Snyder Space-Time: K-Loop and Lie Triple System

Different deformations of the Poincare symmetries have been identified for various non-commutative spaces (e.g. $\kappa$-Minkowski, $sl(2,R)$, Moyal). We present here the deformation of the Poincare symmetries related to Snyder space-time. The notions of smooth"K-loop", a non-associative generalization of Abelian Lie groups, and its infinitesimal counterpart given by the Lie triple system are the key objects in the construction.


Introduction
Snyder space-time has been introduced in 1947, and is one of the first examples of non-commutative geometry [1]. The physical features of a quantum field theory living on this space are not very well known since there have been very few attempts to construct such field theory on this space (see [2] and references therein). It is only recently that a star product and a scalar field theory have been constructed [3,4]. Using a different route and doing the analysis at first order, [5] reached similar results. In both of these works, the scalar field action is shown to be invariant under a new type of deformed Poincaré symmetries. The main feature of this deformed symmetry is the non-(co-)associativity, which explains why this deformation did not appear when considering the (co-)associative deformations of the Poincaré group [6]. In this perspective, the natural question to ask is what is the algebraic structure or "quantum group" encoding the deformation of the Poincaré symmetries consistent with the Snyder non-commutativity? We present here the answer to this question by defining a new type of quantum group.
Before explaining the strategy we shall use in the Snyder case, let us recall the construction in the κ-Minkowski non-commutative space which is well understood and can serve as a guiding line. In this case, the relevant deformation of the Poincaré symmetries, the κ-Poincaré deformation [7,8], is based on the Iwasawa decomposition of SO(p, 1) ∼ AN p . SO(p − 1, 1) ∼ AN p ⊲⊳ SO(p−1, 1) [9,10]. The (Abelian nilpotent) group AN p is interpreted as momentum space when dealing with a scalar field theory living in κ-Minkowski. The deformation of the translation symmetry can be traced back to the non-Abelian AN p group structure. The κ-Poincaré (or bicrossproduct) algebra is encoded by B = C(AN p ) ◮¡ kSO(p − 1, 1), where the algebra of functions over AN p coacts on the group algebra kSO(p − 1, 1), and kSO(p − 1, 1) acts on AN p . This quantum group encodes the deformation of the Poincaré Hopf algebra C(R p ) ⋊ kSO(p − 1, 1) We can follow a similar route to construct the relevant non-commutative space associated to Snyder space-time. The Snyder deformation will be based on the decomposition SO(p − 1, 1) ∼ L . SO(p − 1, 1), where L is not a group but a K-loop, a non-associative generalization of Abelian groups [12]. In this case, the deformation of the translation symmetry will be related to the Kloop structure. The goal of the paper is to construct the quantum group S = C(L) ⋊ kSO(p − 1, 1) -where the group algebra kSO(p−1, 1) acts on the algebra of functions C(L) on the K-loopwith the relevant Hopf structures. This quantum group can be seen once again as a deformation of the Poincaré Hopf algebra C(R p ) ⋊ kSO(p − 1, 1) κ −→ C(L) ⋊ kSO(p − 1, 1), but not in the Hopf or quasi-Hopf algebra setting, since the antipode of S will not be antimultiplicative. This means in particular that this deformation has not been identified in Zakrzewski's classification of Poincaré deformations [6]. We will show that this quantum group S encodes the deformed Poincaré symmetry identified in [3,4]. The infinitesimal structure of a K-loop is a Lie triple system and we will explain how Snyder non-commutative structure is naturally encoded in such structure. Lie triple systems provide therefore a new type of non-commutative spaces. This is the other main result of this paper. As a side result, one recovers that the * -product defined in [3,4] is actually a realization of the Lie triple structure.
In the first section we recall the different notions of loop which can be found in the standard literature, emphasising on the K-loop case. In the second section we introduce the main result of the paper that is the construction of the quantum group encoding the symmetries of Snyder spacetime. In the third section, we show how to construct a scalar field action invariant under such symmetry, using the method described in [13]. In the fourth section, we recall the infinitesimal notion of a smooth K-loop given by Lie triple system and explain how this structure can be interpreted as encoding Snyder non-commutative geometry. The concluding section sets a list of interesting points to explore.

