X}\vartriangleright\\!\blacktriangleleft\mathcal{A}$ can always be
seen as the universal enveloping algebra generated by elements $X=x\otimes 1$,
$\mathcal{A}=1\otimes a$, modulo the commutation relations

$[X,A]=x\otimes a-x_{(1)}\otimes(a\triangleleft x_{(2)});$ (B.5)

in fact, from (B.3a)

$\displaystyle XA$ $\displaystyle=(x\otimes 1)\cdot(1\otimes
a)=x\otimes(1\triangleleft 1)a=x\otimes a,$ (B.6a) $\displaystyle AX$
$\displaystyle=(1\otimes a)\cdot(x\otimes 1)=x_{(1)}\otimes(a\triangleleft
x_{(2)}).$ (B.6b)

###  B.1 The classical $r$-matrix deformation method

Here we give a short review of the deformation induced by the classical
$r$-matrix, while referring to [62, 63, 64] for details.

Given a Lie algebra $\mathfrak{g}$ corresponding to a Lie group $G$, a
classical r-matrix is a tensor $r\in\bigwedge^{2}\mathfrak{g}$ satisfying the
modified Yang-Baxter Equation (MYBE)

$[r_{12},r_{13}+r_{23}]+[r_{13},r_{23}]=t,$ (B.7)

with $t\in\otimes^{3}\mathfrak{g}$ a $\mathfrak{g}$-invariant, and
$r_{\alpha\beta}\in\otimes^{3}\mathfrak{g}$, $\alpha,\beta=1,2,3$, defined as

$\displaystyle r_{12}=c_{ij}a_{i}\otimes a_{j}\otimes 1,$ (B.8) $\displaystyle
r_{23}=c_{ij}1\otimes a_{i}\otimes a_{j},$ (B.9) $\displaystyle
r_{13}=c_{ij}a_{i}\otimes 1\otimes a_{j},$ (B.10)

$a_{i}\in\mathfrak{g}$. If $t=0$ Eq. (B.7) is called classical Yang-Baxter
Equation (CYBE).

This element defines a Poisson-Lie group [62] through the following Sklyanin
bracket:

$\\{f,g\\}=r^{\alpha\beta}(X^{R}_{\alpha}fX^{R}_{\beta}g-X^{L}_{\alpha}fX^{L}_{\beta}g),\qquad
f,g\in\mathcal{C}^{\infty}(G),$ (B.11)

where $X^{R},X^{L}$ are the right- and left-invariant vector fields. The
algebra $\mathcal{C}^{\infty}(G)$ is, then, a Poisson-Hopf algebra with
trivial cosector and antipodes.

If there is no order ambiguity, it is then possible to quantize this structure
to a quantum group via the canonical quantization of the Sklyanin bracket
$\\{,\\}\rightarrow\frac{1}{i}[,]$.

In the next section we shall apply (B.11) to the $\kappa$-Poincaré case,
whereas in section 4.1 it shall be applied to the $\varrho$-Poincaré
deformation. Therefore we shall need the left and right invariant vector
fields of the Poincaré group explicitly. They read (see for example [23])

$\begin{array}[]{lll}X_{\alpha\beta}^{L}&={\Lambda^{\mu}}_{\alpha}\frac{\partial}{\partial\Lambda^{\mu\beta}}-{\Lambda^{\mu}}_{\beta}\frac{\partial}{\partial\Lambda^{\mu\alpha}},&X_{\alpha}^{L}={\Lambda^{\mu}}_{\alpha}\frac{\partial}{\partial
a^{\mu}},\\\
X_{\alpha\beta}^{R}&=\Lambda_{\beta\nu}\frac{\partial}{\partial{\Lambda^{\alpha}}_{\nu}}-\Lambda_{\alpha\nu}\frac{\partial}{\partial{\Lambda^{\beta}}_{\nu}}+a_{\beta}\frac{\partial}{\partial
a^{\alpha}}-a_{\alpha}\frac{\partial}{\partial
a^{\beta}},&X_{\alpha}^{R}=\frac{\partial}{\partial a^{\alpha}}.\end{array}$
(B.12)

###  B.2 The Quantum Group $\mathcal{C}_{\kappa}(P)$

In order to obtain the quantum group $\mathcal{C}_{\kappa}(P)$, one could
start from the $\kappa$-Minkowski commutation relations (2.2) and impose the
covariance of the $\kappa$-Minkowski algebra under the action of the quantum
group. However, as discussed in [23], this method leads to some ambiguity in
the mixed commutators between Lorentz and translation parameters. To avoid
this issue, we will work in the framework of classical $r$-matrices.

A classical $r$-matrix for $\mathcal{C}_{\kappa}(P)$ is found to be [59]:

$r=i\lambda M_{0\nu}\wedge P^{\nu}.$ (B.13)

This $r$-matrix satisfies a modified Yang-Baxter equation, which is why the
twist operator requires an enlargement of the Poincaré algebra as mentioned at
the end of App. A.

