Group Momentum Space and Hopf Algebra Symmetries of Point Particles Coupled to 2+1 Gravity

We present an in-depth investigation of the ${\rm SL}(2,\mathbb{R})$ momentum space describing point particles coupled to Einstein gravity in three space-time dimensions. We introduce different sets of coordinates on the group manifold and discuss their properties under Lorentz transformations. In particular we show how a certain set of coordinates exhibits an upper bound on the energy under deformed Lorentz boosts which saturate at the Planck energy. We discuss how this deformed symmetry framework is generally described by a quantum deformation of the Poincar\'e group: the quantum double of ${\rm SL}(2,\mathbb{R})$. We then illustrate how the space of functions on the group manifold momentum space has a dual representation on a non-commutative space of coordinates via a (quantum) group Fourier transform. In this context we explore the connection between Weyl maps and different notions of (quantum) group Fourier transform appeared in the literature in the past years and establish relations between them.


Introduction
General relativity in three space-time dimensions offers an unparalleled insight on how quantum group deformed relativistic symmetries replace ordinary structures linked to the Poincaré group. The simple picture is that relativistic point particles are coupled to the theory as conical defects owing to the topological nature of Einstein gravity in three dimensions [1,2]. For vanishing cosmological constant the resulting space-time is flat everywhere except at the location of the particle, the conical singularity. The parametrization of a moving defect, like e.g. the conical particle, requires their momentum to be described by an element of the isometry group of the ambient space. In our specific case it turns out that momenta are no longer vectors in three dimensional Minkowski space but elements of the double cover of the Lorentz group in three dimensions, SL(2, R) [3,4]. Lorentz transformations are easily implemented for this curved momentum space in terms of the action of SL(2, R) on itself. At the group theoretic level such transition from vector-like to group-like momenta is captured by the transition from the (group algebra of) the Poincaré group to the quantum double of SL(2, R) a non-trivial Hopf algebra in which the three-dimensional Newton constant appears as a deformation parameter [5,6]. Functions on SL(2, R) provide now the space on which representations of the double of SL(2, R) are defined [7,8]. A notion of Fourier transform can be introduced which maps these functions on a group onto functions of non-commutative space in which coordinates obey non-trivial commutation relations given by the brackets of the Lie algebra of SL(2, R) [9]. Alternatively one can map these functions onto ordinary three-dimensional Minkowski space equipped with a non-commutative star-product determined by the non-abelian group structure of SL(2, R) [10]. Thus the study of relativistic point particles coupled to three-dimensional gravity leads us to consider all the important aspects of deformation of relativistic symmetries and non-commutativity within a clear geometric picture. Our aim in this work will be to review the main features discussed above and point out some new results which will help link the structures emerging in this three-dimensional context to other models of deformed symmetries and non-commutativity in four space-time dimensions. We start in the next Section by reviewing how momenta of conical defects in three dimensions are parametrized by SL(2, R) group elements. In particular we show how the usual notion of massshell describing the momentum of physical particles is replaced by the condition that group valued momenta belong to the conjugacy class of a rotation by a deficit angle characterizing the conical defect. In Section 3 we describe coordinate systems on SL(2, R) starting with two choices which are popular in the literature and introducing a new parametrization based on Euler angles. For each parametrization we write down the explicit form of the conjugacy class condition which represents the realization of a deformed mass-shell in each choice of coordinates and show that for the new parametrization introduced on-shell momenta exhibit a maximum energy. In Section 4 we discuss the action of Lorentz transformations on the SL(2, R) momentum space again analyzing the case of each coordinate set separately. We show that the Euler angle coordinates transform non-linearly under the action of a boost and that the energy saturates at the maximum value when the boost parameter is let to infinity. Thus we find a set of momentum coordinates which provide an explicit realization of doubly or deformed special relativity for a particle coupled to three-dimensional gravity [11,12]. In Section 5 we review what is the structure of the underlying quantum group symmetry associated with the SL(2, R) momentum space given by the quantum double of SL(2, R) and we show how such Hopf algebra can be obtained as a deformation of the group algebra of the Poincaré group. Next we discuss noncommutative plane waves with SL(2, R) momenta and the associated notions of (quantum) group Fourier transform. This gives us the opportunity to clarify the connection between different notions of Fourier transform appearing in the literature and to discuss their relation with Weyl maps which have been extensively used in the literature on non-commutative space-times in 3 + 1 dimensions [13,14]. Finally we provide a master equation for the group Fourier transform which does not rely on a specific notion of star-product and it's valid for any parametrization of the momentum group manifold. Section 6 is devoted to brief concluding remarks.
2 The SL(2, R) momentum space of particles coupled to 2+1 dimensional gravity