K-loop: a review
We recall in this section the basic definitions of the most studied loops with particular emphasis on the notion of K-loop, a (midely) non-associative generalization of Abelian group. Most of the content of this section (except Proposition 2, Definition 4 and Example 2) can be found in the standard reference on K-loops [12] (see also [14] for further references on the notion of loops).
Definition 1 (quasi-group and loop). A quasigroup (S, ·) is a set S with a binary operation "·" such that for each a and b in S, there exist unique elements x and y in S such that a · x = b, y · a = b. A quasigroup is a loop L if it has also an identity e such that e · a = a · e = a.
Note that this implies in particular that in a loop, we have a unique identity element and that the left and right inverse are unique.
The right Bol identity is A loop L is said to be a left Bol loop (resp. right Bol loop) if it satisfies the left Bol identity (resp. the right Bol identity).

Proposition 1 (alternative loop). Let L be a left Bol loop, then it is left alternative, that is
This proposition follows from the left Bol property with b = 1. The consistency relationships between the product and inverse map of a loop will characterize different types of loop.
Definition 3 (inverse properties). Let L be a loop.
Note that the LIP together with the RIP gives the IP. Let us give now some examples of loops.
Definition 4 (Lie group). A group is a loop satisfying the inverse property and A group is therefore in particular a left and right Bol loop. In [15] the authors focused on the Hopf structures related to the Moufang case, which are a natural non-associative generalization of non-Abelian groups. We are interested instead in the K-loop case. In the following, we shall omit the term "left" since we shall always consider left K-loops, unless specified otherwise. To generate loops, one can consider the factorization of Lie groups.
The groupoid L is a smooth K-loop with product "·". Its inverse map coincides with the group inverse map.
The same proposition holds for the Cartan decomposition SO(p, 1) ∼ L ′ . SO(p), and the decomposition SO(p) ∼ L ′′ . SO(p − 1), i.e. both L ′ and L ′′ are K-loops. The proof for these cases is given in [12]. This proposition and its variants for the other decompositions can also be proved using the notion of Lie triple systems [16].
There is a weak form of commutativity and associativity present in a K-loop. When constructed from a group factorization as in the above decompositions, it takes the shape Hence K-loops can be seen as non-associative generalization of Abelian groups. To conclude this section, let us give two examples of K-loops.
Example 1. Consider the orthochronous Lorentz group SO + (3, 1) and its Cartan decomposition SO + (3, 1) ∼ L ′ . SO (3). L ′ is a K-loop [17,18] which can be identified to the space of 3D speeds in Special Relativity, and the loop product encodes the speed addition used when changing reference frame. Let us recall its structure. L ′ is isomorphic to the 3d upper hyperboloid H + 3 . This space can be embedded in R 4 as H + 3 = {V µ ∈ R 4 , V µ V µ = 1, V 0 > 0} and is generated by the boost a = e i η 2 b· K , where K i ∈ so(3, 1) are the boost generators, η is the boost angle, b is the boost vector ( b 2 = 1), and we are using the spinorial representation of SL(2, C) [19,4]. The product of boosts gives the loop product: The rotation R η i , b i is called the Thomas precession. The 3d speed v in Special Relativity is given in terms of the coordinates on H where c is the speed of light. The loop product (2.2) induces the sum of speeds: It is not difficult to check that the inverse ⊖ coincides with the usual inverse ⊖ v = − v and this addition satisfies the automorphic inverse property . This K-loop can also be interpreted as momentum space in which case one reconstructs an Euclidian Snyder space-time [3,4].
Example 2. Consider the decomposition SO(4, 1) ∼ L . SO (3,1). In this case L is isomorphic to the de Sitter space dS, which can be embedded in R 5 as dS = {π A ∈ R 5 , π A π A = −1}. It is generated by the de Sitter boosts a = e i η 2 B·J 4 , where J 4 ≡ J 4µ ∈ so(4, 1) (µ = 0, . . . , 3), η and B µ are the angle and the boost vector (B 2 = −1) respectively and we are using the spinorial representation of SO(4, 1). The de Sitter space is covered by two coordinate charts given by π 4 > 0 and π 4 < 0. As in the previous example, we restrict to the upper part of the de Sitter space, i.e. the sector with π 4 > 0 (which is stable under the Lorentz action). Snyder used the de Sitter space to define momentum space and momentum is given in terms of the coordinates on de Sitter space [1] κ is the Planck mass ( = c = 1 in this example). He did not define the notion of momenta addition, but we can use the K-loop structure to define it where Λ η i , B i is Lorentz transformation determined by the coordinates (η i , B i ). Using the expression (2.3), we obtain the addition of momenta p µ (a 1 ) ⊕ p µ (a 2 ) ≡ (p ⊕ q) µ = κ tanh η 12 B 12 = p µ (a 1 · a 2 ) given in the appendix. Since it is quite complicated we give here the first order contribution in κ 2 (with p · q = p µ q µ , Once again, it is easy to check that −p is the inverse of the addition ⊕. A direct calculation shows that the addition ⊕ satisfies the automorphic inverse property law, that is −(