Following [23] we compute the Poisson brackets for the group-parameters using
Eqs. (B.11) and (B.12) and quantize them via canonical quantization to obtain
the following quantum group structure:

$\displaystyle[a^{\mu},a^{\nu}]$
$\displaystyle=i\lambda({\delta^{\mu}}_{0}a^{\nu}-{\delta^{\nu}}_{0}a^{\mu}),$
(B.14a) $\displaystyle[{\Lambda^{\alpha}}_{\beta},a^{\rho}]$
$\displaystyle=-i\lambda(({\Lambda^{\alpha}}_{0}-{\delta^{\alpha}}_{0}){\Lambda^{\rho}}_{\beta}+(\Lambda_{0\beta}-g_{0\beta})g^{\alpha\rho}),$
(B.14b)
$\displaystyle[{\Lambda^{\alpha}}_{\beta},{\Lambda^{\gamma}}_{\delta}]$
$\displaystyle=0,$ (B.14c) $\displaystyle\Delta(a^{\mu})$
$\displaystyle={\Lambda^{\mu}}_{\nu}\otimes a^{\nu}+a^{\mu}\otimes 1,$ (B.14d)
$\displaystyle\Delta({\Lambda^{\mu}}_{\nu})$
$\displaystyle={\Lambda^{\mu}}_{\alpha}\otimes{\Lambda^{\alpha}}_{\nu},$
(B.14e) $\displaystyle\varepsilon(a^{\mu})$ $\displaystyle=0,$ (B.14f)
$\displaystyle\varepsilon({\Lambda^{\mu}}_{\nu})$
$\displaystyle={\delta^{\mu}}_{\nu},$ (B.14g) $\displaystyle S(a^{\mu})$
$\displaystyle=-a^{\nu}{(\Lambda^{-1})^{\mu}}_{\nu},$ (B.14h) $\displaystyle
S({\Lambda^{\mu}}_{\nu})$ $\displaystyle={(\Lambda^{-1})^{\mu}}_{\nu}.$
(B.14i)

This quantum group has a well-known bicrossproduct structure given in terms of
the decomposition

$\mathcal{C}_{\kappa}(P)=\mathcal{T}_{\kappa}^{*}\vartriangleright\\!\blacktriangleleft\mathcal{C}(SO(1,3)),$
(B.15)

where $C(SO(1,3))$ is the classical commutative algebra of continuous
functions on the Lorentz group and $\mathcal{T}_{\kappa}^{*}$ the algebra of
functions on the dual of the translational sector, defined by

$\displaystyle[a^{\mu},a^{\nu}]$
$\displaystyle=i\lambda({\delta^{\mu}}_{0}a^{\nu}-{\delta^{\nu}}_{0}a^{\mu}),$
(B.16a) $\displaystyle\Delta(a^{\mu})$ $\displaystyle=a^{\mu}\otimes
1+1\otimes a^{\mu},$ (B.16b) $\displaystyle S(a^{\mu})$
$\displaystyle=-a^{\mu},$ (B.16c) $\displaystyle\varepsilon(a^{\mu})$
$\displaystyle=0.$ (B.16d)

The left coaction and the right action are respectively given by

$\displaystyle\beta_{L}(x^{\mu})$ $\displaystyle={\Lambda^{\mu}}_{\nu}\otimes
x^{\nu},$ (B.17a) $\displaystyle{\Lambda^{\alpha}}_{\beta}\vartriangleleft
x^{\varrho}$
$\displaystyle=-{i}\lambda(({\Lambda^{\alpha}}_{0}-{\delta^{\alpha}}_{0}){\Lambda^{\rho}}_{\beta}+(\Lambda_{0\beta}-g_{0\beta})g^{\alpha\rho}),$
(B.17b)

where the right action is read off by the mixed commutators, and the left
coaction is implicitly defined by the coproduct, as discussed in [1]. In this
way we can explicitly see that
$\mathcal{T}_{\kappa}^{*}\sim\mathcal{M}_{\kappa}$, and we can obtain the
noncommutative spacetime from the quantum group via a quotient procedure
similar to that of the classical case, noting that in this quantum version the
quotient should be performed with respect to the deformed action.

Since this result is well-known in the literature, we will omit the details.
We exhibit, in this paper, explicit calculations only for the novel case of
$\mathcal{C}_{\varrho}(P)$.

##  Appendix C Proof of Proposition 4.1

In order to complete the proof of Proposition 4.1 we have to verify that the
action and the coaction satisfy (B.4a)-(B.4d).

We start proving (B.4a). From (4.7) and (4.3f), and since the counit map is a
homomorphism:

$\varepsilon({\Lambda^{\mu}}_{\nu}\triangleleft
x^{\rho})=i\varrho\left[{\delta^{\rho}}_{0}(g_{2\nu}{\delta^{\mu}}_{1}-g_{1\nu}{\delta^{\mu}}_{2})-{\delta^{\rho}}_{0}({\delta^{\mu}}_{1}g_{2\nu}-{\delta^{\mu}}_{2}g_{1\nu})\right]=0,$
(C.1)

while, from (4.3i),

$\varepsilon({\Lambda^{\mu}}_{\nu})\varepsilon(x^{\rho})=0,$ (C.2)

so that (B.4a) is true.

We next consider (B.4b). Starting from the left-hand side we have that
(omitting $i\varrho$ factors)

$\Delta({\Lambda^{\alpha}}_{\beta}\triangleleft
x^{\rho})=\Delta({\Lambda^{\rho}}_{0})[\Delta({\Lambda^{\alpha}}_{1})g_{2\beta}-\Delta({\Lambda^{\alpha}}_{2})g_{1\beta}]-{\delta^{\rho}}_{0}[\Delta(\Lambda_{2\beta}){\delta^{\alpha}}_{1}-\Delta(\Lambda_{1\beta}){\delta^{\alpha}}_{2}],$
(C.3)

since the coproduct is an homomorphism.

