Kinematical versus dynamical contractions of the de Sitter Lie algebras

We start with a Lie algebra $({ \mathcal G },\varphi )$ where ${ \mathcal G }$ is a vector space generated by X_i and φ is a skew symmetric mapping $\varphi :{ \mathcal G }\times { \mathcal G }\to { \mathcal G }$ defined by $\varphi ({X}_{i},{X}_{j})={X}_{k}{C}_{{ij}}^{k}$ and satisfying the Jacobi identity

$$\begin{eqnarray}&&\varphi ({X}_{i},\varphi ({X}_{j},{X}_{k}))+\varphi ({X}_{j},\varphi ({X}_{k},{X}_{i}))+\varphi ({X}_{k},\varphi ({X}_{i},{X}_{j}))=0,\,\forall {X}_{i},{X}_{j},{X}_{k}\in { \mathcal G }.\end{eqnarray} \tag{ 1 }$$

The C^k_ij are called the structure constants of the Lie algebra $({ \mathcal G },\varphi )$. The Jacobi identity shows that a Lie algebra is non associative algebra.

If the mapping ${\psi }_{\epsilon }:{ \mathcal G }\to { \mathcal G }$ is singular for a certain value ₀ of and if the mapping ${\varphi }^{{\prime} }:{ \mathcal G }\times { \mathcal G }\to { \mathcal G }$ is defined by

$$\begin{eqnarray}&&{\varphi }^{{\prime} }(X,Y)=\mathop{\mathrm{lim}}\limits_{\epsilon \to {\epsilon }_{0}}{\psi }_{\epsilon }^{-1}[\varphi ({\psi }_{\epsilon }(X),{\psi }_{\epsilon }(Y))]\end{eqnarray} \tag{ 2 }$$

then $({ \mathcal G },{\varphi }^{{\prime} })$ is a new Lie algebra called the contraction of the Lie algebra $({ \mathcal G },\varphi )$ [5].

The pioneering contraction method is that of Inonu and Wigner [1] which starts with a Lie algebra ${ \mathcal G }={ \mathcal H }+{ \mathcal P }$ where ${ \mathcal H }$ is generated by X_a, ${ \mathcal P }$ is generated by X_α; the structure of ${ \mathcal G }$ being a priori given by

$$\begin{eqnarray*}&&\varphi ({X}_{a},{X}_{b})={X}_{c}{C}_{{ab}}^{c}+{X}_{\gamma }{C}_{{ab}}^{\gamma },\,\varphi ({X}_{a},{X}_{\alpha })={X}_{c}{C}_{a\alpha }^{c}+{X}_{\gamma }{C}_{a\alpha }^{\gamma }\,,\,\varphi ({X}_{\alpha },{X}_{\beta })={X}_{c}{C}_{\alpha \beta }^{c}+{X}_{\gamma }{C}_{\alpha \beta }^{\gamma }\end{eqnarray*}$$

where $a,b,c=1,\ldots ,{\dim }({ \mathcal H })$ and $\alpha ,\beta ,\gamma =1,\ldots ,{\dim }({ \mathcal P })$.

The Inonu-Wigner method uses the parameterized change of basis ${\psi }_{\epsilon }:({X}_{a},{X}_{\alpha })\to ({Y}_{a},{Y}_{\alpha })$ defined by ${Y}_{a}={X}_{a}\,,\,{Y}_{\alpha }=\epsilon {X}_{\alpha }$. The structure of the Lie algebra ${ \mathcal G }$ is given in the new basis by

$$\begin{eqnarray*}&&\varphi ({Y}_{a},{Y}_{b})={Y}_{c}{C}_{{ab}}^{c}+{\epsilon }^{-1}{Y}_{\gamma }{C}_{{ab}}^{\gamma },\,\varphi ({Y}_{a},{Y}_{\alpha })=\epsilon {Y}_{c}{C}_{a\alpha }^{c}+{Y}_{\gamma }{C}_{a\alpha }^{\gamma }\,,\,\varphi ({Y}_{\alpha },{Y}_{\beta })={\epsilon }^{2}{Y}_{c}{C}_{\alpha \beta }^{c}+\epsilon {Y}_{\gamma }{C}_{\alpha \beta }^{\gamma }\end{eqnarray*}$$

In which condition an Inonu-Wigner contraction is it possible? In the limit $\epsilon \to 0$ , the term ${\epsilon }^{-1}{Y}_{\gamma }{C}_{{ab}}^{\gamma }$ diverges. A limit will exist if only if the structure constants ${C}_{{ab}}^{\gamma }$ vanish. Hence to get a Inonu-Wigner contraction, the structure of ${ \mathcal G }$ in the basis $({X}_{a},{X}_{\alpha })$ must be

$$\begin{eqnarray}&&\varphi ({X}_{a},{X}_{b})={X}_{c}{C}_{{ab}}^{c},\,\varphi ({X}_{a},{X}_{\alpha })={X}_{c}{C}_{a\alpha }^{c}+{X}_{\gamma }{C}_{a\alpha }^{\gamma }\,,\,\varphi ({X}_{\alpha },{X}_{\beta })={X}_{c}{C}_{\alpha \beta }^{c}+{X}_{\gamma }{C}_{\alpha \beta }^{\gamma }\end{eqnarray} \tag{ 3 }$$

and

$$\begin{eqnarray*}&&\varphi ({Y}_{a},{Y}_{b})={Y}_{c}{C}_{{ab}}^{c},\,\varphi ({Y}_{a},{Y}_{\alpha })=\epsilon {Y}_{c}{C}_{a\alpha }^{c}+{Y}_{\gamma }{C}_{a\alpha }^{\gamma }\,,\,\varphi ({Y}_{\alpha },{Y}_{\beta })={\epsilon }^{2}{Y}_{c}{C}_{\alpha \beta }^{c}+\epsilon {Y}_{\gamma }{C}_{\alpha \beta }^{\gamma }\end{eqnarray*}$$

in the basis (Y_a, Y_α), i.e.that ${ \mathcal H }$ must be a subalgebra of ${ \mathcal G }$. The structure of the contracted Lie algebra is then

$$\begin{eqnarray}&&{\varphi }^{{\prime} }({Y}_{a},{Y}_{b})={Y}_{c}{C}_{{ab}}^{c}\,,\,{\varphi }^{{\prime} }({Y}_{a},{Y}_{\alpha })={Y}_{\beta }{C}_{a\alpha }^{\beta }\,,\,{\varphi }^{{\prime} }({Y}_{\alpha },{Y}_{\beta })=0\end{eqnarray} \tag{ 4 }$$

The Lie algebra $({ \mathcal G },{\varphi }^{{\prime} })$ defined by (4) is a Inonu-Wigner contraction of the mother Lie algebra $({ \mathcal G },\varphi )$ with respect to the Lie subalgebra ${ \mathcal H }$. It is a semi-direct sum of $({ \mathcal H },{\varphi }^{{\prime} })$ and the abelian Lie algebra $({ \mathcal P },{\varphi }^{{\prime} })$.

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.

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 ${O}_{\pm }(5)$, i.e. that ${{dS}}_{+}(3)$ $[{{dS}}_{-}(3)]$ is isomorhic to $O(1,4)$ [O(2,3)] Lie algebra. The aim of this section is to better clarify velocity-space 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

$$\begin{eqnarray}&&{{ds}}^{2}={\delta }_{{ij}}{{dx}}^{i}{{dx}}^{j}-{\left({{dx}}^{4}\right)}^{2}\pm {\left({{dx}}^{5}\right)}^{2}\equiv {\eta }_{{ab}}{{dx}}^{a}{{dx}}^{b},\,\,i,j=1,2,3\end{eqnarray} \tag{ 5 }$$

where the dimension of the x^a is that of length. The matrix elements η_ab form the diagonal matrix ${\eta }_{\pm }={diag}({I}_{3\times 3},-1,\pm 1)$. Let denote by G_± the group of transformations ${x}^{{\prime} a}={g}_{b}^{a}{x}^{b}$ keeping ds² invariant. It is the group of real square matrices g with order five satisfying ${g}^{t}{\eta }_{\pm }g={\eta }_{\pm }$. The Lie group ${{SO}}_{0}(4,1)$ is the connected component of G₊ while ${{SO}}_{0}(3,2)$ is the connected component of ${G}_{-}$. The Lie algebra ${{ \mathcal O }}_{\pm }(5)$ is the set of the real square matrices X of order 5 satisfying ${}^{t}X{\eta }_{\pm }+{\eta }_{\pm }X=0$. We easily verify that $X={J}_{k}{\theta }^{k}+{A}_{k}{\alpha }^{k}+{B}_{k}{\beta }^{k}+\gamma {\rm{\Gamma }},\,\,\,k=1,2,3,\,$ is the dimensionless matrix

$$\begin{eqnarray}X=\left(\begin{array}{ccc}{\epsilon }_{{kj}}^{i}{\theta }^{k} & {\alpha }^{i} & {\beta }^{i}\\ {\alpha }_{j} & 0 & \gamma \\ \mp {\beta }_{j} & \pm \gamma & 0\end{array}\right)\end{eqnarray} \tag{ 6 }$$

and that $({J}_{k},\,{A}_{k}\,{B}_{k},\,{\rm{\Gamma }})$ is a basis of ${{ \mathcal O }}_{\pm }(5)$. The Lie algebra ${{ \mathcal O }}_{\pm }(5)$ structure is defined by the Lie brackets

$$\begin{eqnarray}&&[{J}_{i},{J}_{j}]={J}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},{A}_{j}]={A}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},{B}_{j}]={B}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},{\rm{\Gamma }}]=0\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&[{A}_{i},{A}_{j}]=-{J}_{k}{\epsilon }_{{ij}}^{k},\,[{A}_{i},{B}_{j}]={\rm{\Gamma }}{\delta }_{{ij}},\,[{A}_{i},{\rm{\Gamma }}]={B}_{i}\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}&&[{B}_{i},{B}_{j}]=\pm {J}_{k}{\epsilon }_{{ij}}^{k},\,[{B}_{i},{\rm{\Gamma }}]=\pm {A}_{i}\end{eqnarray} \tag{ 9 }$$

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.