Snyder quantum group
In this section we present the main result of this paper, that is the construction of the quantum group encoding the symmetries of Snyder spacetime. Firstly, we define the notion of K-Hopf loop, which will be used to define the K-loop algebra kL, the analogue of the group algebra. Secondly, we define K-Hopf coloop which is the dual notion of K-Hopf loop. A specific example will be the algebra of functions on the K-loop C(L). Finally, we introduce the Snyder quantum group C(L) ⋊ kSO(p − 1, 1) built out from cross product of the Lorentz group algebra with the algebra of functions on the K-loop.

K-Hopf loop
We define here the relevant Hopf structure for the K-loop. We use the Sweedler notation for the coproduct △a = a (1) ⊗ a (2) .
Definition 7 (K-Hopf loop). Let k be a field of characteristic = 2, 3 and a, b ∈ H. A K-Hopf loop is a unital algebra H, equipped with algebra homomorphisms △ : H→H ⊗ H, ǫ : H→k forming a coassociative coalgebra, and a map S : H→H such that The equations (3.2) are the alter-ego of the left inverse property, whereas (3.1) is related to the left Bol property. The antipode S is furthermore "multiplicative" and "comultiplicative" (2) ).
This antipode is very different to the one met in a Hopf algebra [9] and in a Moufang Hopf quasigroup [15], since in these cases it is "antimultiplicative" and "anticomultiplicative".

Proposition 3. A K-Hopf loop satisfies
3) says that the K-Hopf loop is left alternative [15]. This is obtained from (3.1) by The group algebra kG for a group G provides a natural example of Hopf algebra [9]. In a similar way, the K-loop algebra kL provides an example of K-Hopf loop.

Proposition 4. If L is a K-loop, then H = kL is a K-Hopf loop with linear extension of the product and on the basis elements a, b we have
and the unit is given by e.
The proof is straightforward. We notice in particular that kL is both co-commutative and co-associative. The infinitesimal version of smooth K-loops is a Lie triple system [16]. The enveloping algebra for a Lie triple system has been constructed in [20]. This is another example of K-Hopf loop and can be seen as the infinitesimal version of kL, when L is a smooth K-loop.

K-Hopf coloop
Once we have linearised the concept of K-loop into the K-loop algebra kL, one can reverse the arrows on all the maps, and obtain a K-Hopf coloop. We note m the multiplication.  (2) .
A K-Hopf coloop satisfies the analogue of Proposition 3, where the arrows are reversed.