For the right-hand side we have

$\displaystyle({\Lambda^{\alpha}}_{\lambda}\triangleleft
x^{\rho})\otimes{\Lambda^{\lambda}}_{\beta}+{\Lambda^{\alpha}}_{\lambda}{\Lambda^{\rho}}_{\sigma}\otimes({\Lambda^{\lambda}}_{\beta}\triangleleft
x^{\sigma})$
$\displaystyle={{\Lambda^{\alpha}}_{\lambda}{\Lambda^{\rho}}_{\sigma}\otimes{\Lambda^{\sigma}}_{0}({\Lambda^{\lambda}}_{1}g_{2\beta}-{\Lambda^{\lambda}}_{2}g_{1\beta})-\delta^{\rho}}_{0}(\Lambda_{2\lambda}{\delta^{\alpha}}_{1}-\Lambda_{1\lambda}{\delta^{\alpha}}_{2})\otimes{\Lambda^{\lambda}}_{\beta}.$
(C.4)

The terms factorized by ${\delta^{\rho}}_{0}$ are exactly equal to those
coming from the coproducts factorized by ${\delta^{\rho}}_{0}$ in the left-
hand side; moreover, the remaining terms can be checked to be equal to the
remaining ones of the left-hand side, thus proving (B.4b).

For (B.4c), from the right-hand side we obtain

$\beta([x^{\mu},x^{\nu}])=[{\delta^{\mu}}_{0}(\Lambda_{2\alpha}{\delta^{\nu}}_{1}-\Lambda_{1\alpha}{\delta^{\nu}}_{2})-{\delta^{\nu}}_{0}(\Lambda_{2\alpha}{\delta^{\mu}}_{1}-\Lambda_{1\alpha}{\delta^{\mu}}_{2})]\otimes
x^{\alpha},$ (C.5)

while the right-hand side is

$\displaystyle\beta([x^{\mu},x^{\nu}])=$
$\displaystyle({\Lambda^{\mu}}_{\alpha}\triangleleft
x^{\nu}-{\Lambda^{\nu}}_{\alpha}\triangleleft x^{\mu})\otimes
x^{\alpha}+{\Lambda^{\mu}}_{\alpha}{\Lambda^{\nu}}_{\beta}\otimes[x^{\alpha},x^{\beta}]$
$\displaystyle=$
$\displaystyle[{\Lambda^{\nu}}_{0}({\Lambda^{\mu}}_{1}g_{2\alpha}-{\Lambda^{\mu}}_{2}g_{1\alpha})-{\delta^{\nu}}_{0}(\Lambda_{2\alpha}{\delta^{\mu}}_{1}-\Lambda_{1\alpha}{\delta^{\mu}}_{2})$
$\displaystyle+{\delta^{\mu}}_{0}(\Lambda_{2\alpha}{\delta^{\nu}}_{1}+\Lambda_{1\alpha}{\delta^{\nu}}_{2})-{\Lambda^{\mu}}_{0}({\Lambda^{\nu}}_{1}g_{2\alpha}-{\Lambda^{\nu}}_{2}g_{1\alpha})]\otimes
x^{\alpha}$
$\displaystyle+{\Lambda^{\mu}}_{\alpha}{\Lambda^{\nu}}_{\beta}\otimes[{\delta^{\alpha}}_{0}(x_{2}{\delta^{\beta}}_{1}-x_{1}{\delta^{\beta}}_{2})-{\delta^{\beta}}_{0}(x_{2}{\delta^{\alpha}}_{1}-x_{1}{\delta^{\alpha}}_{2})];$
(C.6)

terms factorized by ${\delta^{\nu}}_{0}$ and ${\delta^{\mu}}_{0}$ are equal to
(C.5), and the remaining ones cancel out.

Let us turn our attention to (B.4d). To perform the calculation we recall
(4.5b) and (4.9). Computing the left-hand side we have:

${x^{\nu}_{(1)}}^{(\bar{1})}({\Lambda^{\alpha}}_{\beta}\triangleleft
x^{\nu}_{(2)})\otimes{x^{\nu}_{(1)}}^{(\bar{2})}={\Lambda^{\nu}}_{\lambda}({\Lambda^{\alpha}}_{\beta}\triangleleft
1)\otimes x^{\lambda}+({\Lambda^{\alpha}}_{\beta}\triangleleft x^{\nu})\otimes
1,$ (C.7)

while, for the right-hand side:

$({\Lambda^{\alpha}}_{\beta}\triangleleft
x^{\nu}_{(1)}){x^{\nu}_{(2)}}^{(\bar{1})}\otimes{x^{\nu}_{(2)}}^{(\bar{2})}=({\Lambda^{\alpha}}_{\beta}\triangleleft
1){\Lambda^{\nu}}_{\lambda}\otimes
x^{\lambda}+({\Lambda^{\alpha}}_{\beta}\triangleleft x^{\nu})\otimes 1,$ (C.8)

and by means of the commutativity of $\Lambda$’s, the compatibility request is
proved.

##  Appendix D Some explicit calculations of the commutation relations for
$\mathcal{U}_{\varrho}(\mathfrak{p})$

In this appendix we provide explicit details of the computations needed to
prove that the algebra sector of the novel $\varrho$-Poincaré basis remains
undeformed.

We refer to the change of basis (4.25) and remind the reader that the un-
tilded generators close the standard Poincaré algebra. We want to prove that,
after the change of basis, the algebra sector remains the same.