Particles as conical defects in 2+1 gravity
General relativity in three space-time dimensions has the well known property of not possessing local degrees of freedom. All solutions of Einstein's equations are locally flat and the only nontrivial degrees of freedom have to be of topological nature. This must be taken into account when coupling particles to the theory. Indeed, as already shown by Staruszkiewicz in his seminal work [1], later thoroughly extended by Deser, Jackiw and 't Hooft [2], point-like degrees of freedom carrying mass and spin can be added introducing "punctures" on the space-time (flat) manifold.
In the simplest case of a spinless particle, which will be of interest for the present work, one can picture such a puncture as creating a conical singularity resulting in a space-time which is flat everywhere except at the location of the particle. This conical space can be characterized by a deficit angle α = 8πGm where m is the mass of the particle and G is Newton's constant in three dimensions. The deficit angle is simply the angle formed by the missing wedge obtained when "opening up" the conical space on a flat Minkowski space. The metric of such conical space-time is given by with the azimuthal angle ϕ having period 2π − α. Alternatively we can write the metric keeping the angular ϕ-coordinate with a period 2π as: The deficit angle can be measured by calculating the holonomy of a loop around the location of the particle, describing the parallel transport of a vector around the singularity [3], which results in a rotation by an angle α = 8πGm. Roughly speaking we have that the mass i.e. the four-momentum at rest of the point particle/defect is characterized by a rotation proportional to the mass of the particle multiplied by Newton's constant. To characterize the physical momentum of a moving defect let us briefly recall how physical momenta of a relativistic point particle are described in ordinary Minkowski space. In three space-time dimensions Minkowski space R 2,1 is isomorphic, as a vector space, to sl(2, R), the Lie algebra of SL(2, R) the double cover 1 of the proper Lorentz group SO + (1, 2). If γ µ is a basis of traceless 2 × 2 -matrices [3], given an element p ∈ sl(2, R) and a vector p ∈ R 2,1 we can write Momenta are thus given by the following sl(2, R) algebra element whose determinant is det p = (p 0 ) 2 − |p| 2 . We can describe physical momenta by boosting the momentum at rest of a particle. The latter will be given in matrix representation by p = mγ 0 = 0 m −m 0 and the boost is achieved via the adjoint action of SL(2, R) on sl(2, R) so that a physical momentum will be given by Such action preserves the determinant and the physical momenta, obtained by boosting the three momentum at rest, will be characterized by the mass-shell condition We thus have that in three-dimensional Minkowski space the extended momentum space of a relativistic point particle can be identified with sl(2, R) and the physical momenta belong to orbits of SL(2, R) on sl(2, R) which determine the mass-shell.
We can now have an intuitive characterization of the momentum space of a moving conical defect in three dimensions (for more formal discussions we refer the reader to [3,8]). As discussed above the momentum at rest of a conical defect can be characterized by a rotation g = e 4πGmγ 0 ∈ SL(2, R) i.e. a group element rather than the vector p = mγ 0 . The action of a Lorentz boost on the rest momentum will be described just by an action of SL(2, R) on itself and thus physical momenta of a defect of mass m will be given by elements of SL(2, R) belonging to the conjugacy class of a rotation by an angle 4πGm. Therefore, when gravity is switched on, the extended momentum space of a point particle is given by the group manifold SL(2, R) (to be contrasted with the vector space sl(2, R) in the ordinary Minkowski case) while its physical momentum space is given by the action by conjugation of SL(2, R) on the momentum at rest g = e 4πGmγ 0 (to be contrasted with the adjoint action in the ordinary case) [3,8]. In the next Section we give a detailed account of these constructions and of the classification of conjugacy classes.