Proposition 5. A K-Hopf coloop satisfies
On the other hand, we have from (3.4) As an example, we construct the K-Hopf coloop k(L) -which is identified to the algebra of functions on L -by duality, from the K-Hopf loop kL. We use the duality (f, a) ≡ f (a), ∀f ∈ k(L), a basis element in kL, and it is extended by linearity to all elements in kL.
with a, b elements in the basis of kL and f i ∈ k(L), and the unit function is I such that I(a) = 1, ∀ a ∈ L.
The proof is straightforward. We notice that k(L) is a commutative and associative algebra, but is not co-commutative nor co-associative.
The general Hopf coloop satisfies similar properties as in Proposition 3, which can be simply shown by reversing the arrows of the maps.
Example 3. Consider the K-Hopf loop generated from L ∼ SO(4, 1)/SO(3, 1) as given in Example 2. The dual structure is given by the set of functions C(L). The coordinate functions p µ ∈ C(L) is a natural example to consider. The coproduct △p µ is then the dual of the sum (2.4), (2.5)

Snyder Hopf loop
We are now presenting the algebraic structure that is encoding the deformation of the Poincaré symmetry consistent with the Snyder space. We focus on the field k = C. Considering the decomposition SO(p, 1) ∼ L . SO(p − 1, 1), L is isomorphic to the de Sitter space. There is a natural action of SO(p − 1, 1) on L given in terms of the adjoint action a→uau −1 = u ⊲ a, ∀ a ∈ L, u ∈ SO(p − 1, 1). This action is naturally lifted to the action of kSO(p − 1, 1) on k(L), with (u ⊲ f, a) ≡ f (u ⊲ a) and u basis element of kSO(p − 1, 1). We recall that the group Hopf algebra kSO(p − 1, 1) is specified by the coproduct, antipode, counity and unity respectively defined on basis element u ∈ SO(p − 1, 1) (and extended by linearity), with u basis elements of kSO(p − 1, 1) and we used (3.7). This is naturally extended by linearity. We have moreover Proof . We shall consider a, b basis elements of kL before extending naturally by linearity. First we prove (3.8), We have moreover that where ǫ(u ⊲ a) is the counity on kL. Then we prove (3.9) We have also Finally the proof of (3.10) goes as follows where we used that the inverse of L and SO(p − 1, 1) actually coincides with the inverse in SO(p, 1).
Since k(L) is a kSO(p − 1, 1) module and co-module, we can consider the semi-direct product [9].
This construction can be extended to the other cases generated by the decompositions we mentioned above: C(L ′′ ) ⋊ kSO(p − 1) and C(L ′ ) ⋊ kSO(p) are also Snyder Hopf loops. We have the following proposition Proposition 8. The antipode in the Snyder Hopf loop is antimultitplicative but comultiplicative Proof . s where we used Proposition 7, and the properties of the action This is equal to (3.12) if we recall that the algebra k(L) is commutative. Finally we show that the the antipode is comultiplicative where we used the consistency relations between the coproduct and antipode for k(L) and kSO(p, 1) inherited from Proposition 7. The proof of (3.11) follows directly from Proposition 5.

Scalar f ield theory
Once we have defined the fundamental algebraic structure, we can proceed to construct a scalar field theory, invariant under such symmetry group, following the same route as [13]. We focus here on the 4d Lorentzian case, i.e. L ∼ dS and we are going to show how to recover the results of [3,4]. As in Example 2, we focus on the upper part of the de Sitter space L + . We consider therefore the Snyder Hopf loop S + = C(L + ) ⋊ kSO(3, 1) which is naturally obtained from S by restricting the set of functions C(L) to C(L + ). This restriction is consistent with the Hopf structures given in Definition 9.
The scalar field is an element of the dual D * of the coset k(L + ) ∼ S + /kSO(3, 1), which can be specified as the set of distributions with compact support k * (L + ). ∀ f ⊗ u ∈ S, v ∈ SO(3, 1) This set of distributions is equipped with the convolution product which makes k * (L + ) an algebra. The convolution product is constructed using the coproduct on S + , The scalar field can be seen as a function on L + , which is then interpreted as momentum space. Hence, it is convenient to consider the subalgebra of distributions given by the algebra of functions (with compact support) k(L + ) ⊂ D * . For φ i ∈ k(L + ), we have where we are using the measure [da] = d 5 π δ(π A π A + 1)θ(π 4 ) on L + , induced from the group decomposition and the loop product. This convolution product between functions is used to generate the relevant terms in the scalar field action. The propagating term and the φ 3 -like interaction term are given as Ψ • φ(e) = [da] 2 p 2 (a 1 ) + m 2 φ(a 1 )φ(a 2 )δ(a 1 · a 2 ), Ψ(a) = p 2 (a) + m 2 φ(a), (4.1) where the delta function encodes the generalization of the momentum conservation law and p µ (a) is the Snyder coordinates associated to the loop element a. Note that the convolution product is not associative, since it is based on the loop product. It is therefore important to keep track of the order of the brackets. The action Σ(φ) for the scalar field φ reads explicitly In the second line, we have expressed the action in terms of the Snyder coordinates, in particular the measures reads κ 4 [da] = [dp] = 1 The sum of momenta is given by (2.4), (2.5).
We need to check that the Snyder Hopf loop S + encodes the right symmetry for this scalar action. We need therefore to define how the field φ is transforming under the quantum group action. The symmetry action of S + on k * (L + ) is induced by its action on This means concretely that the Lorentz group SO(3, 1) sector acts by the adjoint action, whereas the k(L + ) sector acts by multiplication. This last sector encodes the deformed translation symmetry