The commutators involving only rotations and momenta are easily retrieved. As
an example, let us compute

$\displaystyle=$
$\displaystyle\left[R_{1}\cos(\frac{\varrho}{2}P_{0})-R_{2}\sin(\frac{\varrho}{2}P_{0}),P_{2}\cos(\frac{\varrho}{2}P_{0})+P_{1}\sin(\frac{\varrho}{2}P_{0})\right]$
(D.1) $\displaystyle=$ $\displaystyle
iP_{3}\cos^{2}(\frac{\varrho}{2}P_{0})+iP_{3}\sin^{2}(\frac{\varrho}{2}P_{0})=iP_{3}=i\tilde{P_{3}}.$

When the the third component of momentum is involved we have,

$\displaystyle=$
$\displaystyle[R_{1}\cos(\frac{\varrho}{2}P_{0})-R_{2}\sin(\frac{\varrho}{2}P_{0}),P_{3}]$
(D.2) $\displaystyle=$ $\displaystyle-
iP_{2}\cos(\frac{\varrho}{2}P_{0})-iP_{1}\sin(\frac{\varrho}{2}P_{0})=-i\tilde{P_{2}},$

giving the standard Poincaré result. With analogous calculations we conclude
that

$[\tilde{R_{i}},\tilde{P_{j}}]=i\varepsilon_{ij}{}^{k}P_{k}.$ (D.3)

Lie brackets involving the boost generators are more involved, since they
imply an action of the boost generators on the trigonometric functions. Let us
compute, as an example,

$\displaystyle=\left[N_{1}\cos(\frac{\varrho}{2}P_{0})-N_{2}\sin(\frac{\varrho}{2}P_{0})+\frac{\varrho}{2}R_{3}\tilde{P_{1}},P_{1}\cos(\frac{\varrho}{2}P_{0})-P_{2}\sin(\frac{\varrho}{2}P_{0})\right].$
(D.4)

By exploiting the following result (easily proven by induction)

$[N_{1},(P_{0})^{k}]=ik(P_{0})^{k-1}P_{1},\qquad k\in\mathbb{N},$

the commutators involving boosts and trigonometric functions of $P_{0}$ can be
verified to be

$\displaystyle=-i\frac{\varrho}{2}\sin(\frac{\varrho}{2}P_{0})P_{1},\qquad[N_{1},\sin(\frac{\varrho}{2}P_{0})]=i\frac{\varrho}{2}\cos(\frac{\varrho}{2}P_{0})P_{1}.$
(D.5)

An analogous calculation is also valid for $N_{2}$. Threefore we can compute
all non-trivial terms in (D.4).

$\displaystyle[N_{1}\cos(\frac{\varrho}{2}P_{0}),P_{1}\cos(\frac{\varrho}{2}P_{0})]=iP_{0}\cos^{2}(\frac{\varrho}{2}P_{0})-i\frac{\varrho}{2}P_{1}^{2}\cos(\frac{\varrho}{2}P_{0})\sin(\frac{\varrho}{2}P_{0}),$
(D.6)
$\displaystyle[N_{1}\cos(\frac{\varrho}{2}P_{0}),P_{2}\sin(\frac{\varrho}{2}P_{0})]=i\frac{\varrho}{2}P_{1}P_{2}\cos^{2}(\frac{\varrho}{2}P_{0}),$
$\displaystyle[N_{2}\sin(\frac{\varrho}{2}P_{0}),P_{1}\cos(\frac{\varrho}{2}P_{0})]=-i\frac{\varrho}{2}P_{1}P_{2}\sin^{2}(\frac{\varrho}{2}P_{0}),$
$\displaystyle[N_{2}\sin(\frac{\varrho}{2}P_{0}),P_{2}\sin(\frac{\varrho}{2}P_{0})]=iP_{0}\cos^{2}(\frac{\varrho}{2}P_{0})+i\frac{\varrho}{2}P_{2}^{2}\cos(\frac{\varrho}{2}P_{0})\sin(\frac{\varrho}{2}P_{0}).$

Putting these results together with the correct signs displayed in (D.4) we
obtain

$i(P_{0}-\frac{\varrho}{2}(P_{1}\cos(\frac{\varrho}{2}P_{0})-P_{2}\sin(\frac{\varrho}{2}P_{0}))(P_{2}\cos(\frac{\varrho}{2}P_{0})+P_{1}\sin(\frac{\varrho}{2}P_{0})))=i(\tilde{P_{0}}-\frac{\varrho}{2}\tilde{P_{1}}\tilde{P_{2}}).$
(D.7)

The remaining term in the commutator involving the rotation generator yields

$[\frac{\varrho}{2}\tilde{P_{1}}R_{3},\tilde{P_{1}}]=i\frac{\varrho}{2}\tilde{P_{1}}\tilde{P_{2}},$
(D.8)

exactly canceling the other term proportional to $\frac{\varrho}{2}$.
Therefore we conclude that

$[\tilde{N_{1}},\tilde{P_{1}}]=i\tilde{P_{0}}.$ (D.9)

Similar techniques can be used to show that the commutators between the novel
boost generators and the novel translation generators are the standard
Poincaré ones.

As for the Lorentz sector, analogous computations show that the commutators
between rotations and boosts are also undeformed. The tricky ones are those
involving only boost generators. Let us analyze the one between
$\tilde{N_{1}}$ and $\tilde{N_{2}}$:

$\displaystyle[\tilde{N_{1}},\tilde{N_{2}}]=$ (D.10)
$\displaystyle=[N_{1}\cos(\frac{\varrho}{2}P_{0})-N_{2}\sin(\frac{\varrho}{2}P_{0})+\frac{\varrho}{2}R_{3}\tilde{P_{1}},N_{2}\cos(\frac{\varrho}{2}P_{0})+N_{1}\sin(\frac{\varrho}{2}P_{0})+\frac{\varrho}{2}R_{3}\tilde{P_{2}}].$

Introducing the shorthand notation

$\displaystyle
A=N_{1}\cos(\frac{\varrho}{2}P_{0})-N_{2}\sin(\frac{\varrho}{2}P_{0}),\qquad
B=\frac{\varrho}{2}R_{3}\tilde{P_{1}},$ (D.11) $\displaystyle
C=N_{2}\cos(\frac{\varrho}{2}P_{0})+N_{1}\sin(\frac{\varrho}{2}P_{0}),\qquad
D=\frac{\varrho}{2}R_{3}\tilde{P_{2}},$

we need to compute the quantity $[A,C]+[B,C]+[A,D]+[B,D]$. We start with
$[A,C]$ and further break it down in commutators involving the boosts with no
tilde.