Conjugacy classes as deformed mass-shell
In order to introduce the conjugacy classes of the group SL(2, R) we start by recalling the expansion of the generic group element g ∈ SL(2, R) in terms of the unit and the γ matrices: which can be written in the following matrix form: Now the action by conjugation of SL(2, R) on itself leaves the trace invariant, therefore the number u is an invariant under action by conjugation. In particular, the group SL(2, R) is composed by five different subclasses, called conjugacy classes, which are invariant under conjugation [15].
These subclasses can be described by the eigenvalues of the matrix (9). The secular equation for the generic element of SL(2, R) reads whose solutions are given by We can now classify group elements according to their trace, in fact we see that the above equation has different behaviours according to the value of the discriminant [Tr(g)] 2 − 4, in particular the sign of this term will determine the different sets of solutions, i.e. group elements.
To make contact with the ordinary classification of particles according to the sign of their mass squared, we have here that a group element obtained by conjugating the momentum at rest of a gravitating particle g = cos(4πGm)1 + sin(4πGm)γ 0 = cos(4πGm) sin(4πGm) − sin(4πGm) cos(4πGm) , belongs to the class of elliptic transformations, which represent massive particles. Analogously, hyperbolic elements will represent tachyons and the parabolic elements photons as we can deduce by taking the G → 0 limit and checking that for the former we have a negative mass squared, while for the latter the mass is vanishing. The deformed mass-shell condition assigning a group element to the conjugacy class describing massive particles can be obtained by putting a constraint on its trace which, in light of what we said above, will be given by 1 2 Tr(g) = cos(4πGm) (16) with 0 < m ≤ 1 8G = κπ 2 where we defined a new deformation parameter with dimensions of mass κ = 1 4πG for later convenience. This ends our review of the momentum space of a conical particle described by the SL(2, R) group manifold. In what follows, we will give concrete examples of group parametrizations and their (deformed) mass-shell, before moving on to the description of how relativistic symmetries are implemented for such particles.

Coordinates on SL(2, R)
In this section we recall some known parametrizations of SL(2, R) group elements/momenta and introduce others which will help us get insight on the kinematical properties associated to each set of coordinates.

Cartesian coordinates
Cartesian coordinates on the SL(2, R) group manifold are the most used in the literature. They are based on the parametrization (8) which we introduced in Section 2.2 We define where we introduced the notation p µ κ . = p µ κ (µ = 0, 1, 2) so that the group parameters ξ µ are dimensionless and the coordinates p µ have the dimensions of energy. The deformed mass-shell constraint (16) in Cartesian coordinates reads 1 2 which after easy manipulations gives In the limit κ → ∞ the dispersion relation (20) reproduces the usual mass-shell relation valid for an ordinary (flat) momentum space of a massive particle, i.e. (p 0 ) 2 = |p| 2 +m 2 . The deformation given by the gravity effects, using this parametrization, appears as a renormalization of the mass which is no longer m but M . = κ sin m κ .

Exponential coordinates
This set of coordinates is widely used in the works on Euclidean models (see e.g. [16]) in which SU (2) is considered instead of SL(2, R). The parametrization is defined as follows It is trivial to check that for this parametrization the determinant constraint is satisfied and a generic group element can be written in terms of k -coordinates as where the last equality shows that this parametrization is equivalent to considering the exponential of the sl(2, R) generators, from this the name "exponential coordinates". The conjugacy class constraint this time leads to the mass-shell condition which is just an ordinary mass-shell relation where, however, the values of the mass m are limited by κ π 2 due to the limits on the deficit angle.