φ(a)→φ(u ⊲ a), φ(a)→f (a)φ(a). (4.3)
To determine the exact realization of the translation symmetry, one needs to introduce spacetime, through a Fourier transform. We consider therefore the c-numbers x µ ∈ R 4 and the plane-wave e ip·x . To keep track of the modified sum of momenta (2.4), (2.5), we introduce the modified product * between plane-waves e ip 1 ·x * e ip 2 ·x ≡ e i(p 1 ⊕p 2 )·x .
We can then introduce the Fourier transform of the field φ ∈ C(L + ) The product between fields is given by the * -product, which is the dual of the convolution product With this Fourier transform, the scalar field action becomes The translation action is now given as x → x + ε and the field φ(p) is transforming therefore as φ(p) → e ip·ε φ(p), which gives us therefore f (p) = e ip·ε for the symmetry action (4.3).
To prove invariance of the action under S + , we also need to know how it is acting on convolution product. This is constructed naturally using the coproduct of S + and the coproduct on k(L + ) as in (3.6) It is straightforward to check that the action Σ(φ) constructed from the terms (4.1) and (4.2) is then invariant under such transformations. The coproduct of S + also provides the action of the Lorentz group and the translations on a tensor product of fields The deformation of the translation symmetry is therefore naturally encoded in the loop product. Before closing this section, we can evaluate explicitely the star product between different typical functions, using the properties of the Fourier transform we have introduced. For example, we can calculate the commutator of coordinates with the * -product [3,4] [x µ , x ν ] * = 0. (4.4) The commutator of monomes of the first degree is zero. This is quite different than the usual noncommutative spaces. However, this does not mean that we have a commutative space. Indeed, for example the commutator of a monome of the second degree with one of the first degree is not zero due to the non-associativity. As an example, let us look at the more complicated combination: where J µν ∈ so(3, 1) acts in the usual way on the coordinates x α . By introducing a Weyl map, we are going to show in the next section how this star product is the realization of the noncommutative structure implemented by a Lie triple system, the infinitesimal version of a smooth K-loop.

Snyder space as a Lie triple system
In the first subsection, we first recall the different examples of Bol algebras which are the infinitesimal version of the different smooth Bol loops we considered in Section 2. We focus on the Lie triple system case and show how given a Lie triple system, we can recover Snyder commutation relations, through Jacobson's embedding theorem. In the second subsection, we define the Weyl map and recover the star product we have introduced.