$\displaystyle[N_{1}\cos(\frac{\varrho}{2}P_{0}),N_{2}\cos(\frac{\varrho}{2}P_{0})]=[N_{1},N_{2}]\cos(\frac{\varrho}{2}P_{0})+N_{1}[\cos(\frac{\varrho}{2}P_{0}),N_{2}]\cos(\frac{\varrho}{2}P_{0})$
(D.12)
$\displaystyle+N_{2}[N_{1},\cos(\frac{\varrho}{2}P_{0})]\cos(\frac{\varrho}{2}P_{0})=i\bigl{(}-R_{3}\cos^{2}(\frac{\varrho}{2}P_{0})+\frac{\varrho}{2}\cos(\frac{\varrho}{2}P_{0})\sin(\frac{\varrho}{2}P_{0})(N_{1}P_{2}-N_{2}P_{1})\bigr{)},$
$\displaystyle[N_{1}\cos(\frac{\varrho}{2}P_{0}),N_{1}\sin(\frac{\varrho}{2}P_{0})]=i\frac{\varrho}{2}N_{1}P_{1},$
$\displaystyle[N_{2}\sin(\frac{\varrho}{2}P_{0}),N_{2}\cos(\frac{\varrho}{2}P_{0})]=-i\frac{\varrho}{2}N_{2}P_{2},$
$\displaystyle[N_{2}\sin(\frac{\varrho}{2}P_{0}),N_{1}\sin(\frac{\varrho}{2}P_{0})]=i\bigl{(}R_{3}\sin^{2}(\frac{\varrho}{2}P_{0})+\frac{\varrho}{2}\sin(\frac{\varrho}{2}P_{0})\cos(\frac{\varrho}{2}P_{0})(N_{1}P_{2}-N_{2}P_{1}).$

Adding these up with the signs dictated by (LABEL:n1n2), we get

$[A,C]=i\bigl{(}-R_{3}+\frac{\varrho}{2}{N_{1}P_{1}}+\frac{\varrho}{2}{N_{2}P_{2}}\bigr{)}.$
(D.13)

Next, we compute $[A+B,D]$, given by

$[\tilde{N_{1}},\frac{\varrho}{2}R_{3}\tilde{P_{2}}]=\frac{\varrho}{2}[\tilde{N_{1}},R_{3}]\tilde{P_{2}}+\frac{\varrho}{2}[\tilde{N_{1}},\tilde{P_{2}}]=-i\frac{\varrho}{2}\tilde{N_{2}}\tilde{P_{2}}.$
(D.14)

Then, $[B,C]$ can also be easily calculated

$\displaystyle=[\frac{\varrho}{2}R_{3}\tilde{P_{1}},\tilde{N_{2}}-\frac{\varrho}{2}R_{3}\tilde{P_{2}}]=i\bigl{(}-\frac{\varrho}{2}\tilde{N_{1}}\tilde{P_{1}}+\frac{\varrho^{2}}{4}R_{3}\tilde{P_{1}}^{2}+\frac{\varrho^{2}}{4}R_{3}\tilde{P_{2}}^{2}\bigr{)}.$
(D.15)

Collecting the results we have

$[\tilde{N_{1}},\tilde{N_{2}}]=i\bigl{(}-R_{3}+\frac{\varrho}{2}N_{1}P_{1}+\frac{\varrho}{2}N_{2}P_{2}-\frac{\varrho}{2}\tilde{N_{2}}\tilde{P_{2}}-\frac{\varrho}{2}\tilde{N_{1}}\tilde{P_{1}}+\frac{\varrho^{2}}{4}R_{3}\tilde{P_{1}}^{2}+\frac{\varrho^{2}}{4}R_{3}\tilde{P_{2}}^{2}\bigr{)}.$
(D.16)

Using (4.25), we find

$\tilde{N_{1}}\tilde{P_{1}}+\tilde{N_{2}}\tilde{P_{2}}=N_{1}P_{1}+N_{2}P_{2}+\frac{\varrho}{2}R_{3}\tilde{P_{2}}^{2}++\frac{\varrho}{2}R_{3}\tilde{P_{1}}^{2},$
(D.17)

so that

$[\tilde{N_{1}},\tilde{N_{2}}]=-iR_{3}=-i\tilde{R_{3}},$ (D.18)

like the in the standard Poincaré case. Commutators between
$\tilde{N_{1}},\tilde{N_{2}}$ with $\tilde{N_{3}}$ are computed in a similar
fashion, but with fewer intermediate steps.

##  Appendix E Plane waves in $\kappa$-Minkowski

The $\kappa$-Minkowski spacetime is characterized by a noncommutativity
between coordinates given in (2.2). Following [52, 53], we represent the
coordinates in terms of $5\times 5$ matrices as follows:

$\hat{x}^{0}=-i\ell\begin{pmatrix}0&0&0&0&1\\\ 0&0&0&0&0\\\ 0&0&0&0&0\\\
1&0&0&0&0\end{pmatrix},\quad\hat{x}^{i}=i\ell\begin{pmatrix}0&-\mathbf{e}^{i}&0\\\
-{\mathbf{e}^{i}}^{T}&\mathbf{0}_{3\crossproduct 3}&-{\mathbf{e}^{i}}^{T}\\\
0&\mathbf{e}^{i}&0\end{pmatrix},$ (E.1)

where $\\{\mathbf{e}^{i}\\}_{i=1,2,3}$ is the canonical vector basis for
$\mathbb{R}^{3}$.