Time-symmetric parametrization: Euler coordinates
In this section we develop a set of coordinates on SL(2, R) based on a choice of energy and linear momentum introduced in earlier works on the coupling of point particles to 2 + 1 gravity [3,17,18]. The idea is to describe the momenta of a gravitating particle using a parametrization in terms of angles. If we consider the equation describing an hyperboloid embedded in R 2,2 , i.e. Anti-de Sitter (AdS) space we can introduce two different sets of polar coordinates in the planes u − ξ 0 and ξ 1 − ξ 2 , namely: where the radial coordinate χ ∈ [0, ∞) and the angular coordinates ρ, φ ∈ [0, 2π) are closely related to the Euler angles parametrizing SU (2) [3]. Inserting these expressions in equation (25) we obtain: which, choosing F(χ) = cosh(χ) and G(χ) = sinh(χ), is satisfied. In conclusion our Euler coordinates are given by where χ ∈ [0, ∞) and ρ, φ ∈ [0, 2π). Now, using this set of coordinates, we can rewrite the expansion of the group element g in terms of the γ matrices as: which, using the properties of the γ matrices, can be expressed in the equivalent form [3]: This choice of coordinates shows clearly that the Lorentz transformation represented by g is decomposed in a spatial rotation of angle ρ + φ in the plane γ 1 − γ 2 , a boost in γ 2 direction with the χ parameter and another rotation in the same plane of the previous but by angle ρ − φ.
In order to interpret our coordinates as momenta we rewrite the expansion of the group element (29) using ρ and χ as dimensionful parameters ρ → ρ κ and χ → χ κ , obtaining: Now, as in the previous cases, if we impose the condition which projects the group element into the conjugacy class of rotations, i.e. 1 2 Tr(g) = cos m κ and obtain: which is the deformed mass-shell relation in our new coordinates. If we take the limit κ → ∞ which, simplifying the leading terms and identifying the terms proportional to κ −2 , gives: Such equation reproduces an ordinary relativistic mass-shell relation if one identifies ρ and χ respectively with the energy and the norm of the spatial momentum. This leads us to define the following momenta for the gravitating particles In terms of the new Π -coordinates we have Using this parametrization, the generic group element can be written as as shown in detail in Appendix B. The mass-shell relation (32) can be rewritten in terms of the Π momenta as which solved for the energy gives showing that the energy Π 0 is limited by the range values Π 0 ∈ [m, κπ 2 ]. This was somewhat an expected result, since the energy in this set of coordinates is proportional to the deficit angle but we will have more to say about this feature when considering Lorentz transformations in the next Sections. Figure 4. Deformed mass-shell in terms of the parameters Π µ (µ = 0, 1, 2). Note that when m reaches its maximum value κπ 2 the momentum space collapses into two parallel infinite planes at Π 0 = ± κπ 2 .

Symmetries of SL(2, R) momentum space
At this point we would like to introduce symmetries in the picture. We start by implementing Lorentz transformations in terms of the action of SL(2, R) on itself and discuss how these are realized for each choice of momentum coordinates introduced in the previous Sections. We then discuss the mathematical structure of the quantum double of SL(2, R) which is the quantum group that replaces the Poincaré group when momentum space is SL(2, R) and translation generators are modified accordingly.