4.2.1. Relative space Lie algebras:de Sitter, Newton-Hooke, Poincare and Galilei

Let set ${K}_{i}=\tfrac{1}{c}{A}_{i},\,{P}_{i}=\tfrac{\omega }{c}{B}_{i}$ and H = ω Γ where c is a speed while $\omega =\tfrac{1}{\tau }$ is a frequency. The Lie brackets (7), (8) and (8) become

$$\begin{eqnarray}&&[{J}_{i},{J}_{j}]={J}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},{K}_{j}]={K}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},{P}_{j}]={P}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},H]=0\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&[{K}_{i},{K}_{j}]=-\displaystyle \frac{1}{{c}^{2}}{J}_{k}{\epsilon }_{{ij}}^{k},\,[{K}_{i},{P}_{j}]=\displaystyle \frac{1}{{c}^{2}}H{\delta }_{{ij}},\,[{K}_{i},H]={P}_{i}\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&[{P}_{i},{P}_{j}]=\pm \displaystyle \frac{{\omega }^{2}}{{c}^{2}}{J}_{k}{\epsilon }_{{ij}}^{k},\,[{P}_{i},H]=\pm {\omega }^{2}{K}_{i}\end{eqnarray} \tag{ 12 }$$

They define the de Sitter Lie algebras dS_± in the basis $({J}_{i},{K}_{i},{P}_{i},H)$. The general element of the Lie algebra dS_± is $X={J}_{i}{\theta }^{i}+{K}_{i}{v}^{i}+{P}_{i}{x}^{i}+{Ht}$, θⁱ being dimmensionless, the physical dimensions of vⁱ, xⁱ and t being a velocity, a length and a time respectively. Also the general element of the dual of dS_± is $\alpha ={j}_{i}{J}^{* i}+{k}_{i}{K}^{* i}+{p}_{i}{P}^{* i}+{{EH}}^{* }$ 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 $\epsilon =\tfrac{1}{c}\to 0$, the Lie brackets (10), (11) and (12) become

$$\begin{eqnarray}&&[{J}_{i},{J}_{j}]={J}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},{K}_{j}]={K}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},{P}_{j}]={P}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},H]=0\end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray}&&[{K}_{i},{K}_{j}]=0,\,[{K}_{i},{P}_{j}]=0,\,[{K}_{i},H]={P}_{i}\end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray}&&[{P}_{i},{P}_{j}]=0,\,[{P}_{i},H]=\pm {\omega }^{2}{K}_{i}\end{eqnarray} \tag{ 15 }$$

They define the Newton-Hooke Lie algebras ${ \mathcal N }{{ \mathcal H }}_{\pm }$ in the basis $({J}_{i},{K}_{i},{P}_{i},H)$.

In the limit $\omega =\tfrac{1}{\tau }\to 0$ , the Lie brackets (10), (11) and (12) become

$$\begin{eqnarray}&&[{J}_{i},{J}_{j}]={J}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},{K}_{j}]={K}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},{P}_{j}]={P}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},H]=0\end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray}&&[{K}_{i},{K}_{j}]=-\displaystyle \frac{1}{{c}^{2}}{J}_{k}{\epsilon }_{{ij}}^{k},\,[{K}_{i},{P}_{j}]=\displaystyle \frac{1}{{c}^{2}}H{\delta }_{{ij}},\,[{K}_{i},H]={P}_{i}\end{eqnarray} \tag{ 17 }$$

$$\begin{eqnarray}&&[{P}_{i},{P}_{j}]=0,\,[{P}_{i},H]=0\end{eqnarray} \tag{ 18 }$$

They define the Poincare Lie algebra ${ \mathcal P }$ in in the basis (J_i, K_i, P_i, H).

As $\tfrac{1}{c}$ 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

$$\begin{eqnarray}&&[{J}_{i},{J}_{j}]={J}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},{K}_{j}]={K}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},{P}_{j}]={P}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},H]=0\end{eqnarray} \tag{ 19 }$$

$$\begin{eqnarray}&&[{K}_{i},{K}_{j}]=0,\,[{K}_{i},{P}_{j}]=0,\,[{K}_{i},H]={P}_{i}\end{eqnarray} \tag{ 20 }$$

$$\begin{eqnarray}&&[{P}_{i},{P}_{j}]=0,\,[{P}_{i},H]=0\end{eqnarray} \tag{ 21 }$$

which defines the Galilei Lie algebra ${ \mathcal G }$ in the basis (J_i, K_i, P_i, H), is obtained from (13), (14) and (15) when $\omega \to 0$ or from (16), (17) and (18) when $\epsilon =\tfrac{1}{c}\to 0$. The Galilei Lie algebra is then a velocity-space contraction of the Poincare Lie algebra and a space-time contraction of the Newton-Hooke Lie algebras.

4.2.2. Relative time Lie algebras:de Sitter, Poincare, Para-Poincare and Carroll

Let set ${K}_{i}=\tfrac{1}{c}{A}_{i},\,{P}_{i}=\kappa {B}_{i}$ and $M=\tfrac{\kappa }{c}{\rm{\Gamma }}$ where c is a speed while $\kappa =\tfrac{1}{r}$ is a curvature. The Lie brackets (7), (8) and (8) become

$$\begin{eqnarray}&&[{J}_{i},{J}_{j}]={J}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},{K}_{j}]={K}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},{P}_{j}]={P}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},M]=0\end{eqnarray} \tag{ 22 }$$

$$\begin{eqnarray}&&[{K}_{i},{K}_{j}]=-\displaystyle \frac{1}{{c}^{2}}{J}_{k}{\epsilon }_{{ij}}^{k},\,[{K}_{i},{P}_{j}]=M{\delta }_{{ij}},\,[{K}_{i},M]=\displaystyle \frac{1}{{c}^{2}}{P}_{i}\end{eqnarray} \tag{ 23 }$$

$$\begin{eqnarray}&&[{P}_{i},{P}_{j}]=\pm \displaystyle \frac{{\kappa }^{2}}{{c}^{2}}{J}_{k}{\epsilon }_{{ij}}^{k},\,[{P}_{i},M]=\pm {\kappa }^{2}{K}_{i}\end{eqnarray} \tag{ 24 }$$

They define the de Sitter Lie algebras dS_± in the basis (J_i, K_i, P_i, M). The general element of the Lie algebra dS_± is $X={J}_{i}{\theta }^{i}+{K}_{i}{v}^{i}+{P}_{i}{x}^{i}+M\xi$, θⁱ being dimmensionless, the physical dimensions of vⁱ, xⁱ and ξ being a velocity, a length and a a specific action respectively. Also the general element of the dual of dS_± is $\alpha ={j}_{i}{J}^{* i}+{k}_{i}{K}^{* i}+{p}_{i}{P}^{* i}+{{mM}}^{* }$ 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 $\epsilon =\tfrac{1}{c}\to 0$,the Lie brackets (22), (23) and (24) become

$$\begin{eqnarray}&&[{J}_{i},{J}_{j}]={J}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},{K}_{j}]={K}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},{P}_{j}]={P}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},M]=0\end{eqnarray} \tag{ 25 }$$

$$\begin{eqnarray}&&[{K}_{i},{K}_{j}]=0,\,[{K}_{i},{P}_{j}]=M{\delta }_{{ij}},\,[{K}_{i},M]=0\end{eqnarray} \tag{ 26 }$$

$$\begin{eqnarray}&&[{P}_{i},{P}_{j}]=\pm \displaystyle \frac{{\kappa }^{2}}{{c}^{2}}{J}_{k}{\epsilon }_{{ij}}^{k},\,[{P}_{i},M]=\pm {\kappa }^{2}{K}_{i}\end{eqnarray} \tag{ 27 }$$

They define the Para-Poincare Lie algebras ${ \mathcal N }{{ \mathcal H }}_{\pm }$ in the basis (J_i, K_i, P_i, M).

In the limit $\kappa \to 0$ , the Lie brackets (22), (23) and (24) become

$$\begin{eqnarray}&&[{J}_{i},{J}_{j}]={J}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},{K}_{j}]={K}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},{P}_{j}]={P}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},M]=0\end{eqnarray} \tag{ 28 }$$

$$\begin{eqnarray}&&[{K}_{i},{K}_{j}]=-\displaystyle \frac{1}{{c}^{2}}{J}_{k}{\epsilon }_{{ij}}^{k},\,[{K}_{i},{P}_{j}]=M{\delta }_{{ij}},\,[{K}_{i},M]=\displaystyle \frac{1}{{c}^{2}}{P}_{i}\end{eqnarray} \tag{ 29 }$$

$$\begin{eqnarray}&&[{P}_{i},{P}_{j}]=0,\,[{P}_{i},M]=0\end{eqnarray} \tag{ 30 }$$

They define the Poincare Lie algebra ${ \mathcal P }$ in the basis (J_i, K_i, P_i, M).

As $\tfrac{1}{c}$ multiplies A_i and Γ while κ mulipliplies B_i and Γ, it follows that the Para-Poincare lie algebras and the Poincare Lie algebra are velocity-time contractions and space-time contraction of the de Sitter Lie algebras respectively.