We find that also in this case the time-to-the-right ordering of plane waves
is linked to the translation generators in the bicrossproduct basis. Indeed,
noncommutative $\kappa$-plane waves in the time-to-the-right ordering,

$\hat{\phi}_{R}(p)=e^{ip_{i}\hat{x}^{i}}e^{ip_{0}\hat{x}^{0}},$ (E.2)

lead to the composition law of momenta

$\begin{cases}(p\oplus_{R}k)_{0}=p_{0}+k_{0},\\\
(p\oplus_{R}k)_{i}=p_{i}+e^{-\ell p_{0}}k_{i}.\end{cases}$ (E.3)

Defining the time-to-the-right translation operator $P^{R}_{\mu}$ such that
$\hat{\phi}(p\oplus_{R}k)$ be an eigenfunction,

$P^{R}_{\mu}\hat{\phi}(p\oplus_{R}k)=(p\oplus_{R}k)_{\mu}\hat{\phi}(p\oplus_{R}k),$
(E.4)

one can show that the coproducts of the time-to-the-right translation
generators are the well-known bicrossproduct ones

$\displaystyle\Delta P^{R}_{0}=P^{R}_{0}\otimes 1+1\otimes P^{R}_{0},$ (E.5)
$\displaystyle\Delta P^{R}_{i}=P^{R}_{i}\otimes 1+e^{-\ell P^{R}_{0}}\otimes
P_{i}^{R}\,.$

On the other hand, noncommutative $\kappa$-plane waves in the time-symmetric
ordering

$\hat{\phi}_{S}(p)=e^{\frac{ip_{0}\hat{x}^{0}}{2}}e^{ip_{i}\hat{x}^{i}}e^{\frac{ip_{0}\hat{x}^{0}}{2}},$
(E.6)

lead to the composition law

$\begin{cases}(p\oplus_{S}k)_{0}=p_{0}+k_{0},\\\
(p\oplus_{S}k)_{i}=p_{i}e^{\frac{\ell k_{0}}{2}}+e^{\frac{-\ell
p_{0}}{2}}k_{i}\,.\end{cases}$ (E.7)

This is compatible with the coproducts for the time-symmetric translation
generators,

$P^{S}_{\mu}\hat{\phi}(p\oplus_{S}k)=(p\oplus_{S}k)_{\mu}\hat{\phi}(p\oplus_{S}k)\,,$
(E.8)

of the form

$\displaystyle\Delta P^{S}_{0}=P^{S}_{0}\otimes 1+1\otimes 1\otimes
P^{S}_{0},$ (E.9) $\displaystyle\Delta P_{i}^{S}=P_{i}^{S}\otimes
e^{\frac{\ell P^{S}_{0}}{2}}+e^{\frac{{-\ell P^{S}_{0}}}{2}}\otimes
P_{i}^{S}.$

These are the coproducts of the $\kappa$-Poincaré translations in the so-
called standard basis [6] and are related to the bicrossproduct ones via the
map [1]

$\displaystyle P_{0}^{R}=P_{0}^{S},$ (E.10) $\displaystyle
P_{i}^{R}=P_{i}^{S}e^{-\frac{\ell P_{0}^{S}}{2}}.$

