Kinematical versus dynamical contractions of the de Sitter Lie algebras

We present two kinematical Lie algebras contraction processes to improve the Bacry and Lévy-Leblond contractions (H Bacry, et al, 1968 J. Math. Phys., 9, 1605–1614) :(speed-time, speed-space and space-time contraction). For the first one, we introduce kinematical parameters, namely the radius r of the Universe, the period τ of the Universe and the speed of light c = r τ − 1 . Next we present them as static, Newtonian and flat limits through the use of the dynamical parameters, namely the mass, m, the energy, E0 and the compliance C, all depending on mass as well as length and time. We consider that the second one as the best. To give a little physical taste for each kinematical Lie algebra, we set up the equations of change with respect each group parameter through the use of the Poisson brackets defined by the Kirillov form.


Introduction
Group (algebra) contraction is a method which allows to construct a new group (algebra) from an old one. Contraction of Lie groups and Lie algebras started sixty six years ago with nonu and Wigner [1] in 1953, when they were trying to connect Galilean relativity and special relativity. Eight years later, in 1961, Saletan [2] provided a mathematical foundation for the Inonu-Wigner method. Since then, various papers have been produced and the method of contraction has been applied to various Lie groups and Lie algebras [3][4][5][6][7][8][9][10].
The method has also been used by Bacry and Lévy-Leblond [11] to connect the de Sitter Lie algebras to all other kinematical Lie algebras through three kinds of contractions: speed-space contractions, speed-time contractions and space-time contractions. The terminology is related to the fact that Bacry and Lévy-Leblond have, first of all, scaled the velocity-space generators, the velocity-time translation generators and the space-time translation generators by a parameter ò to obtain, in the limit 0   , the respective contractions that we prefer to call velocity-space contractions, velocity-time contractions and space-time contractions. The Lévy-Leblond contraction approach has been also extended to supersymmetry [12] and kinematical superalgebras [13][14][15].
Within the corresponding eleven Lie groups, four of them, namely the Galilei group G governing the Newtonian physics (Galilean relativity), the Poincaré group P governing the Einstein physics (special relativity), the Newton-Hooke groups NH ± describing Galilean relativity in the presence of a cosmological constant and the de Sitter Lie groups dS ± governing the de Sitter relativity of a space-time in expansion or oscillating universe, are well known in physics literature.
Within the remaining five ones, the Para-Poincaré groups P ± and the Static S are still unknown in physics, but the Para-Galilei group G ± and the Carroll group C are gaining more interest in recent times.
The Para-Galilei group has been identified as governing a light spring [16].
The Carroll group has been associated to tachyon dynamics [17][18][19], to Carrollian electromagnetism [20] versus Galilean electromagnetism [21] or to the dynamics of Carroll particles [22] and Carroll strings [23]. The anisotropic Carroll group in two space dimensions (i.e. without rotations) has been identified as the isometry group of gravitational plane waves [24,25]. The Carroll group has also been used recently in the study of ultrarelatistic gravity [26] and for the generalization of Newton-Cartan gravity [27,28]. The Carroll group has been compared to the Galilei group in the study of gravitational waves [29,30], of confined dynamical systems [31], of gravity [32] and of covariant hydrodynamics [33].
The purpose of this paper is, first of all, to clarify the origin of the names given to the three Lévy-Leblond types of contraction and then improve the Levy-Leblond method further.
The main purpose of this paper is to improve the contraction process conducting from the de Sitter Lie algebra to other kinematical Lie algebras. It presents a contraction process in terms of a new set of dynamical parameters (a mass m, an energy E 0 and a compliance C) related to the kinematical parameters (a speed c, a radius r and a period τ) by r C E m , , where c r 1 t = -. Note that the kinematic descriptions are associated only with lengths and times, while the dynamic descriptions are associated with the mass as well as with lengths and time.
In section 2 we recall the Inonu-Wigner contraction, while section 3 recalls the Bacry-Lévy-Leblond method and uses it to establish the twelve kinematical Lie algebras as obtained by Ngendakumana et al [10]. We end the section by setting the raison d'être of a need of improvement the Lévy-Leblond contraction process. It is the purpose of the section 4. With the subsection 4.2, we revisit the Lévy-Leblond method by replacing the dimensional basis by a dimensioned one by scaling some of the vector basis, according the kind of Lie algebra we want to obtain, by either We recover the all the kinematical Lie, except the static Lie algebra, as results of a velocity-space contraction, velocity-time contraction or a space-time contraction. The static Lie algebra is obtained as a velocity-space-time contraction (that Levy-Leblond call the general one) of the de Sitter Lie algebras. The process of this section fails to find the Static Lie algebra as a velocity-space contraction of the Carroll Lie algebra, as a velocity-time contraction of the Galilei Lie algebra or as a space-time contraction of the Para-Galilei Lie algebra (see figure 1 in [11]). With the subsection 4.3 we solve the problem by working with the kinematical parameters which are radius r of the Universe, related to the cosmological constant by r 2 3 = L , the period τ of the Universe, and the velocity c of light defined by c r 1 t = -. In doing so, all the kinematical Lie algebras are found by the contraction process which consists in keeping one parameter finite and letting the remaining two tend to infinity, their ratio being kept finite. The results are summarized in the figure 1 in this paper. However, with the method used in subsection 4.3, we have to keep one parameter finite and let the remaining two tend to infinity, their ratio being kept finite.
To not have to take the precaution above we introduce with section 5 the dynamical contractions by first parameterizing the de Sitter Lie algebras by the dynamical parameters mass m, compliance C (inverse of stiffness or of Hooke constant or of force constant), and energy, E 0 . The dynamical parameters and the kinematical parameters are related by r C E m , , The corresponding contraction consist in letting only of the dynamical parameters go to infinity without constraining the remaining ones, contrary to the kinematical contractions process. The three Bacry-Lévy-Leblond contractions, i.e. the velocity-space contraction, the velocity-time contraction and the space-time contraction correspond then respectively to an infinite energy E 0 , an infinite mass m and an infinite compliance C. They are the Newtonian limit, the static limit and the flat limit of Dyson [34]. This why we claim that the dynamical contraction process is the best one.
Finally in section 6, the Kirillov method is used to establish, for each kinematical Lie algebra, a Poisson-Lie algebra and the equations of change with respect any parameter of the Lie group. Those equations clarify the relationships and differences between the twelve kinematical Lie algebras according the up-down, right-left and frontward-backward contractions (see figure 2). They also permit to split the dual vector space of a kinematical Lie algebra in direct sum of irreducible vector subspaces with respect the operator d ds , s being a parameter of the Lie group.