The Lie algebra structure

$$\begin{eqnarray}&&[{J}_{i},{J}_{j}]={J}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},{K}_{j}]={K}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},{P}_{j}]={P}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},M]=0\end{eqnarray} \tag{ 31 }$$

$$\begin{eqnarray}&&[{K}_{i},{K}_{j}]=0,\,[{K}_{i},{P}_{j}]=M{\delta }_{{ij}},\,[{K}_{i},M]=0\end{eqnarray} \tag{ 32 }$$

$$\begin{eqnarray}&&[{P}_{i},{P}_{j}]=0,\,[{P}_{i},M]=0\end{eqnarray} \tag{ 33 }$$

which defines the Carroll Lie algebra ${ \mathcal C }$ in the basis (J_i, K_i, P_i, M), is obtained from (25), (26) and (27) when $\kappa \to 0$ or from (28), (29) and (30) when $\epsilon =\tfrac{1}{c}\to 0$. The Carroll Lie algebra is then a velocity-time contraction of the Poincare Lie algebra and a space-time contraction of the Para-Poincare Lie algebras.

4.2.3. Cosmological Lie algebras:de Sitter, Newton-Hooke, Para-Poincare and Para-Galilei

Let set ${F}_{i}=\tfrac{\omega }{r}{A}_{i},\,{P}_{i}=\tfrac{1}{r}{B}_{i}$ and H = ω Γ where r is a radius while $\omega =\tfrac{1}{\tau }$ is a frequency. The Lie brackets (7), (8) and (8) become

$$\begin{eqnarray}&&[{J}_{i},{J}_{j}]={J}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},{K}_{j}]={K}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},{P}_{j}]={P}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},H]=0\end{eqnarray} \tag{ 34 }$$

$$\begin{eqnarray}&&[{F}_{i},{F}_{j}]=-\displaystyle \frac{{\omega }^{2}}{{r}^{2}}{J}_{k}{\epsilon }_{{ij}}^{k},\,[{F}_{i},{P}_{j}]=\displaystyle \frac{1}{{r}^{2}}H{\delta }_{{ij}},\,[{F}_{i},H]={\omega }^{2}{P}_{i}\end{eqnarray} \tag{ 35 }$$

$$\begin{eqnarray}&&[{P}_{i},{P}_{j}]=\pm \displaystyle \frac{1}{{r}^{2}}{J}_{k}{\epsilon }_{{ij}}^{k},\,[{P}_{i},H]=\pm {F}_{i}\end{eqnarray} \tag{ 36 }$$

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 $X={J}_{i}{\theta }^{i}+{F}_{i}{\zeta }^{i}+{P}_{i}{x}^{i}+{Ht}$, θⁱ being dimmensionless, the physical dimensions of ${\zeta }^{i},{x}^{i}$ and t being a velocity, a length and a time respectively. Also the general element of the dual of dS_± is $\alpha ={j}_{i}{J}^{* i}+{f}_{i}{F}^{* i}+{p}_{i}{P}^{* i}+{{EH}}^{* }$ where j_i is the ith component of the angular momentum, f_i is the ith component of a force, p_i is the ith component of the linear momentum and E is an energy.

In the limit $\omega =\tfrac{1}{\tau }\to 0$,the Lie brackets (34), (35) and (36) become

$$\begin{eqnarray}&&[{J}_{i},{J}_{j}]={J}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},{F}_{j}]={F}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},{P}_{j}]={P}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},H]=0\end{eqnarray} \tag{ 37 }$$

$$\begin{eqnarray}&&[{F}_{i},{F}_{j}]=0,\,[{F}_{i},{P}_{j}]=\displaystyle \frac{1}{{r}^{2}}H{\delta }_{{ij}},\,[{F}_{i},H]=0\end{eqnarray} \tag{ 38 }$$

$$\begin{eqnarray}&&[{P}_{i},{P}_{j}]=\pm \displaystyle \frac{1}{{r}^{2}}{J}_{k}{\epsilon }_{{ij}}^{k},\,[{P}_{i},H]=\pm {F}_{i}\end{eqnarray} \tag{ 39 }$$

They define the Para-Poincare Lie algebras ${{ \mathcal P }}_{\pm }$ in the basis (J_i, F_i, P_i, H)

In the limit $\kappa =\tfrac{1}{r}\to 0$ , the Lie brackets (34), (35) and (36) become

$$\begin{eqnarray}&&[{J}_{i},{J}_{j}]={J}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},{F}_{j}]={F}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},{P}_{j}]={P}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},H]=0\end{eqnarray} \tag{ 40 }$$

$$\begin{eqnarray}&&[{F}_{i},{F}_{j}]=0,\,[{F}_{i},{P}_{j}]=0,\,[{F}_{i},H]={\omega }^{2}{P}_{i}\end{eqnarray} \tag{ 41 }$$

$$\begin{eqnarray}&&[{P}_{i},{P}_{j}]=0,\,[{P}_{i},H]=\pm {F}_{i}\end{eqnarray} \tag{ 42 }$$

They define the Newton-Hooke Lie algebra ${ \mathcal N }{{ \mathcal H }}_{\pm }$ in the basis $({J}_{i},{F}_{i},{P}_{i},H)$.

As ω multiplies A_i and Γ while $\kappa =\tfrac{1}{r}$ mulipliplies A_i and B_i, it follows that the Newton-Hooke lie algebras and the Para-Poincare Lie algebras are velocity-space contractions and space-time contraction of the de Sitter Lie algebras respectively.

The Lie algebra structure

$$\begin{eqnarray}&&[{J}_{i},{J}_{j}]={J}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},{F}_{j}]={F}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},{P}_{j}]={P}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},H]=0\end{eqnarray} \tag{ 43 }$$

$$\begin{eqnarray}&&[{F}_{i},{F}_{j}]=0,\,[{F}_{i},{P}_{j}]=0,\,[{F}_{i},H]=0\end{eqnarray} \tag{ 44 }$$

$$\begin{eqnarray}&&[{P}_{i},{P}_{j}]=0,\,[{P}_{i},H]=\pm {F}_{i}\end{eqnarray} \tag{ 45 }$$

which defines the Para-Galilei Lie algebra ${ \mathcal G }\pm$ in the basis (J_i, F_i, P_i, H), is obtained from (37), (38) and (39) when $\kappa =\tfrac{1}{r}\to 0$ or from (40), (41) and (42) when $\omega =\tfrac{1}{\tau }\to 0$. The Para-Galilei Lie algebra is then a velocity-space contraction of the Newton-Hooke Lie algebra and a velocity-space contraction of the Para-Poincare Lie algebras.

4.2.4. The Static Lie algebra

By setting ${K}_{i}=\tfrac{\kappa }{\omega }{A}_{i}$, ${P}_{i}=\kappa {B}_{i}$ and $M=\tfrac{\kappa }{\omega }{\rm{\Gamma }}$, the structure of the de Sitter Lie dS_± becomes

$$\begin{eqnarray}&&[{J}_{i},{J}_{j}]={J}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},{K}_{j}]={K}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},{P}_{j}]={P}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},M]=0\end{eqnarray} \tag{ 46 }$$

$$\begin{eqnarray}&&[{K}_{i},{K}_{j}]=-\displaystyle \frac{{\kappa }^{2}}{{\omega }^{2}}{J}_{k}{\epsilon }_{{ij}}^{k},\,[{K}_{i},{P}_{j}]=M{\delta }_{{ij}},\,[{K}_{i},M]=\displaystyle \frac{{\kappa }^{2}}{{\omega }^{2}}{P}_{i}\end{eqnarray} \tag{ 47 }$$

$$\begin{eqnarray}&&[{P}_{i},{P}_{j}]=\pm {\kappa }^{2}{J}_{k}{\epsilon }_{{ij}}^{k},\,[{P}_{i},M]=\pm {\kappa }^{2}{K}_{i}\end{eqnarray} \tag{ 48 }$$

In the limit $\kappa \to 0$, the structure defined by (46), (47) and (48) becomes the structure

$$\begin{eqnarray}&&[{J}_{i},{J}_{j}]={J}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},{K}_{j}]={K}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},{P}_{j}]={P}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},M]=0\end{eqnarray} \tag{ 49 }$$

$$\begin{eqnarray}&&[{K}_{i},{K}_{j}]=-\displaystyle \frac{{\kappa }^{2}}{{\omega }^{2}}{J}_{k}{\epsilon }_{{ij}}^{k},\,[{K}_{i},{P}_{j}]=M{\delta }_{{ij}},\,[{K}_{i},M]=\displaystyle \frac{{\kappa }^{2}}{{\omega }^{2}}{P}_{i}\end{eqnarray} \tag{ 50 }$$

$$\begin{eqnarray}&&[{P}_{i},{P}_{j}]=\pm {\kappa }^{2}{J}_{k}{\epsilon }_{{ij}}^{k},\,[{P}_{i},M]=\pm {\kappa }^{2}{K}_{i}\end{eqnarray} \tag{ 51 }$$

defining the Static Lie algebra ${ \mathcal S }$ 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.

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 $c=r{\tau }^{-1}$ during all the process.