## References

  * [1] S. Majid and H. Ruegg, _Bicrossproduct structure of kappa Poincaré group and noncommutative geometry_ , _Phys. Lett. B_ 334 (1994) 348 [hep-th/9405107].
  * [2] V. G. Drinfeld, _Constant quasiclassical solutions of the Yang–Baxter quantum equation_ , _Dokl. Akad. Nauk SSSR_ 273 (1983) 531 [http://mi.mathnet.ru/dan9865].
  * [3] T. Jurić, S. Meljanac and A. Samsarov, _Light-like $\kappa$-deformations and scalar field theory via Drinfel’d twist_, _J. Phys. Conf. Ser._ 634 (2015) 012005 [1506.02475].
  * [4] J. Lukierski, H. Ruegg and W. J. Zakrzewski, _Classical quantum mechanics of free kappa relativistic systems_ , _Annals Phys._ 243 (1995) 90 [hep-th/9312153].
  * [5] J. Lukierski, H. Ruegg, A. Nowicki and V. N. Tolstoi, _Q deformation of Poincaré algebra_ , _Phys. Lett. B_ 264 (1991) 331.
  * [6] J. Lukierski, A. Nowicki and H. Ruegg, _New quantum poincaré algebra and $\kappa$-deformed field theory_, _Physics Letters B_ 293 (1992) 344.
  * [7] E. Celeghini, R. Giachetti, E. Sorace and M. Tarlini, _Three dimensional quantum groups from contraction of SU(2)_q_ , _J. Math. Phys._ 31 (1990) 2548.
  * [8] P. P. Kulish, V. D. Lyakhovsky and A. I. Mudrov, _Extended jordanian twists for lie algebras_ , _Journal of Mathematical Physics_ 40 (1999) 4569 [https://doi.org/10.1063/1.532987].
  * [9] M. Gerstenhaber, A. Giaquinto and S. D. Schack, _Quantum symmetry_ , in _Quantum Groups_ , P. P. Kulish, ed., (Berlin, Heidelberg), pp. 9–46, Springer Berlin Heidelberg, 1992.
  * [10] V. N. Tolstoy, _Chains of extended jordanian twists for lie superalgebras_ , 2004.
  * [11] A. Borowiec and A. Pachol, _$\kappa$ -minkowski spacetime as the result of jordanian twist deformation_, _Phys. Rev. D_ 79 (2009) 045012.
  * [12] N. Reshetikhin, _Multiparameter quantum groups and twisted quasitriangular hopf algebras_ , _Letters in Mathematical Physics_ 20 (1990) 331.
  * [13] J.-G. Bu, H.-C. Kim, Y. Lee, C. H. Vac and J. H. Yee, _kappa-deformed spacetime from twist_ , _Phys. Lett. B_ 665 (2008) 95 [hep-th/0611175].
  * [14] T. R. Govindarajan, K. S. Gupta, E. Harikumar, S. Meljanac and D. Meljanac, _Twisted statistics in kappa-Minkowski spacetime_ , _Phys. Rev. D_ 77 (2008) 105010 [0802.1576].
  * [15] P. Kosinski and P. Maslanka, _Lorentz-invariant interpretation of noncommutative space-time: Global version_ , hep-th/0408100.
  * [16] J. Lukierski and M. Woronowicz, _New lie-algebraic and quadratic deformations of minkowski space from twisted poincaré symmetries_ , _Physics Letters B_ 633 (2006) 116.
  * [17] R. Oeckl, _Untwisting noncommutative R**d and the equivalence of quantum field theories_ , _Nucl. Phys. B_ 581 (2000) 559 [hep-th/0003018].
  * [18] J. Wess, _Deformed coordinate spaces: Derivatives_ , in _1st Balkan Workshop on Mathematical, Theoretical and Phenomenological Challenges Beyond the Standard Model: Perspectives of Balkans Collaboration_ , pp. 122–128, 2003, hep-th/0408080, DOI.
  * [19] M. Chaichian, P. P. Kulish, K. Nishijima and A. Tureanu, _On a Lorentz-invariant interpretation of noncommutative space-time and its implications on noncommutative QFT_ , _Phys. Lett. B_ 604 (2004) 98 [hep-th/0408069].
  * [20] M. Chaichian, P. Presnajder and A. Tureanu, _New concept of relativistic invariance in NC space-time: Twisted Poincare symmetry and its implications_ , _Phys. Rev. Lett._ 94 (2005) 151602 [hep-th/0409096].
  * [21] M. Dimitrijević Ćirić, N. Konjik and A. Samsarov, _Noncommutative scalar field in the nonextremal Reissner-Nordström background: Quasinormal mode spectrum_ , _Phys. Rev. D_ 101 (2020) 116009 [1904.04053].
  * [22] F. Lizzi and P. Vitale, _Time discretization from noncommutativity_ , _Phys. Lett. B_ 818 (2021) 136372 [2101.06633].
  * [23] F. Lizzi, L. Scala and P. Vitale, _Localization and observers in $\varrho$-Minkowski spacetime_, _Phys. Rev. D_ 106 (2022) 025023 [2205.10862].
  * [24] M. Dimitrijevic Ciric, N. Konjik, M. A. Kurkov, F. Lizzi and P. Vitale, _Noncommutative field theory from angular twist_ , _Phys. Rev. D_ 98 (2018) 085011 [1806.06678].
  * [25] F. Lizzi, M. Manfredonia and F. Mercati, _The momentum spaces of $\kappa$-Minkowski noncommutative spacetime_, _Nucl. Phys. B_ 958 (2020) 115117 [2001.08756].
  * [26] A. Sitarz, _Noncommutative differential calculus on the kappa Minkowski space_ , _Phys. Lett. B_ 349 (1995) 42 [hep-th/9409014].
  * [27] J. Kowalski-Glikman and S. Nowak, _Noncommutative space-time of doubly special relativity theories_ , _Int. J. Mod. Phys. D_ 12 (2003) 299 [hep-th/0204245].
  * [28] M. Dimitrijevic, L. Jonke, L. Moller, E. Tsouchnika, J. Wess and M. Wohlgenannt, _Deformed field theory on kappa space-time_ , _Eur. Phys. J. C_ 31 (2003) 129 [hep-th/0307149].
  * [29] G. Amelino-Camelia, G. Gubitosi, A. Marciano, P. Martinetti and F. Mercati, _A no-pure-boost uncertainty principle from spacetime noncommutativity_ , _Phys. Lett. B_ 671 (2009) 298 [0707.1863].
  * [30] M. Arzano, J. Kowalski-Glikman and A. Walkus, _Lorentz invariant field theory on kappa-Minkowski space_ , _Class. Quant. Grav._ 27 (2010) 025012 [0908.1974].