Contraction of Lie algebras
We start with a Lie algebra ,  j ( )where  is a vector space generated by X i and j is a skew symmetric mapping ) and satisfying the Jacobi identity The C k ij are called the structure constants of the Lie algebra ,  j ( ). The Jacobi identity shows that a Lie algebra is non associative algebra.
If the mapping :    y  is singular for a certain value ò 0 of ò and if the mapping ) is a new Lie algebra called the contraction of the Lie algebra ,  j ( ) [5].

Inonu-Wigner contrations
The pioneering contraction method is that of Inonu and Wigner [1] which starts with a Lie algebra    = + where  is generated by X a ,  is generated by X α ; the structure of  being a priori given by The structure of the Lie algebra  is given in the new basis by In which condition an Inonu-Wigner contraction is it possible? In the limit 0   , the term Y C ab

diverges.
A limit will exist if only if the structure constants C ab g vanish. Hence to get a Inonu-Wigner contraction, the structure of  in the basis X X , i.e.that  must be a subalgebra of  . The structure of the contracted Lie algebra is then The Lie algebra ,  j¢ ( ) defined by (4) is a Inonu-Wigner contraction of the mother Lie algebra ,  j ( )with respect to the Lie subalgebra . It is a semi-direct sum of ,  j¢ ( ) and the abelian Lie algebra ,  j¢ ( ). This process has been used by Bacry and Levy-Leblond [11]. In the next section we briefly recall the results and point why we need to improve some aspect of that paper.
The remaining Lie brackets are given by the table 1 [10]. The ParaPoincaré Lie algebra ,  + which is isomorphic to the Euclidean Lie algebra 4 ( ) where the 'translations' generated by K i and H form an abelian Lie subalgebra does not appear in the list of kinematical ones by Bacry and Lévy-Leblond [11]. The argument is that the inertial transformations are compact. However they are noncompact and only space translations are compact.
Using the Inonu-Wigner contraction method [1], Bacry and Lévy-Leblond [11] have established that these Lie algebras are approximations of the de Sitter Lie algebras. Their links are summarized by the contractions scheme (see figure 1 on page 1610 of [11]). We will refer to these nomenclature in the next two sections.
The parameter used by Bacry and Levy-Leblond to scale the subalgebra  is dimensionless. This is certainly due to the fact those authors have set c=1 and r=1, c being the speed of light while r is the radius of the Universe. The table at page 1608 shows that the generators of the kinematical Lie algebras in question seem to be dimensionless. This physics interpretation behind is then a bit difficult to follow. We propose to improve this situation in the next two sections. Table 1. The kinematical Lie algebras in term of c,r and τ.