4.3.1. The de Sitter Lie algebras

Let ${K}_{i}=\tfrac{1}{c}{A}_{i}$, ${P}_{i}=\tfrac{1}{r}{B}_{i}$ and $H=\tfrac{1}{\tau }{\rm{\Gamma }}$ where $\sigma =\tfrac{1}{c}$ is a slowness, $\kappa =\tfrac{1}{r}$ is a curvature while $\omega =\tfrac{1}{\tau }$ is a frequency. In the new basis $({J}_{k},\,{K}_{k}\,{P}_{k},\,H)$ of ${{ \mathcal O }}_{\pm }(5)$ the matrix X above becomes

$$\begin{eqnarray}X=\left(\begin{array}{ccc}{\epsilon }_{{kj}}^{i}{\theta }^{k} & \tfrac{{v}^{i}}{c} & \tfrac{{x}^{i}}{r}\\ \tfrac{{v}_{j}}{c} & 0 & \tfrac{t}{\tau }\\ \mp \tfrac{{x}_{j}}{r} & \pm \tfrac{t}{\tau } & 0\end{array}\right)\end{eqnarray} \tag{ 52 }$$

where ${\alpha }^{i}=\tfrac{{v}^{i}}{c}\,,{\beta }^{i}=\tfrac{{x}^{i}}{r}\,,\gamma =\tfrac{t}{\tau }$. 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

$$\begin{eqnarray}&&[{J}_{i},{J}_{j}]={J}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},{K}_{j}]={K}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},{P}_{j}]={P}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},H]=0\end{eqnarray} \tag{ 53 }$$

$$\begin{eqnarray}&&[{K}_{i},{K}_{j}]=-\displaystyle \frac{1}{{c}^{2}}{J}_{k}{\epsilon }_{{ij}}^{k},\,[{K}_{i},{P}_{j}]=\displaystyle \frac{\tau }{{cr}}H{\delta }_{{ij}},\,[{K}_{i},H]=\displaystyle \frac{r}{c\tau }{P}_{i}\end{eqnarray} \tag{ 54 }$$

$$\begin{eqnarray}&&[{P}_{i},{P}_{j}]=\pm \displaystyle \frac{1}{{r}^{2}}{J}_{k}{\epsilon }_{{ij}}^{k},\,[{P}_{i},H]=\pm \displaystyle \frac{c}{r\tau }{K}_{i}\end{eqnarray} \tag{ 55 }$$

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 $c=r{\tau }^{-1}$. 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).

4.3.2. The Newton-Hooke, Poincaré and Para-Poincaré Lie algebras

4.3.3. The Galilei, Para-Galilei and Carroll Lie algebras

4.3.4. The static Lie algebra

The limit of the Lie brackets (62) and (63) as the speed c and the period τ tend to $\infty$ while $\tfrac{c}{\tau }$ and r are kept finite ,the limit (64) and (65) as the radius r and the period τ tend to $\infty$ while $\tfrac{r}{\tau }$ 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 $\tfrac{c}{r}$ and τ are kept finite are the same; i.e

$$\begin{eqnarray}&&[{K}_{i},{K}_{j}]=0,\,[{K}_{i},{P}_{j}]=0,\,[{K}_{i},H]=0\end{eqnarray} \tag{ 68 }$$

$$\begin{eqnarray}&&[{P}_{i},{P}_{j}]=0,\,[{P}_{i},H]=0\end{eqnarray} \tag{ 69 }$$

The Lie brackets (53), (68) and (69) define the Static Lie algebra ${ \mathcal S }$.

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 $c,\tau \to \infty$, $\tfrac{c}{\tau }$ and r finite (velocity-time contractions ), the vertical arrows represent the contractions as $c,r\to \infty$, $\tfrac{r}{c}$ and τ finite (velocity-space contractions) and the oblique arrows represent the contractions as $r,\tau \to \infty$, $\tfrac{r}{\tau }$ and c finite (space-time contractions).

The de Sitter Lie algebras dS_± are then defined in the basis (J_i, Q_i, P_i, H), by the Lie brackets

$$\begin{eqnarray}&&[{J}_{i},{J}_{j}]={J}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},{Q}_{j}]={Q}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},{P}_{j}]={P}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},H]=0\end{eqnarray} \tag{ 70 }$$

$$\begin{eqnarray}&&[{Q}_{i},{Q}_{j}]=-\displaystyle \frac{1}{{{mE}}_{0}}{J}_{k}{\epsilon }_{{ij}}^{k},\,[{Q}_{i},{P}_{j}]=\displaystyle \frac{1}{{E}_{0}}H{\delta }_{{ij}},\,[{Q}_{i},H]=\displaystyle \frac{1}{m}{P}_{i}\end{eqnarray} \tag{ 71 }$$

$$\begin{eqnarray}&&[{P}_{i},{P}_{j}]=\pm \displaystyle \frac{1}{{{CE}}_{0}}{J}_{k}{\epsilon }_{{ij}}^{k},\,[{P}_{i},H]=\pm \displaystyle \frac{1}{C}{Q}_{i}\end{eqnarray} \tag{ 72 }$$

The de Sitter Lie algebras dS_± are then characterized by the three dynamical parameters m, C and E₀. 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 $\omega =\tfrac{1}{\sqrt{{mC}}}$ (time $\tau =\sqrt{{mC}}$), the speed $c=\sqrt{\tfrac{{E}_{0}}{m}}$ (slowness $s=\sqrt{\tfrac{m}{{E}_{0}}}$) and the curvature $\kappa =\tfrac{1}{\sqrt{{{CE}}_{0}}}$ (the radius $r=\sqrt{{{CE}}_{0}}$).

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 ${{ \mathcal O }}_{\pm }(3)$), these algebras are the solvable ones.

5.2.1. Mass-Compliance Lie algebras: Newton-Hooke

When the energy E₀ tends to infinity, the structure of the de Sitter Lie algebras given by (70), (71) and (72) becomes

$$\begin{eqnarray}&&[{J}_{i},{J}_{j}]={J}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},{Q}_{j}]={Q}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},{P}_{j}]={P}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},H]=0\end{eqnarray} \tag{ 73 }$$

$$\begin{eqnarray}&&[{Q}_{i},{Q}_{j}]=0,\,[{Q}_{i},{P}_{j}]=0,\,[{Q}_{i},H]=\displaystyle \frac{1}{m}{P}_{i}\end{eqnarray} \tag{ 74 }$$

$$\begin{eqnarray}&&[{P}_{i},{P}_{j}]=0,\,[{P}_{i},H]=\pm \displaystyle \frac{1}{C}{Q}_{i}\end{eqnarray} \tag{ 75 }$$

and defines the Newton-Hooke Lie algebras characterized by the mass m and the compliance C related to the frequency $\omega =\tfrac{1}{\sqrt{{mC}}}$ (time $\tau =\sqrt{{mC}}$). 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.

5.2.2. 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

$$\begin{eqnarray}&&[{J}_{i},{J}_{j}]={J}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},{Q}_{j}]={Q}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},{P}_{j}]={P}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},H]=0\end{eqnarray} \tag{ 76 }$$

$$\begin{eqnarray}&&[{Q}_{i},{Q}_{j}]=-\displaystyle \frac{1}{{{mE}}_{0}}{J}_{k}{\epsilon }_{{ij}}^{k},\,[{Q}_{i},{P}_{j}]=\displaystyle \frac{1}{{E}_{0}}H{\delta }_{{ij}},\,[{Q}_{i},H]=\displaystyle \frac{1}{m}{P}_{i}\end{eqnarray} \tag{ 77 }$$

$$\begin{eqnarray}&&[{P}_{i},{P}_{j}]=0,\,[{P}_{i},H]=0\end{eqnarray} \tag{ 78 }$$

and defines the Poincare Lie algebra charcterized by the mass m and the energy E₀ related to the speed $c=\sqrt{\tfrac{{E}_{0}}{m}}$ (slowness $s=\sqrt{\tfrac{m}{{E}_{0}}}$). The Poincare Lie group is then a semi direct product of the direct product of the Lorentz group on the abelian group of space-time translations.

5.2.3. Compliance-Energy Lie algebras: Para-Poincare

When the mass m tends to infinity, the structure of the de Sitter Lie algebras given by (70), (71) and (72) becomes

$$\begin{eqnarray}&&[{J}_{i},{J}_{j}]={J}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},{Q}_{j}]={Q}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},{P}_{j}]={P}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},H]=0\end{eqnarray} \tag{ 79 }$$

$$\begin{eqnarray}&&[{Q}_{i},{Q}_{j}]=0,\,[{Q}_{i},{P}_{j}]=\displaystyle \frac{1}{{E}_{0}}H{\delta }_{{ij}},\,[{Q}_{i},H]=0\end{eqnarray} \tag{ 80 }$$

$$\begin{eqnarray}&&[{P}_{i},{P}_{j}]=\pm \displaystyle \frac{1}{{{CE}}_{0}}{J}_{k}{\epsilon }_{{ij}}^{k},\,[{P}_{i},H]=\pm \displaystyle \frac{1}{C}{Q}_{i}\end{eqnarray} \tag{ 81 }$$

and defines the Para-Poincare Lie algebra charcterized by the compliance C and the energy E₀ related to the curvature $\kappa =\tfrac{1}{\sqrt{{{CE}}_{0}}}$ (the radius $r=\sqrt{{{CE}}_{0}}$). The Para-Poincare Lie group is then a semi direct product of the Para-Lorentz group on the abelian group of impulses-time translations.

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 ${{ \mathcal O }}_{\pm }(3)$), these algebras are the nilpotent ones.

5.3.1. Energy Lie algebra: Carroll

The structure

$$\begin{eqnarray}&&[{J}_{i},{J}_{j}]={J}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},{Q}_{j}]={Q}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},{P}_{j}]={P}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},H]=0\end{eqnarray} \tag{ 82 }$$

$$\begin{eqnarray}&&[{Q}_{i},{Q}_{j}]=0,\,[{Q}_{i},{P}_{j}]=\displaystyle \frac{1}{{E}_{0}}H{\delta }_{{ij}},\,[{Q}_{i},H]=0\end{eqnarray} \tag{ 83 }$$

$$\begin{eqnarray}&&[{P}_{i},{P}_{j}]=0,\,[{P}_{i},H]=0\end{eqnarray} \tag{ 84 }$$

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 E₀.

5.3.2. Compliance Lie algebra: Para-Galilei

The structure

$$\begin{eqnarray}&&[{J}_{i},{J}_{j}]={J}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},{Q}_{j}]={Q}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},{P}_{j}]={P}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},H]=0\end{eqnarray} \tag{ 85 }$$

$$\begin{eqnarray}&&[{Q}_{i},{Q}_{j}]=0,\,[{Q}_{i},{P}_{j}]=0,\,[{Q}_{i},H]=0\end{eqnarray} \tag{ 86 }$$

$$\begin{eqnarray}&&[{P}_{i},{P}_{j}]=0,\,[{P}_{i},H]=\pm \displaystyle \frac{1}{C}{Q}_{i}\end{eqnarray} \tag{ 87 }$$

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.