  * [31] F. Lizzi, M. Manfredonia, F. Mercati and T. Poulain, _Localization and reference frames in $\kappa$-Minkowski spacetime_, _Phys. Rev. D_ 99 (2019) 085003 [1811.08409].
  * [32] A. Ballesteros, G. Gubitosi, I. Gutierrez-Sagredo and F. Mercati, _Fuzzy worldlines with $\kappa$-Poincaré symmetries_, _JHEP_ 12 (2021) 080 [2109.09699].
  * [33] M. Arzano and J. Kowalski-Glikman, _A group theoretic description of the $\kappa$-Poincaré Hopf algebra_, _Phys. Lett. B_ 835 (2022) 137535 [2204.09394].
  * [34] G. Amelino-Camelia, _Quantum-spacetime phenomenology_ , _Living Rev. Rel._ 16 (2013) 5 [0806.0339].
  * [35] J. Lukierski and H. Ruegg, _Quantum kappa Poincaré in any dimension_ , _Phys. Lett. B_ 329 (1994) 189 [hep-th/9310117].
  * [36] J. Kowalski-Glikman and S. Nowak, _Doubly special relativity theories as different bases of kappa Poincaré algebra_ , _Phys. Lett. B_ 539 (2002) 126 [hep-th/0203040].
  * [37] G. Gubitosi and F. Mercati, _Relative locality in $\kappa$-Poincaré_, _Class. Quant. Grav._ 30 (2013) 145002 [1106.5710].
  * [38] G. Gubitosi and S. Heefer, _Relativistic compatibility of the interacting $\kappa$-Poincaré model and implications for the relative locality framework_, _Phys. Rev. D_ 99 (2019) 086019 [1903.04593].
  * [39] S. Gutt, _An explicit $*$-product on the cotangent bundle of a lie group_, _Letters in Mathematical Physics_ 7 (1983) 249.
  * [40] J. M. Gracia-Bondia, F. Lizzi, G. Marmo and P. Vitale, _Infinitely many star products to play with_ , _JHEP_ 04 (2002) 026 [hep-th/0112092].
  * [41] G. Amelino-Camelia, L. Barcaroli, S. Bianco and L. Pensato, _Planck-scale dual-curvature lensing and spacetime noncommutativity_ , _Adv. High Energy Phys._ 2017 (2017) 6075920 [1708.02429].
  * [42] M. D. Ćirić, N. Konjik and A. Samsarov, _Noncommutative scalar quasinormal modes of the Reissner–Nordström black hole_ , _Class. Quant. Grav._ 35 (2018) 175005 [1708.04066].
  * [43] M. D. Ćirić, N. Konjik and A. Samsarov, _Search for footprints of quantum spacetime in black hole QNM spectrum_ , 10, 2019, 1910.13342.
  * [44] G. Amelino-Camelia, L. Barcaroli and N. Loret, _Modeling transverse relative locality_ , _Int. J. Theor. Phys._ 51 (2012) 3359 [1107.3334].
  * [45] K. Hersent and J.-C. Wallet, _Field theories on $\rho$-deformed Minkowski space-time_, 2304.05787.
  * [46] M. Kurkov and P. Vitale, _Four-dimensional noncommutative deformations of U(1) gauge theory and $L_{\infty}$ bootstrap._, _JHEP_ 01 (2022) 032 [2108.04856].
  * [47] G. Gubitosi, F. Lizzi, J. J. Relancio and P. Vitale, _Double quantization_ , _Phys. Rev. D_ 105 (2022) 126013 [2112.11401].
  * [48] S. Majid, _Physics for algebraists: Non-commutative and non-cocommutative hopf algebras by a bicrossproduct construction_ , _Journal of Algebra_ 130 (1990) 17.
  * [49] S. Majid, _Matched pairs of Lie groups associated to solutions of the Yang-Baxter equations._ , _Pacific Journal of Mathematics_ 141 (1990) 311 .
  * [50] W. M. Singer, _Extension theory for connected hopf algebras_ , _Journal of Algebra_ 21 (1972) 1.
  * [51] G. Amelino-Camelia and S. Majid, _Waves on noncommutative space-time and gamma-ray bursts_ , _Int. J. Mod. Phys. A_ 15 (2000) 4301 [hep-th/9907110].
  * [52] L. Freidel, J. Kowalski-Glikman and S. Nowak, _Field theory on kappa-Minkowski space revisited: Noether charges and breaking of Lorentz symmetry_ , _Int. J. Mod. Phys. A_ 23 (2008) 2687 [0706.3658].
  * [53] J. Kowalski-Glikman, _Living in Curved momentum Space_ , _Int. J. Mod. Phys. A_ 28 (2013) 1330014 [1303.0195].
  * [54] G. Felder and B. Shoikhet, _Deformation quantization with traces_ , math/0002057.
  * [55] S. Galluccio, F. Lizzi and P. Vitale, _Twisted Noncommutative Field Theory with the Wick-Voros and Moyal Products_ , _Phys. Rev. D_ 78 (2008) 085007 [0810.2095].
  * [56] S. Galluccio, F. Lizzi and P. Vitale, _Translation Invariance, Commutation Relations and Ultraviolet/Infrared Mixing,_ _JHEP_ 09 (2009), 054.
  * [57] V. G. Kupriyanov and P. Vitale, _Noncommutative ${\mathrm{\mathbb{R}}}^{d}$ via closed star product,_ _JHEP_ 08 (2015), 024.
  * [58] G. Amelino-Camelia, L. Barcaroli, G. D’Amico, N. Loret and G. Rosati, _Quantum-gravity-induced dual lensing and IceCube neutrinos_ , _Int. J. Mod. Phys. D_ 26 (2017) 1750076 [1609.03982].
  * [59] P. Kosinski and P. Maslanka, _The $\kappa$-Weyl group and its algebra_, in _From Field Theory to Quantum Groups_ , pp. 41–51, World Scientific, (jun, 1996), DOI.
  * [60] P. Zaugg, _The quantum Poincaré group from quantum group contraction_ , _J. Phys._ 28 (1995) 2589 [hep-th/9409100].
  * [61] A. Agostini, _Fields and symmetries in kappa-Minkowski noncommutative spacetime_ , Università di Napoli Federico II thesis, 2003.  hep-th/0312305.
  * [62] V. Chari and A. Pressley, _A Guide to Quantum Groups_. Cambridge University Press, 1995.
  * [63] S. Zakrzewski, _Quantum Poincaré group related to the kappa -Poincaré algebra_ , _Journal of Physics A: Mathematical and General_ 27 (1994) 2075.
  * [64] J. Lukierski and H. Ruegg, _Quantum $\kappa$-Poincaré in any dimension_, _Physics Letters B_ 329 (1994) 189.