Lie symbol
Lie algebra name K H , We propose to recover the Bacry-Lévy-Leblond contractions scheme by using the kinematical parameters r, τ and c which are respectively the radius of universe, the period of the Universe and speed of light. We first introduce the de Sitter Lie algebras dS ± as isomorphic to the pseudo-orthogonal Lie algebras O 5  ( ), i.e. that dS 3 3)] Lie algebra. The aim of this section is to better clarify velocityspace contractions, velocity-time contractions and space-time contractions of Bacry and Lévy-Leblond [11]. Let V be a five dimensional manifold equipped with the metric where the dimension of the x a is that of length. The matrix elements η ab form the diagonal matrix diag I , 1, 1 It is the group of real square matrices g with order five satisfying g g . We easily verify that , , The dimensionless generator A i and B i play the role of velocity generator and space translation generator in the ith direction respectively while the dimensionless Γ plays the role of time translations generator. and H=ω Γ where c is a speed while 1 w = t is a frequency. The Lie brackets (7), (8) and (8) become They define the de Sitter Lie algebras dS ± in the basis J K P H , , , and t being a velocity, a length and a time respectively. Also the general element of the dual of dS ± is where j i is the ith component of the angular momentum, k i is the ith component of the static momentum, p i is the ith component of the linear momentum and E is an energy. In the limit 0 They define the Newton-Hooke Lie algebras   in the basis J K P H , , , the Lie brackets (10), (11) and (12) become They define the Poincare Lie algebra  in in the basis (J i , K i , P i , H). As c 1 multiplies A i and B i while ω mulipliplies B i and Γ, it follows that the Newton-Hooke lie algebras and the Poincare Lie algebra are velocity-space contractions and space-time contraction of the de Sitter Lie algebras respectively. The Lie algebra structure (13), (14) and (15) when 0 w  or from (16), (17) and (18) The Lie brackets (7), (8) and (8) become , θ i being dimmensionless, the physical dimensions of v i , x i and ξ being a velocity, a length and a a specific action respectively. Also the general element of the dual of dS ± is where j i is the ith component of the angular momentum, k i is the ith component of the static momentum, p i is the ith component of the linear momentum and m is a mass.
In the limit 0 c 1  =  ,the Lie brackets (22), (23) and (24) become They define the Para-Poincare Lie algebras   in the basis (J i , K i , P i , M).
In the limit 0 k  , the Lie brackets (22), (23) and (24) become They define the Poincare Lie algebra  in the basis (J i , K i , P i , M). As   and H=ω Γ where r is a radius while 1 w = t is a frequency. The Lie brackets (7), (8) and (8) become They define the de Sitter Lie algebras dS ± in the basis (J i , F i , P i , H). The general element of the Lie algebra dS ± is X J F Px Ht They define the Para-Poincare Lie algebras   in the basis (J i , F i , P i , H) In the limit 0 r 1 k =  , the Lie brackets (34), (35) and (36) become They define the Newton-Hooke Lie algebra   in the basis J F P H , , , As ω multiplies A i and Γ while In the limit 0 k  , the structure defined by (46), (47) and (48) becomes the structure defining the Static Lie algebra  in the relative time kinematical Lie algebras. As κ multiplies A i , B i and Γ, the static Lie algebra is a velocity-space-time (the general according Levy-Leblond [11]) contraction of the de Sitter Lie algebras.