Boosts for Cartesian coordinates
We start by considering the Cartesian parametrization: Without loss of generality we consider a Lorentz boost in γ 1 -direction. This is represented by the group element b = e 1 2 ηγ 2 where η is the boost rapidity. The action of SL(2, R) on itself is realized by conjugation: which in components will read In order to find an explicit form for the boosted momenta we rewrite the exponentials above as b = e 1 2 ηγ 2 = cosh η in this way equation (42) becomes Writing down the product on the right hand side above, using the multiplication properties of the γ matrices and putting together the terms proportional to the same γ matrix we obtain the expressions for the boosted parameters p µ in terms of the old parameters p µ We thus obtain the ordinary action of a Lorentz boost in the x 1 direction.

Boosts for exponential coordinates
As in the previous case we consider the action by conjugation of the Lorentz boost in the γ 1 -direction. In this case, considering the generic group element parametrized by exponential coordinates the action by conjugation results: In analogy with the previous case we can find an explicit expression for the boosted parameters using the matrix expansion of the exponentials Using the multiplication properties of the γ matrices and identifying the terms multiplying the same γ-matrix one finds and thus also the exponential coordinates transform as ordinary momenta in Minkowski space under a boost.

Boosts on Euler coordinates
In this parametrization the generic group element g can be written as: therefore the action by conjugation of the Lorentz boost b results: Again, we can compute this expression using the same steps of the previous cases. In this case, the result we obtain is: which leads us to write the following system: whose solution is given by: We thus have that Euler coordinates on the SL(2, R) provide a non-linear realization of Lorentz boosts. Obviously in the limit κ → ∞ the transformations (54) become just ordinary Lorentz boosts. Consider now a particle initially at rest (namely Π 0 = m, Π 1 = Π 2 = 0) the system of equations (54) simplifies to: We see that in the limit when the boost parameter goes to infinity, η → ∞, the energy does not grow arbitrarily but saturates at the value κπ 2 . We thus have that the non-linear realization of Lorentz boosts given in Euler coordinates is just an example of deformed or doubly special relativity [11,19] in which we have a maximum energy built in the deformed structure of Lorentz transformations.

The quantum double of SL(2, R)
In this Section we show how the transition from a Minkowski to SL(2, R) momentum space translates for the structure of relativistic symmetries in a deformation of the Poincaré group to the quantum double of SL(2, R), a quantum group denoted as D(SL(2, R)). We start by recalling the definition and Hopf algebra properties of D (SL(2, R)). For a gentle introduction to the necessary notions of Hopf algebras we refer the reader to [20] while a rigorous introduction to D(SL(2, R)) can be found in [7]. The quantum double D(SL(2, R) is defined, as a vector space, by the tensor product [21] D(SL(2, R)) = C(SL(2, R)) ⊗ C(SL(2, R)) where C (SL(2, R)) is the space of complex functions on SL(2, R) and C(SL(2, R)) the group algebra of SL(2, R) which, roughly speaking, can be thought of as a vector space comprised by the elements of the group itself. Let us denote an element of the double as (f ⊗g) ∈ D(SL(2, R)), the Hopf algebra structure is given by Complex conjugate : where . The explicit expression for the co-product on C(SL(2, R)) is given by telling us that for such space the co-product is simply a way of extending the notion of function on the group to a tensor product of the latter in a way compatible with the composition of the group.

D(SL(2, R)) as a deformation of C(ISL(2, R))
The group algebra C(ISL(2, R)) is defined as a vector space by the tensor product where R 2,1 is the group of translations which we denote as T x with x ∈ R 2,1 . The group algebra C(R 2,1 ) is the set of elements d 3 xf (x)T x , withf (x) a function (or distribution) with compact support. C(R 2,1 ) can be identified with a sub-algebra C(R 2,1 ) of functions on R 2,1 .
This means that for any element d 3 xf (x)T x of the group algebra we can associate a function f (p) = d 3 xf (x)e −ipx ∈ C(R 2,1 ) whose Fourier transform has compact support. This amounts to identify In conclusion, the group algebra C(ISL(2, R)) can be identified with the tensor product C(ISL(2, R)) = C(R 2,1 ) ⊗ C(SL(2, R)) C(R 2,1 ) ⊗ C(SL(2, R)) .
At this point it is evident that the difference with the quantum double is that we have C(R 2,1 ) instead of C (SL(2, R)). Thus the connection between C(ISL(2, R)) and D(SL(2, R)) is a "lift" or "deformation map" from C(R 2,1 ) to C (SL(2, R)). This will turn out to be an algebra but not a co-algebra morphism [21] since the structure of the co-product for C(R 2,1 ) will be different than the one for C(SL(2, R)). To obtain this map we write down the Hopf algebra structure of