Lie triple system
In Section 2, we have first introduced the concept of Bol loop. The concept of Bol algebra is the infinitesimal version of a smooth Bol loop.  [21]: 3)  This can be derived from Proposition 9 using the loop given by a Lie group. We recover that a Lie algebra is the infinitesimal version of a Lie group.
Definition 11 (Mal'tsev algebra, see [20] and references therein). A Mal'tsev algebra is a left Bol algebra such that Such algebra can be seen as the infinitesimal version of a smooth Moufang loop as it can be checked using Proposition 9.
We introduce now the relevant example for studying Snyder space-time.
Definition 12 (Lie triple system, see [20,22,16] and references therein). A Lie triple system ℓ is a left Bol algebra such that It is therefore totally determined in terms of the trilinear product which has the following properties In fact a direct calculation from (5.3) shows that we have This structure can be seen as the infinitesimal version of a smooth K-loop [16], once again using Proposition 9.
Lie triple systems have been studied by mathematicians and most of the results found in the context of Lie algebras have been analysed in the Lie triple system context, see [16,23,24] and references therein.
The property (5.5) indicates that the trilinear product can be expressed in terms of some Lie algebra bracket. The following theorem makes this statement more precise.
Theorem 1 (standard imbedding [22,24]). Let ℓ a Lie triple system, and for X, Y ∈ ℓ, we note δ X,Y (Z) the linear transformation Z→ X, Y, Z and m the set of all such linear transformations. Then m is a Lie algebra and moreover the vector space G ≡ ℓ ⊕ m is a Lie algebra with bracket The map σ : G→G, such that σ(X) = −X, σ(A) = A is an involutive automorphism of G.
Notice that the bracket appearing in this construction has nothing to do with the Bol bracket [ , ] B . Remark 1. From the property (5.2), we notice that δ X,Y is a derivation. One can restate the previous theorem by saying that one can make Der ℓ ⊕ ℓ a Lie algebra, where Der ℓ is the set of (inner) derivations of ℓ.
Proof . To prove the Theorem 1, one has to check that the Lie bracket satisfies the Jacobi identity. Following the property (5.4), it is clear that we have for example The relation states that A = δ X,Y is a derivation. This relation is shown by straightforward calculations. The other relations follow also by direct calculations.
Conversely, given a Lie algebra G and an involutive automorphism σ, one can easily construct a Lie triple system. Indeed, since σ is an involution, it has two eigenvalues −1, 1, with respective eigenspaces noted ℓ, m, such that G = ℓ ⊕ m. σ is an automorphism, therefore using the compatibility with the product, it is easy to check that This theorem and its converse are the key to understand the nature of Snyder spacetime. The following example shows how to construct a Lie triple system from so(p, 1) and recover Snyder non-commutative structure.
The bilinear and trilinear products are then constructed in a similar way as in Proposition 9 This means in particular that there is two types of bracket one should not confuse. On one hand there is the Bol bracket [X µ , X ν ] B = 0 which is trivially zero since we are dealing with a K-loop (and especially the automorphic property). On the other hand, there is the bracket associated with the imbedding Lie algebra [X µ , X ν ] ≡ δ Xµ,Xν from which the trilinear product is constructed as in (5.5). The Snyder commutation relations are encoded in this bracket and the natural algebraic framework to describe the Snyder space-time given by the Lie triple system structure.

Weyl map
We have seen in the last section that given a Lie triple system, Jacobson's theorem allows to recover the Snyder commutation relation. It is therefore natural to consider the set of functions on Snyder spacetime as the set of functions on the Lie triple system. The notion of enveloping algebra U (B) has been constructed for Bol algebras, and in particular it satisfies the Poincaré-Birkhoff-Witt theorem; we refer to [20] for the details. As a consequence, such enveloping algebra U (ℓ) exists for a Lie triple system ℓ. This enveloping algebra U (ℓ) can be interpreted as the algebra of non-commutative functions C(ℓ) on Snyder space-time, generated from the coordinates operators X µ .
We want to check now that the star product we have introduced in Section 4 is related to the non-commutative structure encoded in the Lie triple system. For this we need to introduce a Weyl map, an algebra isomorphism between the algebra of functions on the Lie triple system C(ℓ) and the algebra of functions with the star product C * (R 4 ) .
where p is a momentum coordinate choice on the upper part of the de Sitter space, [dp] the measure on the upper part of the de Sitter space, e ip·X ∈ L and f (p) ∈ C(L). For example, for the generator X µ , we have W(X µ )(x) ≡ x µ . The Weyl map is an isomorphism of algebra therefore by definition we need to have W(f ·ĥ)(x) ≡ (W(f ) * W(ĥ))(x). The definition of the Weyl map (5.6) makes sure that the * -product we are using is precisely the one defined in Section 4. We have in particular It is then immediate to check that in accordance with (4.4). To consider the mapping of a product of three operators, we need to be careful since we have to take into account the non-associativity. Indeed, the position operator X µ acts by ( * -)multiplication of x µ and we use the natural action on the left, so that when considering the product of operators, we have This is important to keep in mind since we are using now a non-associative structure: order does matter. The product of operators is therefore naturally ordered from the right to the left. We have then When considering the triple product, the commutator does contribute in a non trivial way due to the non-associativity where we have used (4.4), so that the last term in (5.7) is zero and we have used the result in (4.5). The Weyl map preserves, as required, the Lie triple system structure. The Snyder commutation relation [X µ , X ν ] can also be obtained from the * -product using its action a functionf (x), which we take to be x α for simplicity [3,4] This shows that the commutator of the position operators does satisfy