Kinematical improvment
Even if the previous section clarifies better the Inonu-Wigner contractions of the de Sitter Lie algebras, there remain the following problem: the structures of relative space Lie algebras, the relative time Lie algebras and the cosmological Lie algebras are defined in different bases. We propose to remove that situation in the next section. We use three kinematical parameters: the speed c, the radius r and the period τ. We ignore the relation c r 1 t =during all the process.
Hence the parameters associated with K i ,P i and H have velocity, length and time as respective physical dimension. The Lie brackets (7), (8) and (8) become then Let us now study the limits of the de Sitter Lie algebras as the constants tend to infinity. Normally the three constants are constrained by c r 1 t = -. However, we ignore it for a moment. We use it at the end of the section to show that our way of doing has recovered the results of table 1. It is first of all evident that (53) does not change.
We are then only interested in the behavior of (54) and (55).

The Newton-Hooke, Poincaré and Para-Poincaré Lie algebras
In this section we look for the limits of the de Sitter Lie algebras as two of the constants tend to infinity while their ratio is kept finite.
We verify that the limits of (54) and (55), as the speed c and the radius r tend to infinity while their ratio r c and τ are kept finite, are The Lie brackets (53), (60) and (61) define the Para-Poincaré Lie algebra   . We then notice that the Newton-Hooke Lie algebras, the Poincaré Lie algebra and the Para-Poincaré Lie algebras are respectively the velocity-space, space-time and velocity-time contractions of the de Sitter Lie algebras as in [11].

The Galilei, Para-Galilei and Carroll Lie algebras
(a) Galilei Lie algebra.
The limit of the Lie brackets (56) and (57) as the radius r and the period τ tend to infinity while r t and c are kept finite and the limit (58) and (59) as the radius r and the speed c tend to infinity while c r and τ are kept finite are the same, i.e.
The limit of the Lie brackets (56) and (57) as the speed c and the period τ tend to infinity while c t and r are kept finite and the limit (60) and (61) as the radius r and the speed c tend to infinity while c r and τ are kept finite are the same, i.e.
(c) Carroll Lie algebra. The limit of the Lie brackets (58) and (59) as the speed c and the period τ tend to infinity while c t is kept finite and the limit (60) and (61) as the radius r and the period τ tend to infinity while r t is kept finite are the same, i.e.
The Lie brackets (53), (66) and (67) define the Carroll Lie algebra . Hence the Galilei, the Para-Galilei, the Carroll Lie algebras are respective contractions of the Newton-Hooke or Poincaré Lie algebras, the Newton-Hooke or the Para-Poincaré Lie algebras, the Poincaré or the Para-Poincaré Lie algebras respectively.

The static Lie algebra
The limit of the Lie brackets (62) and (63) as the speed c and the period τ tend to ¥ while c t and r are kept finite , the limit (64) and (65) as the radius r and the period τ tend to ¥ while r t and c are kept finite and the limit of the Lie brackets (66) and (67) as the speed c and the radius r tend infinity while c r and τ are kept finite are the same; i.e K K K P K H , 0, , 0 The Lie brackets (53), (68) and (69) define the Static Lie algebra  . When the constraint c=rτ −1 is taken in account, the Lie brackets in the table 1 are recovered. These approximations through kinematical parameters are summarized in the following cube (see figure 1). On the cube, the horizontal arrows represent the contractions as c, t  ¥,

Dynamical improvement of the Levy-Leblond approach
In the process of the previous section, we were sending two parameters at infinity while keeping finite their ratio and the third one. We introduce in this section the dynamical parameters compliance C, mass m and energy E 0 . These dynamical parameters enter the de Sitter Lie algebras structure by replacing the boost generators K i by the momentum generators Q K i m i 1 = , m being a mass and by defining the compliance C and the energy E 0 respectively by C m 2 = t and E mc In the contraction process, only one parameter will be sent to infinity without any precaution on the other two. Similarly to the kinematical contraction process, at the end of the dynamical contraction process, the kinematical Lie algebras will distributed on a cube. Two opposite faces with finite versus infinite mass, two opposite faces with a finite versus infinite energy and two faces with a finite versus infinite compliance.

Three finite parameters Lie algebras: the de Sitter
The de Sitter Lie algebras dS ± are then defined in the basis (J i , Q i , P i , H), by the Lie brackets The de Sitter Lie algebras dS ± are then characterized by the three dynamical parameters m, C and E 0 . They are at the edge of the finite mass, finite energy and finite compliance. The de Sitter Lie algebras are then characterized by three finite kinematical parameters: the frequency

Two finite parameters Lie algebras: Newton-Hooke, Poincare and Para-Poioncare
When one of the three parameters becomes infinite, the structure of the de Sitter Lie algebras gives rise to a Lie algebra characterized by the remaining two. In one dimension of space (case of 3   ( )), these algebras are the solvable ones.