5.3.3. Mass Lie algebra: Galilei

The structure

$$\begin{eqnarray}&&[{J}_{i},{J}_{j}]={J}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},{Q}_{j}]={Q}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},{P}_{j}]={P}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},H]=0\end{eqnarray} \tag{ 88 }$$

$$\begin{eqnarray}&&[{Q}_{i},{Q}_{j}]=0,\,[{Q}_{i},{P}_{j}]=0,\,[{Q}_{i},H]=\displaystyle \frac{1}{m}{P}_{i}\end{eqnarray} \tag{ 89 }$$

$$\begin{eqnarray}&&[{P}_{i},{P}_{j}]=0,\,[{P}_{i},H]=0\end{eqnarray} \tag{ 90 }$$

defining the Galilei Lie algebra is obtained from that of the Newton-Hooke Lie algebras defined by (73), (74) and (75) when the compliance C tends to infinity or from that of the Poincare Lie algebras defined (76), (77) and (78)when the energy tends to infinity. The Galilei Lie algebras characterized by a finite mass m.

We can say that the mass m is galilean, the compliance C is para-galilean and the energy E₀ is carrollian.

The zero parameters Lie algebra is the static Lie algebra which is obtained from the Galilei Lie algebra as the mass tends to infinity, from the Carroll Lie algebra as the energy tends to infinity or from the Para-Galilei Lie algebra as the compliance tends to infinity.Its structure is

$$\begin{eqnarray}&&[{J}_{i},{J}_{j}]={J}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},{Q}_{j}]={Q}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},{P}_{j}]={P}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},H]=0\end{eqnarray} \tag{ 91 }$$

$$\begin{eqnarray}&&[{Q}_{i},{Q}_{j}]=0,\,[{Q}_{i},{P}_{j}]=0,\,[{Q}_{i},H]=0\end{eqnarray} \tag{ 92 }$$

$$\begin{eqnarray}&&[{P}_{i},{P}_{j}]=0,\,[{P}_{i},H]=0\end{eqnarray} \tag{ 93 }$$

It is the abelian Lie algebra in the case of the dimension one of space (case of ${{ \mathcal O }}_{\pm }(3)$).

All the Lie brackets defining these Lie algebras have in common the Lie brackets

$$\begin{eqnarray*}&&[{J}_{i},{J}_{j}]={J}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},{Q}_{j}]={Q}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},{P}_{j}]={P}_{k}{\epsilon }_{{ij}}^{k},\,[{J}_{i},H]=0\end{eqnarray*}$$

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 $m\to \infty$ (static limit), the vertical arrows represent the contractions as the energy ${E}_{0}\to \infty$(Newtonian limit) and the oblique arrows represent the contractions as the compliance $C\to \infty$ (flat limit). If we use coordinates $\left(\tfrac{1}{m},\tfrac{1}{C},\tfrac{1}{{E}_{0}}\right)$, the kinematical Lie algebras constitute then the cube below (figure 2). 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.

Table 2.  The kinematical Lie algebras in term of the mass, the compliance and the energy.

Lie symbol	Lie algebra name	$[{Q}_{i},H]$	$[{Q}_{i},{Q}_{j}]$	$[{Q}_{i},{P}_{j}]$	$[{P}_{i},{P}_{j}]$	$[{P}_{i},H]$
${{dS}}_{\pm }$	${de}\,{Sitter}$	$\tfrac{1}{m}{P}_{i}$	$-\tfrac{1}{{{mE}}_{0}}{J}_{k}{\epsilon }_{{ij}}^{k}$	$\tfrac{1}{{E}_{0}}H{\delta }_{{ij}}$	$\pm \tfrac{1}{{{CE}}_{0}}{J}_{k}{\epsilon }_{{ij}}^{k}$	$\pm \tfrac{1}{C}{Q}_{i}$
P	Poincare	$\tfrac{1}{m}{P}_{i}$	$-\tfrac{1}{{{mE}}_{0}}{J}_{k}{\epsilon }_{{ij}}^{k}$	$\tfrac{1}{{E}_{0}}H{\delta }_{{ij}}$	0	0
${{NH}}_{\pm }$	Newton—Hooke	$\tfrac{1}{m}{P}_{i}$	0	0	0	$\pm \tfrac{1}{C}{Q}_{i}$
P±	${Para}\,{Poincare}$	0	0	$\tfrac{1}{{E}_{0}}H{\delta }_{{ij}}$	$\pm \tfrac{1}{{{CE}}_{0}}{J}_{k}{\epsilon }_{{ij}}^{k}$	$\pm \tfrac{1}{C}{Q}_{i}$
G	Galilei	$\tfrac{1}{m}{P}_{i}$	0	0	0	0
G±	Para—Galilei	0	0	0	0	$\pm \tfrac{1}{C}{Q}_{i}$
C	Carroll	0		$\tfrac{1}{{E}_{0}}H{\delta }_{{ij}}$	0	0
S	Static	0	0	0	0	0

Table 3.  Distribution of kinematical Lie algebras through the kinematical versus dynamical contractions.

Kinematical process	Dynamical process	Kinematical Lie algebras
Relative time groups	finite energy groups	${{dS}}_{\pm },P,{P}_{\pm },C$
Absolute time groups	infinite energy groups	${{NH}}_{\pm },G,{G}_{\pm },S$
Relative space groups	finite mass groups	${{dS}}_{\pm },{{NH}}_{\pm },P,G$
Absolute space	infinite mass groups	${P}_{\pm },{G}_{\pm },C,S$
Cosmological groups	finite compliance groups	${{dS}}_{\pm },{{NH}}_{\pm },{P}_{\pm },{G}_{\pm }$
Local groups	infinite compliance groups	$P,G,C,S$

We know that the Poisson bracket of two functions defined on the dual ${{ \mathcal G }}^{* }$ of any Lie algebra ${ \mathcal G }$ is defined by

$$\begin{eqnarray}&&\left\{f,g\right\}={K}_{{ij}}(a)\displaystyle \frac{\partial f}{\partial {a}_{i}}\displaystyle \frac{\partial g}{\partial {a}_{j}}\end{eqnarray} \tag{ 94 }$$

where a_i are the coordinates on ${{ \mathcal G }}^{* }$ and ${K}_{{ij}}(a)=-{a}_{k}{C}_{{ij}}^{k}$ are the matrix elements of the Kirillov form.

Let the general element of the dual of a kinematical Lie algebra be ${j}_{k}{J}^{k* }+{q}_{k}{Q}^{k* }+{\pi }_{k}{P}^{k* }+{{EH}}^{* }$ where j_k are the components of the angular momentum conjugated to the angle θ^k, π_k are the components of the linear momentum conjugate to the space translation x^k, q_k are the components of the position conjugated to the impulse p^k while E is an energy conjugated to the time translation t.

From (70), (71) and (72) follows that the Kirillov matrix associated to the de Sitter groups is

$$\begin{eqnarray}K=\left(\begin{array}{cccc}-{j}_{k}{\epsilon }_{{ij}}^{k} & -{q}_{k}{\epsilon }_{{ij}}^{k} & -{\pi }_{k}{\epsilon }_{{ij}}^{k} & 0\\ {q}_{k}{\epsilon }_{{ij}}^{k} & \tfrac{{j}_{k}}{{{mE}}_{0}}{\epsilon }_{{ij}}^{k} & -\tfrac{E}{{E}_{0}}{\delta }_{{ij}} & -\tfrac{{\pi }_{i}}{m}\\ {\pi }_{k}{\epsilon }_{{ij}}^{k} & \tfrac{E}{{E}_{0}}{\delta }_{{ij}} & \mp \tfrac{{j}_{k}}{{{CE}}_{0}}{\epsilon }_{{ij}}^{k} & \mp \tfrac{{q}_{i}}{C}\\ 0 & \tfrac{{\pi }_{j}}{m} & \pm \tfrac{{q}_{j}}{C} & 0\end{array}\right)\end{eqnarray} \tag{ 95 }$$

From the Lie brackets (see previous section) defining the kinematical Lie algebras in function of m,E₀ and C follows then that the kinematical Poisson-Lie algebras are defined by the common Poisson brackets

$$\begin{eqnarray}&&\left\{{j}_{i},{j}_{j}\right\}=-{j}_{k}{\epsilon }_{{ij}}^{k},\,\left\{{j}_{i},{q}_{j}\right\}=-{q}_{k}{\epsilon }_{{ij}}^{k},\,\left\{{j}_{i},{\pi }_{j}\right\}=-{\pi }_{k}{\epsilon }_{{ij}}^{k},\,\left\{{j}_{i},E\right\}=0\end{eqnarray} \tag{ 96 }$$

and the other Poisson brackets given by the table 4.

Table 4.  The kinematical Poisson-Lie algebras.

Lie symbol	Lie algebra name	$\left\{E,{q}_{i}\right\}$	$\left\{{q}_{i},{q}_{j}\right\}$	$\left\{{\pi }_{i},{q}_{j}\right\}$	$\left\{{\pi }_{i},{\pi }_{j}\right\}$	$\left\{E,{\pi }_{i}\right\}$
${{dS}}_{\pm }$	${de}\,{Sitter}$	$\tfrac{1}{m}{\pi }_{i}$	$\tfrac{1}{{{mE}}_{0}}{j}_{k}{\epsilon }_{{ij}}^{k}$	$\tfrac{E}{{E}_{0}}{\delta }_{{ij}}$	$\mp \tfrac{1}{{{CE}}_{0}}{j}_{k}{\epsilon }_{{ij}}^{k}$	$\pm \tfrac{1}{C}{q}_{i}$
${{NH}}_{\pm }$	Newton—Hooke	$\tfrac{1}{m}{\pi }_{i}$	0	0	0	$\pm \tfrac{1}{C}{q}_{i}$
P	Poincare	$\tfrac{1}{m}{\pi }_{i}$	$\tfrac{1}{{{mE}}_{0}}{j}_{k}{\epsilon }_{{ij}}^{k}$	$\tfrac{E}{{E}_{0}}{\delta }_{{ij}}$	0	0
G	Galilei	$\tfrac{1}{m}{\pi }_{i}$	0	0	0	0
P±	${Para}\,{Poincare}$	0	0	$\tfrac{E}{{E}_{0}}{\delta }_{{ij}}$	$\mp \tfrac{1}{{{CE}}_{0}}{j}_{k}{\epsilon }_{{ij}}^{k}$	$\pm \tfrac{1}{C}{q}_{i}$
G±	Para—Galilei	0	0	0	0	$\pm \tfrac{1}{C}{q}_{i}$
C	Carroll	0		$\tfrac{E}{{E}_{0}}{\delta }_{{ij}}$	0	0
S	Static	0	0	0	0	0

We rewrite (94) as $\left\{f,g\right\}={X}_{f}(g)$ where the vector field X_f is defined by

$$\begin{eqnarray}&&{X}_{f}={K}_{{ij}}(a)\displaystyle \frac{\partial f}{\partial {a}_{i}}\displaystyle \frac{\partial }{\partial {a}_{j}}\end{eqnarray} \tag{ 97 }$$

and verifies $[{X}_{f},{X}_{g}]={X}_{\left\{f,g\right\}}$. It is known that the mapping ρ defined by ρ(f) = X_f is a realization of the Lie algebra ${ \mathcal G }$.

Use of the structure of de Sitter Lie algebras $d{{ \mathcal S }}_{\pm }$ defined by the Lie brackets (70), (71) and (72) permits to verify that the de Sitter is the realized by the vector fields

$$\begin{eqnarray}&&{X}_{{j}_{i}}=-{j}_{k}{\epsilon }_{{ij}}^{k}\displaystyle \frac{\partial }{\partial {j}_{j}}-{q}_{k}{\epsilon }_{{ij}}^{k}\displaystyle \frac{\partial }{\partial {q}_{j}}-{\pi }_{k}{\epsilon }_{{ij}}^{k}\displaystyle \frac{\partial }{\partial {\pi }_{j}}\end{eqnarray} \tag{ 98 }$$

$$\begin{eqnarray}&&{X}_{{q}_{i}}={q}_{k}{\epsilon }_{{ij}}^{k}\displaystyle \frac{\partial }{\partial {j}_{j}}+\displaystyle \frac{{j}_{k}}{{{mE}}_{0}}{\epsilon }_{{ij}}^{k}\displaystyle \frac{\partial }{\partial {q}_{j}}-\displaystyle \frac{E}{{E}_{0}}\displaystyle \frac{\partial }{\partial {\pi }_{i}}-\displaystyle \frac{{\pi }_{i}}{m}\displaystyle \frac{\partial }{\partial E}\end{eqnarray} \tag{ 99 }$$

$$\begin{eqnarray}&&{X}_{{\pi }_{i}}={\pi }_{k}{\epsilon }_{{ij}}^{k}\displaystyle \frac{\partial }{\partial {j}_{j}}+\displaystyle \frac{E}{{E}_{0}}\displaystyle \frac{\partial }{\partial {q}_{i}}\mp \displaystyle \frac{{j}_{k}}{{{CE}}_{0}}{\epsilon }_{{ij}}^{k}\displaystyle \frac{\partial }{\partial {\pi }_{j}}\mp \displaystyle \frac{{q}_{i}}{C}\displaystyle \frac{\partial }{\partial E}\end{eqnarray} \tag{ 100 }$$

$$\begin{eqnarray}&&{X}_{E}=\displaystyle \frac{{\pi }_{i}}{m}\displaystyle \frac{\partial }{\partial {q}_{i}}\pm \displaystyle \frac{{q}_{i}}{C}\displaystyle \frac{\partial }{\partial {\pi }_{i}}\end{eqnarray} \tag{ 101 }$$

The realizations of the other kinematical Lie algebras are obtained from (98) to (101) through the dynamical contraction process defined in the previous section. No need to make them explicit here.

If X_f is the generating function of the one parameter diffeomorphism ${{\rm{\Phi }}}_{s}:{{ \mathcal G }}^{* }\to {{ \mathcal G }}^{* }$ and if X_g is the generating function of the one parameter diffeomorphism ${{\rm{\Phi }}}_{\lambda }:{{ \mathcal G }}^{* }\to {{ \mathcal G }}^{* }$, then the equation of change of the function g with respect to s is $\tfrac{{dg}}{{ds}}={X}_{f}(g)$ while the equation of change of f with respect λ is $\tfrac{{df}}{d\lambda }={X}_{g}(f)$, $\tfrac{{dg}}{{ds}}=-\tfrac{{df}}{d\lambda }$.

For each parameter s, let define F_s as ${F}_{s}=\left\{f\in {{ \mathcal G }}^{* }:\tfrac{{df}}{{ds}}=0\right\}$ and let ${V}_{s}={{ \mathcal G }}^{* }/{F}_{s}$ be the variables submanifold of ${{ \mathcal G }}^{* }$. The equations of change describe how the coordinates on V_s change with respect the ad hoc parameter.

Note that as ${{\rm{\Phi }}}_{s}\circ {{\rm{\Phi }}}_{t}={{\rm{\Phi }}}_{s+t}$, 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 pⁱ, the space translation xⁱ and the time translation parameter t appearing in the dS_± Lie algebra element $X={J}_{k}{\theta }^{k}+{Q}_{k}{p}^{k}+{P}_{k}{x}^{k}+{Ht}$. 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

• the angle θⁱ are
  $$\begin{eqnarray}&&\displaystyle \frac{{{dj}}_{l}}{d{\theta }^{i}}={j}_{k}{\epsilon }_{{li}}^{k},\,\displaystyle \frac{{{dq}}_{l}}{d{\theta }^{i}}={q}_{k}{\epsilon }_{{li}}^{k},\,\displaystyle \frac{d{\pi }_{l}}{d{\theta }^{i}}={\pi }_{k}{\epsilon }_{{li}}^{k},\,\displaystyle \frac{{dE}}{d{\theta }^{i}}=0\end{eqnarray} \tag{ 102 }$$
  meaning that j_i, q_i, π_i and E are constant with respect θⁱ.It follows that ${F}_{{\theta }^{i}}=\left\{{j}_{i},{q}_{i},{\pi }_{i},E\right\}$ is quadridimensional while ${V}_{{\theta }^{i}}=\left\{{j}_{k},{q}_{k},{\pi }_{k},\,k\ne i\right\}$ is 6-dimensional, a direct sum ${V}_{{\theta }^{i}}={V}_{{\theta }^{i}}({j}_{k})\oplus {V}_{{\theta }^{i}}({q}_{k})\oplus {V}_{{\theta }^{i}}({\pi }_{k}),k\ne i$ of 2-dimensional irreducible subspaces under the operator $\tfrac{d}{d{\theta }^{i}}$.
• the impulse pⁱ are
  $$\begin{eqnarray}&&\displaystyle \frac{{{dj}}_{l}}{{{dp}}^{i}}={q}_{k}{\epsilon }_{{il}}^{k},\,\displaystyle \frac{{{dq}}_{l}}{{{dp}}^{i}}=\displaystyle \frac{{j}_{k}}{{{mE}}_{0}}{\epsilon }_{{il}}^{k},\,\displaystyle \frac{d{\pi }_{l}}{{{dp}}^{i}}=-\displaystyle \frac{E}{{E}_{0}}{\delta }_{l}^{i},\,\displaystyle \frac{{dE}}{{{dp}}^{i}}=-\displaystyle \frac{{\pi }_{i}}{m}\end{eqnarray} \tag{ 103 }$$
  meaning that j_i, q_i are constant with respect pⁱ. We then have that ${F}_{{p}^{i}}=\left\{{j}_{i},{q}_{i}\right\}$ is 2 dimensional and that ${V}_{{p}^{i}}=\left\{{j}_{k},{q}_{k},{\pi }_{l},E\right\},k\ne i$ is 8-dimensional a direct sum ${V}_{{p}^{i}}={V}_{{p}^{i}}({j}_{k},{q}_{k})\oplus {V}_{{p}^{i}}({\pi }_{l},E),k\ne i$, two 4 dimensional irreducible subspaces $\tfrac{d}{{{dp}}^{i}}$.
• the space translation xⁱ are
  $$\begin{eqnarray}&&\displaystyle \frac{{{dj}}_{l}}{{{dx}}^{i}}={\pi }_{k}{\epsilon }_{{il}}^{k},\,\displaystyle \frac{{{dq}}_{l}}{{{dx}}^{i}}=\displaystyle \frac{E}{{E}_{0}}{\delta }_{l}^{i},\,\displaystyle \frac{d{\pi }_{l}}{{{dx}}^{i}}=\mp \displaystyle \frac{{j}_{k}}{{{CE}}_{0}}{\epsilon }_{{il}}^{k},\,\displaystyle \frac{{dE}}{{{dx}}^{i}}=\mp \displaystyle \frac{{q}_{i}}{C}\end{eqnarray} \tag{ 104 }$$
  meaning that j_i, π_i are constant with respect xⁱ. We then have that ${F}_{{x}^{i}}=\left\{{j}_{i},{\pi }_{i}\right\}$ is 2 dimensional and that ${V}_{{x}^{i}}=\left\{{j}_{k},{\pi }_{k},{q}_{l},E\right\},k\ne i$ is 8-dimensional a direct sum ${V}_{{x}^{i}}={V}_{{x}^{i}}({j}_{k},{\pi }_{k})\oplus {V}_{{x}^{i}}({q}_{l},E),k\ne i$, two 4 dimensional irreducible subspaces $\tfrac{d}{{{dx}}^{i}}$.
• the time translation t are
  $$\begin{eqnarray}&&\displaystyle \frac{{{dj}}_{l}}{{dt}}=0,\,\displaystyle \frac{{{dq}}_{i}}{{dt}}=\displaystyle \frac{{\pi }_{i}}{m},\,\displaystyle \frac{d{\pi }_{i}}{{{dx}}^{i}}=\pm \displaystyle \frac{{q}_{i}}{C},\,\displaystyle \frac{{dE}}{{dt}}=0\end{eqnarray} \tag{ 105 }$$
  meaning that j_i, E are constant with respect t. We then have that ${F}_{t}=\left\{{j}_{i},E\right\}$ is 4 dimensional and that ${V}_{t}=\left\{{q}_{i},{\pi }_{i}\right\}$, a 6 dimensional irreducible space $\tfrac{d}{{dt}}$.

The equations of change corresponding to all kinematical Lie algebra are given in the table 5. In this table we illustrate the equations change with respect the longitude angle φ = θ³, the momentum up p = p³, the altitude z = x³ and the time t. Also the greek indices take the value 1 and 2 while the latin indices take the values 1, 2 and 3. The equations of the form $\tfrac{{df}}{{ds}}=0$ do not appear in the table. Moreover the first column of the table contains the dimensions of V_s, the second one indicates the first order differential equations, the third one the second order differential equation when available, finally the last one shows the corresponding kinematical Lie algebras paired as mother-daughter in the parental relations (see figure 2) up-down for the variations with respect time (energy E₀ is absent in the equations), front-backward for the variations with respect the altitude z (mass m is absent in the equations) and right-left for the variations with respect the momentum up p (compliance C is absent in the equations). We also notice that ${V}_{\varphi }={V}_{\varphi }({j}_{\mu })\oplus {V}_{\varphi }({q}_{\mu })\oplus {V}_{\varphi }({\pi }_{\mu })$ (i.e. $6=2+2+2$ as sum of dimensions) under the differential rotation operator $\tfrac{d}{d\varphi }$. Note that ${V}_{\varphi }({j}_{\mu })$ means that the components of j form an irreducible entity under the differential operator $\tfrac{d}{d\varphi }$. Also V_t is irreducible under the differential time operator $\tfrac{d}{{dt}}$ in the de Sitter and Newton-Hooke case, that ${V}_{z}={V}_{z}({j}_{\mu },{\pi }_{\mu })\oplus {V}_{z}(q,E)$ (i.e. $6=4+2$) is irreducible under the differential altitude operator $\tfrac{d}{{dz}}$ in the de Sitter and Para-Poincaré cases, and finally that ${V}_{p}={V}_{p}({j}_{\mu },{q}_{\mu })\oplus {V}_{p}(\pi ,E)$ (i.e.6 = 4 + 2) is irreducible under the differential momentum up operator $\tfrac{d}{{dp}}$ in the de Sitter and Poincaré cases. The reader can also verify that the three dimensional manifolds are irreducible under the operator $\tfrac{d}{{dt}}$. They are direct sums (i.e. 3 = 2 + 1) under the operators $\tfrac{d}{{dz}}$ and $\tfrac{d}{{dp}}$. Finally the two dimensional ones are irreducible and correspond to the pairs containing the static Lie algebra as a daughter. Note also that all the ten coordinates on ${{ \mathcal G }}^{* }$ are constant in the Carroll and static Lie algebras cases.

Table 5.  Equations of change.

${{dimV}}_{s}$	1st order DE	2nd order DE	Lie algebras
6	$\tfrac{d{\alpha }_{\mu }}{d\varphi }={\alpha }_{\nu }{\epsilon }_{\mu }^{\nu };\,{\alpha }_{\mu }={j}_{\mu },{q}_{\mu },{\pi }_{\mu }$	$\tfrac{{d}^{2}{\alpha }_{\mu }}{d{\varphi }^{2}}=-{\alpha }_{\mu }$	All
6	$\tfrac{d}{{dt}}({q}_{i},{\pi }_{i})=\left(\tfrac{{\pi }_{i}}{m},\pm \tfrac{{q}_{i}}{C}\right)$	$\tfrac{{d}^{2}f}{{{dt}}^{2}}=\pm \tfrac{f}{{mC}}$	${{dS}}_{\pm },{{NH}}_{\pm }$
3	$\tfrac{{{dq}}_{i}}{{dt}}=\tfrac{{\pi }_{i}}{m}$,   (cst velocity)	No one	P, G
3	$\tfrac{d{\pi }_{i}}{{dt}}=\pm \tfrac{{q}_{i}}{C}$,  (cst force)	No one	P± ,G±
6	$\tfrac{d}{{dz}}({j}_{\mu },{\pi }_{\mu })=({\pi }_{\nu },\mp \tfrac{{j}_{\nu }}{{{CE}}_{0}}){\epsilon }_{\mu }^{\nu },\tfrac{d}{{dz}}(q,E)=\left(\tfrac{E}{{E}_{0}},\mp \tfrac{q}{C}\right)$	$\tfrac{{d}^{2}f}{{{dz}}^{2}}=\mp \tfrac{f}{{{CE}}_{0}}$	${{dS}}_{\pm }$ ,P±
3	$\tfrac{{{dj}}_{\mu }}{{dz}}={\pi }_{\nu }{\epsilon }_{\mu }^{\nu }$ (cst pseudo-momentum), $\tfrac{{dE}}{{dz}}=\mp \tfrac{q}{C}$ (cst force)	No one	${{NH}}_{\pm }$ ,G±
3	$\tfrac{{{dj}}_{\mu }}{{dz}}={\pi }_{\nu }{\epsilon }_{\mu }^{\nu }$ (cst pseudo-momentum),$\tfrac{{dq}}{{dz}}=\tfrac{E}{{E}_{0}}$ (cst number)	No one	$P,C$
2	$\tfrac{{{dj}}_{\mu }}{{dz}}={\pi }_{\nu }{\epsilon }_{\mu }^{\nu }$ (cst pseudo-momentum)	No one	G, S
6	$\tfrac{d}{{dp}}({j}_{\mu },{q}_{\mu })=({q}_{\nu },\tfrac{{j}_{\nu }}{{{mE}}_{0}}){\epsilon }_{\mu }^{\nu },\tfrac{d}{{dp}}(\pi ,E)=-\left(\tfrac{E}{{E}_{0}},\tfrac{\pi }{m}\right)$	$\tfrac{{d}^{2}f}{{{dp}}^{2}}=-\tfrac{f}{{{mE}}_{0}}$	${{dS}}_{\pm }$ ,P
3	$\tfrac{{{dj}}_{\mu }}{{dp}}={q}_{\nu }{\epsilon }_{\mu }^{\nu }$ (cst pseudo-position), $\tfrac{{dE}}{{dp}}=-\tfrac{\pi }{m}$ (cst velocity)	No one	${{NH}}_{\pm }$, G
3	$\tfrac{{{dj}}_{\mu }}{{dp}}={q}_{\nu }{\epsilon }_{\mu }^{\nu }$ (pseudo-position),$\tfrac{d\pi }{{dp}}=-\tfrac{E}{{E}_{0}}$ (cst number)	No one	P, C
2	$\tfrac{{{dj}}_{\mu }}{{dp}}={q}_{\nu }{\epsilon }_{\mu }^{\nu }$, (pseudo-position)	No one	G± , S

The Poincaré Lie algebra in one space dimension is defined by the Lie brackets

$$\begin{eqnarray}&&[Q,P]=\displaystyle \frac{1}{{E}_{0}}H,\,[Q,H]=\displaystyle \frac{1}{m}P,\,[P,H]=0\end{eqnarray} \tag{ 106 }$$

where Q genrates momenta, P generates space translations while P generates time translations. We verify that $\exp ({x}_{0}^{{\prime} }P+{t}_{0}^{{\prime} }H)={{Ad}}_{\exp ({pQ}+{xP}+{tH})}(\exp ({x}_{0}P+{t}_{0}H))$ gives the Poincare space-time transformations

$$\begin{eqnarray}&&{x}_{0}^{{\prime} }=\cosh \left(\displaystyle \frac{p}{\sqrt{{{mE}}_{0}}}\right){x}_{0}+\sqrt{\displaystyle \frac{{E}_{0}}{m}}\sinh \left(\displaystyle \frac{p}{\sqrt{{{mE}}_{0}}}\right){t}_{0}+x,\,{t}_{0}^{{\prime} }=\sqrt{\displaystyle \frac{m}{{E}_{0}}}\sinh \left(\displaystyle \frac{p}{\sqrt{{{mE}}_{0}}}\right){x}_{0}+\cosh \left(\displaystyle \frac{p}{\sqrt{{{mE}}_{0}}}\right){t}_{0}+t\end{eqnarray} \tag{ 107 }$$

where p is an additive momentum.

If we define the boost by $v=\sqrt{\tfrac{{E}_{0}}{m}}\tanh \left(\tfrac{p}{\sqrt{{{mE}}_{0}}}\right)$, then we recover the corresponding non additive boosts composition law $v^{\prime\prime} =\tfrac{v+v^{\prime} }{1+\tfrac{{mvv}^{\prime} }{{E}_{0}}}$. It is the usual Lorentz one when ${E}_{0}={{mc}}^{2}$. Similary if slowness is defined by $s=\sqrt{\tfrac{m}{{E}_{0}}}\tanh \left(\tfrac{p}{\sqrt{{{mE}}_{0}}}\right)$, then the slowness composition law is $s^{\prime\prime} =\tfrac{s+s^{\prime} }{1+\tfrac{{E}_{0}{vv}^{\prime} }{m}}$.

The limits of (107) are given in the table

$$\begin{eqnarray}\begin{array}{l}\begin{array}{cccc} & & & \\ & {E}_{0}\to \infty & m\to \infty & \\ & {x}_{0}^{{\prime} }={x}_{0}+\tfrac{p}{m}{t}_{0}+x,\,{t}_{0}^{{\prime} }={t}_{0}+t & {x}_{0}^{{\prime} }={x}_{0}+x,\,{t}_{0}^{{\prime} }={t}_{0}+\tfrac{p}{{E}_{0}}{x}_{0}+t & \\ & {\rm{Galilei}}\,{\rm{transformations}} & {\rm{Carroll}}\,{\rm{transformations}} & \\ & {\rm{on}}\,{\rm{space-time}} & {\rm{on}}\,{\rm{space-time}} & \\ & & & \end{array}\end{array}\end{eqnarray} \tag{ 108 }$$

where $v=\tfrac{p}{m}$ is a Galilean boost and $s=\tfrac{p}{{E}_{0}}$ is a Carrollian slowness.

The Lie algebra of one spatial Para-Poincaré Lie algebra P_± is defined by the Lie brackets

$$\begin{eqnarray}&&[Q,P]=\displaystyle \frac{1}{{E}_{0}}H,\,[[Q,H]=0,\,[P,H]=\pm \displaystyle \frac{1}{C}Q.\end{eqnarray} \tag{ 109 }$$

We verify that $\exp ({p}_{0}^{{\prime} }Q+{t}_{0}^{{\prime} }H)={{Ad}}_{\exp ({pQ}+{xP}+{tH})}(\exp ({p}_{0}Q+{t}_{0}H))$ gives the Para-Poincare momentum-time transformations

$$\begin{eqnarray}&&{p}_{0}^{{\prime} }=\cosh \left(\displaystyle \frac{x}{\sqrt{{{CE}}_{0}}}\right){p}_{0}-\sqrt{\displaystyle \frac{{E}_{0}}{C}}\sinh \left(\displaystyle \frac{x}{\sqrt{{{CE}}_{0}}}\right){t}_{0}+p,\,{t}_{0}^{{\prime} }=-\sqrt{\displaystyle \frac{C}{{E}_{0}}}\sinh \left(\displaystyle \frac{x}{\sqrt{{{CE}}_{0}}}\right){p}_{0}+\cosh \left(\displaystyle \frac{x}{\sqrt{{{CE}}_{0}}}\right){t}_{0}+t\end{eqnarray} \tag{ 110 }$$

in the ${P}_{-}$ case and

$$\begin{eqnarray}&&{p}_{0}^{{\prime} }=\cos \left(\displaystyle \frac{x}{\sqrt{{{CE}}_{0}}}\right){p}_{0}+\sqrt{\displaystyle \frac{{E}_{0}}{C}}\sin \left(\displaystyle \frac{x}{\sqrt{{{CE}}_{0}}}\right){t}_{0}+p,\,{t}_{0}^{{\prime} }=-\sqrt{\displaystyle \frac{C}{{E}_{0}}}\sin \left(\displaystyle \frac{x}{\sqrt{{{CE}}_{0}}}\right){p}_{0}+\cos \left(\displaystyle \frac{x}{\sqrt{{{CE}}_{0}}}\right){t}_{0}+t\end{eqnarray} \tag{ 111 }$$

in the ${P}_{+}$ case. The additive parameter x is non compact (compact) in the ${P}_{-}({P}_{+})$ case.

If $f=\sqrt{\tfrac{{E}_{0}}{C}}\tanh \left(\tfrac{x}{\sqrt{{{CE}}_{0}}}\right)$ $\left(f=\sqrt{\tfrac{{E}_{0}}{C}}\tan \left(\tfrac{x}{\sqrt{{{CE}}_{0}}}\right)\right)$ is a force for ${P}_{-}$ (${P}_{+}$) while $\phi =\sqrt{\tfrac{C}{{E}_{0}}}\tanh \left(\tfrac{x}{\sqrt{{{CE}}_{0}}}\right)$ $\left(\phi =\sqrt{\tfrac{C}{{E}_{0}}}\tan \left(\tfrac{x}{\sqrt{{{CE}}_{0}}}\right)\right)$ is an inverse of force for ${P}_{-}$ (${P}_{+}$), then we get the non additive composition laws $f^{\prime\prime} =\tfrac{f+f^{\prime} }{1\mp \tfrac{{Cff}^{\prime} }{{E}_{0}}}$ and $\phi ^{\prime\prime} =\tfrac{\phi +\phi ^{\prime} }{1\mp \tfrac{{E}_{0}\phi \phi ^{\prime} }{C}}$ for the P_± case. Moreover we have that

$$\begin{eqnarray}\begin{array}{l}\begin{array}{cccc} & & & \\ & {E}_{0}\to \infty & C\to \infty & \\ & {p}_{0}^{{\prime} }={p}_{0}\pm \tfrac{x}{C}{t}_{0}+p,\,{t}_{0}^{{\prime} }={t}_{0}+t & {p}_{0}^{{\prime} }={p}_{0}+p,\,{t}_{0}^{{\prime} }={t}_{0}-\tfrac{x}{{E}_{0}}{p}_{0}+t & \\ & {\rm{Para-Galilei}}\,{\rm{transformations}} & {\rm{Carroll}}\,{\rm{transformations}} & \\ & {\rm{on}}\,{\rm{momentum-time}} & {\rm{on}}\,{\rm{momentum-time}} & \\ & & & \end{array}\end{array}\end{eqnarray} \tag{ 112 }$$

where $f=\pm \tfrac{x}{C}$ is a force for the Para-Galilei case P_± an force and $\phi =-\tfrac{x}{{E}_{0}}$ is a Carrollian inverse of force.

The Lie algebra of one spatial Newton-Hooke Lie algebra ${{NH}}_{\pm }$ is defined by the Lie brackets

$$\begin{eqnarray}&&[Q,P]=0,\,[Q,H]=\displaystyle \frac{1}{m}P,\,[P,H]=\pm \displaystyle \frac{1}{C}Q\end{eqnarray} \tag{ 113 }$$

We verify that $\exp ({p}_{0}^{{\prime} }Q+{x}_{0}^{{\prime} }P)={{Ad}}_{\exp ({pQ}+{xP}+{tH})}(\exp ({p}_{0}Q+{x}_{0}P))$ gives the Newton-Hooke momentum-space transformations

$$\begin{eqnarray}&&{p}_{0}^{{\prime} }=\cosh \left(\displaystyle \frac{t}{\sqrt{{mC}}}\right){p}_{0}-\sqrt{\displaystyle \frac{m}{C}}\sinh \left(\displaystyle \frac{t}{\sqrt{{mC}}}\right){x}_{0}+p,\,{x}_{0}^{{\prime} }=-\sqrt{\displaystyle \frac{C}{m}}\sinh \left(\displaystyle \frac{t}{\sqrt{{mC}}}\right){p}_{0}+\cosh \left(\displaystyle \frac{t}{\sqrt{{mC}}}\right){x}_{0}+x\end{eqnarray} \tag{ 114 }$$

in the NH₊ case and

$$\begin{eqnarray}&&{p}_{0}^{{\prime} }=\cos \left(\displaystyle \frac{t}{\sqrt{{mC}}}\right){p}_{0}+\sqrt{\displaystyle \frac{m}{C}}\sin \left(\displaystyle \frac{t}{\sqrt{{mC}}}\right){x}_{0}+p,\,{x}_{0}^{{\prime} }=-\sqrt{\displaystyle \frac{C}{m}}\sin \left(\displaystyle \frac{t}{\sqrt{{mC}}}\right){p}_{0}+\cos \left(\displaystyle \frac{t}{\sqrt{{mC}}}\right){x}_{0}+x\end{eqnarray} \tag{ 115 }$$

in the ${{NH}}_{-}$ case. The additive parameter t is non compact (compact) in the ${{NH}}_{+}({{NH}}_{-})$ case. Moreover the dampinglike coefficient $b=\sqrt{\tfrac{m}{C}}\tanh \left(\tfrac{t}{\sqrt{{mC}}}\right)$ ($b=\sqrt{\tfrac{m}{C}}\tan \left(\tfrac{t}{\sqrt{{mC}}}\right)$) for the NH₊(${{NH}}_{-}$) whose the dimension is MT⁻¹ and $\beta =\sqrt{\tfrac{C}{m}}\tanh \left(\tfrac{t}{\sqrt{{mC}}}\right)$ and an inverse of a dampinglike $\left(\beta =\sqrt{\tfrac{C}{m}}\tan \left(\tfrac{t}{\sqrt{{mC}}}\right)\right)$ for the NH₊$\left({{NH}}_{-}\right)$ whose the dimension is ${M}^{-1}T$ satisfy the non additive composition laws $b^{\prime\prime} =\tfrac{b+b^{\prime} }{1\pm \tfrac{{Cbb}^{\prime} }{m}}$ and $\beta ^{\prime\prime} =\tfrac{\beta +\beta ^{\prime} }{1\pm \tfrac{m\beta \beta ^{\prime} }{C}}$. for the ${{NH}}_{\pm }$ cases.

We finally have

$$\begin{eqnarray}\begin{array}{l}\begin{array}{cccc} & & & \\ & m\to \infty & C\to \infty & \\ & {p}_{0}^{{\prime} }={p}_{0}\mp \tfrac{t}{C}{x}_{0}+p,\,{x}_{0}^{{\prime} }={x}_{0}+x & {p}_{0}^{{\prime} }={p}_{0}+p,\,{x}_{0}^{{\prime} }={x}_{0}-\tfrac{t}{m}{p}_{0}+x & \\ & {\rm{Para-Galilei}}\,{\rm{group}}\,{\rm{action}} & {\rm{Galilei}}\,{\rm{group}}\,{\rm{action}} & \\ & {\rm{on}}\,{\rm{momentum-space}} & {\rm{on}}\,{\rm{momentum-space}} & \\ & & & \end{array}\end{array}\end{eqnarray} \tag{ 116 }$$