Concluding remarks
There are two ways to interpret Snyder commutation relations. The first one is to say that this non-commutative space is actually a subspace of a bigger non-commutative space generated from the coordinates algebra (X α , J µν ). The coordinates J µν are interpreted as coordinates describing extra dimensions. This is the perspective followed in the Doplicher-Fredenhagen-Roberts model [25] which can be seen then as an abelianization of the Snyder model [4,26] with this interpretation. The second possibility is to consider space-time as only generated by the X µ . In this case, we have to face the issue that the coordinates commutations do not close. We argued in this paper that the solution of this issue is to consider Snyder space-time given in terms of a Lie triple system, that is instead of the bilinear product (5.8), one considers the trilinear product In this case, one can construct a meaningful notion of non-commutative algebra of functions using the notion of enveloping algebra for Lie triple system which is dual, using a generalized Fourier transform, to the algebra of functions on the K-loop L ∼ SO(p, 1)/SO(p − 1, 1). On the other hand, given the Lie triple system, Jacobson's theorem tells us how to embedd it into a Lie algebra and to recover the Snyder commutation relations, from the trilinear product Lie triple systems provide therefore a new type of non-commutative geometry. This noncommutative geometry can be seen as flat since one can identify a quantum group, a deformation of the the Poincaré group, which acts on this space in a consistent way. We provided here the definition of this new quantum group, which we called Snyder quantum group, using the structure of K-loop. This latter can be be interpreted as momentum space as in Snyder's initial idea. The K-loop structure provides a rule to add momenta, an addition which is non-commutative and non-associative, just like the speed addition in Special Relativity. We have also presented how we can construct a scalar field action on a Lie triple system using a star product realization and how this action is invariant under the Snyder quantum group, encoding the analogue of the flat symmetries. This work opens up a number of questions which are of interests for either mathematicians or theoretical physicists.
• The Snyder quantum group needs to be studied in details: the analysis of the representation theory should be performed. It would be interesting to check if the weak notion of commutativity (2.1) is related to a notion of braiding.
• The classification of the bi-covariant differential calculus [27] for Snyder space-time should be performed to make the analysis of the conserved currents [11].
• Now that we have a well defined classical action for a scalar field, it would be interesting to check if this non-commutative space fullfils Snyder's hope, that is whether a quantum field theory living in this space has no UV divergences.
Considering the ratio of the equation (A.5) with (A.1), we obtain We can use the inverse map F −1 = cos 2 θ 2 D to determine B in terms of B 0 , p 1 , p 2 and η, since we have that r · B = 0, where V ≡ cos 2 θ 2 D( D( β)) and W ≡ cos 2 θ 2 D( E(p 1 , p 2 )) are vectors totally determined in terms of p 1 and p 2 .
We can used now the value for B identified in (A.7) We can then plug the value of B 0 into (A.7) to obtain B i in terms of the p i tanh The final step to recover the sum is to recall that tanh η 2 = tanh η 1+Γ −1 and Γ can be expressed in terms of Γ i Γ = Γ 1 Γ 2 1 − p 1 · p 2 κ 2 .
Hence we have that