C(ISL(2, R))
Product Complex conugate : where the inverse map ι is defined as in the quantum double R g f ( p) is a vector representation of SL(2, R). The explicit expression for the co-product of C(R 2,1 ) is given by which differs from the co-product for C(SL(2, R)) since the composition law of the arguments of f is abelian while in the co-product for C(SL(2, R)) is not. Notice however that the Hopf algebra structure of C(ISL(2, R)) is very similar to the one of the quantum double of SL(2, R).
In particular instead of the map ad g (·) we now have R g (·), thus to pass from one Hopf algebra to the other we need a connection between the two maps. In other words we are looking for an algebra morphism ϕ such that where f ( p) ∈ C(R 2,1 ) is lifted to ϕ(f ( p)) ≡ f (g) ∈ C(SL(2, R)). Looking at the two Hopf algebra structures it is clear that the morphism should satisfy the relation which written explicitly reads Clearly the morphism depends on the choice of group parametrization [21] indeed each sets of coordinates on SL(2, R) can be used to map C(R 2,1 ) into C(SL(2, R)). Below we will see how group parametrization are associated to different examples of non-commutative plane waves and we will link them to the notion of Weyl map used in the literature on non-commutative space-times.

Non-commutative plane waves and Weyl maps
In order to establish a link between plane waves and Weyl maps we start by reviewing the different notions of non-commutative plane waves that are found in the literature on non-commutative spaces in 2 + 1-dimensions in connection with SL(2, R) momentum space.
The -product, that is customarily introduced on C(R 3 ) in the literature on discrete Euclidean quantum gravity models [9,10], in the Minkowskian case is defined by where x ∈ R 2,1 are coordinates of X = x µ γ µ ∈ sl(2, R) (µ=1,2,3). The group element appearing in the plane wave is taken in the Cartesian parametrization g( p) = u1 + p µ κ γ µ . These plane waves can be obtained through a map E given by where {C(R 2,1 ), } is the set of functions on R 2,1 equipped with the -product (67). In this way we have an ordinary plane wave equipped with a non-commutative star-product determined by the non-abelian composition rule of the group e i p 1 · x e i p 2 · x = e i( p 1 ⊕ p 2 )· x , which at leading order in the deformation parameter reads where p 1 ∧ m p 2 def = diag(−, +, +) p 1 ∧ p 2 . Another set of plane waves [22] labelled by the group element g ∈ SL(2, R) is given in terms of the map whereĈ κ (R 2,1 ) is the non commutative version of C κ (R 2,1 ) i.e. coordinates on R 2,1 equipped with the non trivial Lie bracket ThusĈ κ (R 2,1 ) can be seen as the universal enveloping algebra U (sl(2, R)) [9]. The k µ in (70) are the exponential coordinates introduced before Notice that the map e clearly depends on the group parametrization. Indeed using Cartesian coordinates p µ we have The same could be repeated for other sets of coordinates. The different plane waves will be linked by a map φ such that This map is an isomorphism between algebras, in particular considering the product of two non-commuting functions we have Notice that the map φ −1 def = Ω : C κ (R 2,1 ) →Ĉ κ (R 2,1 ) such that can be seen as a Weyl map [14]. Indeed the -product can be written in terms of Ω as just as in the case of Weyl maps for Lie-algebra non-commutative space-times in 3+1 dimensions as discussed for example in [13].