Mass-Compliance Lie algebras: Newton-Hooke
When the energy E 0 tends to infinity, the structure of the de Sitter Lie algebras given by (70), (71) and (72) becomes and defines the Newton-Hooke Lie algebras characterized by the mass m and the compliance C related to the frequency ). The Newton-Hooke Lie group is then a semi direct product of the direct product of rotations and time translations on the abelian group of impulses-positions.

Mass-Energy Lie algebra: Poincare
When the compliance C tends to infinity, the structure of the de Sitter Lie algebras given by (70), (71) and (72) becomes and defines the Para-Poincare Lie algebra charcterized by the compliance C and the energy E 0 related to the curvature ). The Para-Poincare Lie group is then a semi direct product of the Para-Lorentz group on the abelian group of impulses-time translations.

One finite parameter Lie algebras: Galilei, Para-Galilei and Carroll
When two of the three parameters become infinite, the structure of the de Sitter Lie algebras gives rise to a Lie algebra characterized by the remaining one. In one dimension of space (case of 3   ( )), these algebras are the nilpotent ones.

Energy Lie algebra: Carroll
The structure defining the Carroll Lie algebra is obtained from that of the Poincare Lie algebra defined by (76), (77) and (78) when the mass m tends to infinity or from that of the Para-Poincare Lie algebras defined (88), (86) and (90) when the compliance tends to infinity. The Carroll Lie algebras characterized by a finite energy E 0 .

Compliance Lie algebra: Para-Galilei
The structure defining the Para-Galilei Lie algebras is obtained from that of the Newton-Hooke Lie algebras defined by (73), (74) and (75) when the mass m tends to infinity or from that of the Para-Poincare Lie algebras defined (88), (86) and (90) when the energy tends to infinity. The Para-Galilei Lie algebras characterized by a finite compliance C.

Mass Lie algebra: Galilei
The structure We can say that the mass m is galilean, the compliance C is para-galilean and the energy E 0 is carrollian.
It is the abelian Lie algebra in the case of the dimension one of space (case of the others are summarized in the table 2 while the limiting process is given by the figure 2 where the horizontal arrows represent the contractions as the mass m  ¥ (static limit), the vertical arrows represent the contractions as the energy E 0  ¥(Newtonian limit) and the oblique arrows represent the contractions as the compliance C  ¥ (flat limit). If we use coordinates , ,  The table 3 give a comparison of kinematical Lie algebras distribution obtained through the dynamical contraction process above with that obtained by the kinematical contraction process as given by MacRae [8]. The relative (absolute) time groups in the kinematical contraction process correspond to the fine (infinite) energy groups in the dynamical contraction process. Similarly the relative (absolute) space correspond to the finite (infinite) mass while the cosmological (local) groups correspond to the finite (infinite) compliance.

A glance at the physics associated to the kinematical Lie algebras
Let us have a look at the physics associated to the kinematical Lie algebras in function of the three dynamical parameters:compliance C, mass m and energy E 0 .

Poisson brackets
We know that the Poisson bracket of two functions defined on the dual *  of any Lie algebra  is defined by where j k are the components of the angular momentum conjugated to the angle θ k , π k are the components of the linear     Note that as s t s t F F = F + • , the change parameters must be additive. It is true for the longitudinal angle. We show in the appendix that it is also true for the momentum parameter p i , the space translation x i and the time translation parameter t appearing in the dS ± Lie algebra element X J Q p P x Ht k k k k k k q = + + + . We use the one spatial Poincaré Para-Poincaré and Newton-Hooke, Lie algebras to respectively associate a non additive boost to momentum, a non additive force to space translation and to time translation a non additive dampinglike coefficient.
In the maximal case of the de Sitter case, it follows from tt follows (98) to (101) that the equations of change with where f x C =  is a force for the Para-Galilei case P ± an force and x E 0 f =is a Carrollian inverse of force.

A.3. From time translation to dampinglike coefficient
The Lie algebra of one spatial Newton-Hooke Lie algebra NH  is defined by the Lie brackets gives the Newton-Hooke momentumspace transformations