(Quantum) group Fourier transform
The non-commutative plane wave e g ( x) is associated to a notion of quantum group Fourier transform (QGFT) which maps functions on a n-dimensional Lie group G to functions inĈ κ (R n ) [9]: In the case of SL(2, R) associated to the non-commutative plane wave e g ( x) = e ik µx µ we have where the Haar measure on the group is expressed in terms of "exponential" coordinates k µ (see Appendix D for a review of the Haar measure in the various parametrizations). This Fourier transform is related to the group Fourier transform (GFT) discussed in [10,23,24] via the map φ introduced in the previous Section. Indeed the following diagram holds The composition of maps F • φ (QGFT • phi-map) is the group Fourier transform Notice that the quantization maps Q discussed in [16] in our picture coincides with the Weyl map in (77). Moreover if we change coordinates on the group and pass from the plane wave in exponential coordinates k to cartesian coordinates p the quantum group Fourier transform by construction will read Below we show how to describe these Fourier transforms in a unified framework.

The master Fourier transform
In this section we exhibit an alternative route to construct non-commutative plane waves and the quantum group Fourier transform. This will lead us to a coordinate independent definition of both. We start by considering the group elements written in the three different sets of coordinates introduced above If we take the generators of the algebra sl(2, R) as γ µ def = iκx µ [25], we immediately recover the non-commutative plane waves introduced above, in the cartesian and exponential parametrization, and we also obtain a new plane wave for the Π parametrization where the coordinatesx µ obey the non-trivial Lie bracket (71). Notice also that the plane wave in the parametrization Π is known in the literature on non-commutative spaces as the "time symmetrized" plane wave and has an associated Weyl map [13,26]. The quantum group Fourier transform for the three sets of coordinates reads Now we can introduce a formal expression which defines the quantum group Fourier transform in terms of a generic group element g. We define a master plane wave (for SL(2, R) 2 ) given by Let us show how this definition of plane wave reproduces all the cases above. In the case of exponential coordinates k we have and from which we obtain the following plane wave as expected.
In the case of cartesian coordinates we have and Tr(gγ ν ) = Tr Plugging these expressions in the master plane wave we have which, once again reproduces the plane wave in p parametrization. Notice that the plane wave we obtained is always of the form e i k· x but now k is expressed as a function of p (through the change of coordinates on SL(2, R) which we know).
Lastly for the Euler coordinates Π we have e g( Π) (x) = e i k( Π)· x , where k( Π) is the change of coordinates between k and Π: We can write the plane wave as where p( Π) is the change of coordinates between p and Π: Using the commutation relations for thex we can write the same plane wave in terms of the product of three exponentials We can thus write down our coordinate independent definition of Fourier transform the master quantum group Fourier transform where This expression provides a coordinate independent definition of Fourier transform which for each group parametrization reproduces the corresponding Fourier transforms found in the literature.

Conclusions
In this work we offered a quick, but we hope comprehensive, journey through the notions of deformed symmetry and non-commutativity which are encountered in the study of a relativistic point particle coupled to three-dimensional gravity. In doing so we reviewed and clarified various notions that appear scattered in the literature but also provided some valuable new insights on these models. On one side we showed for the first time, introducing the new set of Euler coordinates, how the framework of doubly or deformed special relativity is realized in the context of SL(2, R) momentum space. On the other hand we clarified several important notions which have appeared in the past in the literature on non-commutative spaces and those recently studied within the community working on group field theory. On this side we were able clarify the status of the different notions of (quantum) group Fourier transforms appearing in the literature and we were able to formulate a master Fourier transform which does not depend on the choice of star-product or Weyl map. Our findings bridge an important gap between the two fields and we believe this work will provide a useful reference for those seeking cross-fertilization between the techniques used in the two different research areas of non-commutative field theory and group field theory.

A.1 Basic general properties
The group SL(2, R) is represented by the set of all 2 × 2 -matrices in which M ij ∈ R and det M = M 11 M 22 − M 21 M 12 = 1. The number of free parameters, considering the determinant constraint, is n = 4 − 1 = 3. A way to represents a group element g as a matrix of SL(2, R) is using the expansion in terms of the unit and the γ matrices previously introduced, namely: where u, ξ 0 , ξ 1 , ξ 2 are real parameters. This expression can be explicitly rewritten as a matrix whose determinant constraint is det g = u 2 − (ξ 1 ) 2 − (ξ 2 ) 2 + (ξ 0 ) 2 = 1. This equation shows that the u parameter can be expressed as a function of the other parameters: We note that in the latter expression we have an ambiguity in the choice of the sign of u. This characteristic is not related to the orthocronous proper Lorentz group SO + (1, 2) but only at its spinor representation SL(2, R), for this reason SL(2, R) is a group larger than SO + (1, 2). In particular, SL(2, R) is described as double covering of SO + (1, 2). In the following, however, we will select a sign to limit ourselves to study the proper Lorentz group transformations 3 . At this point, we make another step, namely characterize geometrically the group SL(2, R).
which is the equation of an elliptic hyperboloid (see Figure 7). Afterwards, if we take the quantity 1 − u 2 = 0 (namely |u| = 1), the equation becomes which is the equation of a cone (see Figure 8). Finally, if we take 1 − u 2 < 0 (namely |u| > 1), the determinant reads which is the equation of an hyperbolic hyperboloid (see Figure 9). These are all the geometries of the parameter space of the group SL(2, R). In particular, we must imagine that for a fixed value of the parameter u, also known as embedding parameter, we generate one of these geometries. We can clarify our ideas observing the Figure 10. Figure 10. The left part of the figure shows the intersection of the Σ-plane, which represents a surface with u = cost, with the AdS-manifold. The value of the constant is such that |u| < 1 and, as seen before, this implies a Lobacevskij geometry, obtained rotating the intersection of the Σ-plane, for the parameter space (right part of the figure).

B Euler coordinates expansion
Starting by the following expression for the group element in terms of the dimensionful parameters ρ κ and χ κ , we can separate the exponentials as follows g = e this is because γ 0 , obviously, commutes with itself and then we can divide the exponential in two parts. Now, we focus on the three central exponentials.
Rewriting the latter in matrix form we have At this point, considering that we defined our spatial momenta as substituting these expressions we obtain e 1 2 φγ 0 e χκγ 1 e − 1 2 φγ 0 = cosh χ κ 1 + cos φ sinh χ κ γ 1 + sin φ sinh χ κ γ 2 where the last equality can be easily obtained using the properties of the γ matrices in the resummation of the exponential. In conclusion, substituting also the Euler angle ρ with our momentum Π 0 and putting together the exponentials, we obtain the expression (37).

C Change of coordinates between different parametrizations
Let us start by writing all together the set of coordinates on the manifold we considered Now, starting by the system (113), we can find the transformation laws from a set of coordinates to another.

D.1 Haar measure in Cartesian parametrization
Using the p -parametrization we have is immediate to write the Haar measure on the group; in fact, substituting the parameters ξ µ in the formula (117), we obtain dµ g( p) = d p κ 1 + | p κ | 2 . (119)

D.2 Haar measure in exponential parametrization
Using the k -parametrization the matter is more complicated. This is because these coordinates appear as a combination of hyperbolic functions and thus the form of the Haar measure (117) is more difficult to obtain. In particular, we have to calculate the determinant of the Jacobian matrix J related to the transformation ξ → k.
Performing the calculation, we obtain which, given equation (117), allows us to write the Haar measure as

D.3 Haar measure in Euler parametrization
The Haar measure in the Π -parametrization can be found in the same way as the previous ones.
In particular the new parametrization leads to the following Jacobian determinant which implies that the Haar measure in this parametrization is given by namely: