As an illustrative starting point we choose the Poincaré algebra in D = d + 1 space-time dimensions M a b , M c d = η b c M a d − η a c M b d − η b d M a c + η a d M b c , M a b , P c = η b c P a − η a c P b , P a , P b = 0 , (2.1) where small Roman indices from the beginning of the alphabet are fundamental s o ( 1 , d ) indices, e.g., a = 0, 1, … , d, and η _ab is the flat Minkowski metric of signature (− + +⋯ +). When separating the time and space indices according to a = (0, i) with i = 1, … , d, we let [12] J i j = M i j , B i = λ 1 / 2 M i 0 , T i = λ 1 / 2 P i , H = P 0 (2.2) which is an invertible change of basis for any λ > 0 and the algebra becomes J i j , J k l = δ j k J i l − δ i j J i k − δ j l J i k + δ i l J j k , J i j , B k = δ j k B i − δ i k B j , J i j , T k = δ j k T i − δ i k T j , J i j , H = 0 , B i , B j = λ J i j , B i , T j = − λ δ i j H , B i , H = − T i , T i , T j = T i , H = 0 . (2.3)

We see that we can take the limit λ → 0 smoothly and obtain a new algebra in that limit. This so-called contracted algebra is the non-relativistic Galilei algebra (c → ∞) where now Galilean boosts commute among themselves and with translations. This is the most famous example of an Inönü–Wigner contraction of a Lie algebra [11]. As is usual, the contracted algebra is no longer isomorphic to the algebra with λ > 0. The square root in Eq. 2.2 arises since we think of λ as 1/c ². ⁶ For future reference we write the resulting contracted Galilei algebra J i j , J k l = δ j k J i l − δ i j J i k − δ j l J i k + δ i l J j k , J i j , B k = δ j k B i − δ i k B j , J i j , T k = δ j k T i − δ i k T j , J i j , H = 0 , B i , B j = 0 , B i , T j = 0 , B i , H = − T i , T i , T j = T i , H = 0 . (2.4)

An alternative contraction of the algebra is obtained by formally interchanging the roles of time and space directions for the translation generators, i.e., letting [12] J i j = M i j , K i = λ 1 / 2 M i 0 , T i = P i , K = λ 1 / 2 P 0 (2.5)

and contracting again λ → 0. This leads to the Carroll algebra J i j , J k l = δ j k J i l − δ i j J i k − δ j l J i k + δ i l J j k , J i j , K k = δ j k K i − δ i k K j , J i j , T k = δ j k T i − δ i k T j , J i j , K = 0 , K i , K j = 0 , K i , T j = − δ i j K , K i , K = 0 , T i , T j = T i , K = 0 . (2.6)

This contraction of the Poincaré algebra is also known as the Carroll limit in which the speed of light tends to zero (c → 0). We see from Eqs 2.2 and 2.5 that there is a duality between the Galilean and the Carrollian contraction that simply exchanges the role of space and time translations in the contraction. Thinking of the time direction as the longitudinal direction of the world-line of a (massive) particle and the space directions as the transverse directions, makes it clear that a similar duality between longitudinal and transverse directions will be present for contractions related to extended objects as was discussed in more detail in Ref. 10.

Let us now formalise the contraction process. We start from an algebra g with generators t _α and structures constants f _αβ ^γ . Then for each λ > 0 we define a Lie algebra isomorphism c λ : g → g λ to another algebra g λ . If the limit λ → 0 is well-defined, we call the limiting algebra g 0 the contracted algebra. Note that at λ = 0 we no longer necessarily have a Lie algebra isomorphism. In the examples above, the isomorphisms for λ > 0 were given in Eqs 2.2 and 2.5, respectively. Contractions preserve the number of generators but the resulting algebra is not necessarily isomorphic to the starting one. Moreover, it is not generally possible to reverse the contraction process directly.

One approach to undoing the contraction perturbatively requires the knowledge of the original algebra g . Writing its generators multiplied by a formal power series in λ t α → ∑ n ≥ 0 t α ⊗ λ n 0 α + n = ∑ n ≥ 0 t α n , (2.7) where the offset n ₀(α) can depend on the generator, we construct an infinite-dimensional algebra of the generators t α ( n ) . We refer to the generator t α ( n ) as “level n” and think of it as the nth order perturbative expansion in the parameter λ. The offset should be chosen in such a way that the commutators of level m with level n only contain generators of level ≥ m + n , where the commutator is defined in using the associative product of power series together with the Lie bracket on g . The lowest order commutators involving only the t α ( 0 ) then correspond to the contracted algebra, but the higher terms capture the perturbative expansion of the original of the original algebra g . We denote by g ( N ) the algebra obtained by keeping terms up to level N. In this way, g ( 0 ) = g 0 , the contraction of g . When we keep all levels, we shall use the notation g ( ∞ ) . ⁷

Let us exemplify this in the case of the Galilei algebra Eq. 2.4. We define for n ≥ 0 J i j n = M i j ⊗ λ n , B i n = M i 0 ⊗ λ 1 / 2 + n , T i n = P i ⊗ λ 1 / 2 + n , H n = P 0 ⊗ λ n , (2.8) where the offsets are taken in accordance with Eq. 2.2. The associated Lie algebra is J i j m , J k l n = δ j k J i l m + n − δ i j J i k m + n − δ j l J i k m + n + δ i l J j k m + n , J i j m , B k n = δ j k B i m + n − δ i k B j m + n , J i j m , T k n = δ j k T i m + n − δ i k T j m + n , J i j m , H n = 0 , B i m , B j n = J i j m + n + 1 , B i m , T j n = − δ i j H m + n + 1 , B i m , H n = − T i m + n , T i m , T j n = T i m , H n = 0 . (2.9)

Since all commutators of generators at levels m and n generate only terms of level at least m + n, we can consistently quotient out all generators above a fixed level N. This leads to a finite-dimensional algebra. Retaining only the generators of level 0 leads to the Galilei algebra that is obtained by contraction. Keeping all generators up to level N then gives a perturbative approximation to the Poincaré algebra up to that order. The algebra Eq. 2.9 was given in Ref. 48, see also Refs. 49–52.

Repeating the same construction for the Carroll contraction Eq. 2.5 one can start with J i j n = M i j ⊗ λ n , K i n = M i 0 ⊗ λ 1 / 2 + n , T i n = P i ⊗ λ n , K n = P 0 ⊗ λ 1 / 2 + n (2.10)

We note that, when comparing Eq. 2.8 for the expanded Galilei algebra with Eq. 2.10 for the Carroll algebra, there is a duality between the two algebras where the λ ^1/2 is changed from P ₀ to P _i . This is a generalisation of the type of duality that has been noted before in Refs. 10, 24, 53.

The definition Eq. 2.10 leads to the infinite-dimensional algebra J i j m , J k l n = δ j k J i l m + n − δ i j J i k m + n − δ j l J i k m + n + δ i l J j k m + n , J i j m , K k n = δ j k K i m + n − δ i k K j m + n , J i j m , T k n = δ j k T i m + n − δ i k T j m + n , J i j m , K n = 0 , K i m , K j n = J i j m + n + 1 , K i m , T j n = − δ i j K m + n , K i m , K n = − T i m + n + 1 , T i m , T j n = T i m , K n = 0 . (2.11)

We note that comparing this formula to Eq. 2.9, there are subtle but important differences in the shifts of the indices by + 1 on the right-hand sides which are due to the placements of λ ^1/2 in the definitions of the algebra, and so ultimately to the physical meaning of the contractions.

The above procedure can also be viewed as a variant of the method of Lie algebra expansions that was originally introduced in Refs. 17–22. For a Lie algebra expansion in its formulation given in Ref. 20 one requires an abelian semi-group S whose elements we call λ _i and the S-expanded Lie algebra g × S has a basis t _α ⊗ λ _i and the Lie bracket t α ⊗ λ i , t β ⊗ λ j = f α β γ t γ ⊗ λ i λ j (2.12)

and commutativity of the product on S ensures the Jacobi identity of the expanded algebra. A simple example of a semi-group is given by S E ( N ) = { λ 0 , … , λ N , λ N + 1 } with abelian product λ i λ j = λ i + j if i + j ≤ N λ N + 1 otherwise (2.13)

The element λ _N+1 serves as a substitute for zero in the multiplication. Tensoring this semi-group with the real numbers corresponds to taking the quotient of the polynomial rings R [ λ ] / ( λ N + 1 R [ λ ] ) , i.e., working perturbatively in λ up to order N. In this identification we have simply λ _i = λ ⁱ , i.e., the ith basis of the semi-group should be identified with the ith power of the expansion parameter λ. One can also take the limit N → ∞ and work with formal power series.

A more refined version of the Lie algebra expansion method can be obtained when the original Lie algebra has a decomposition. We here restrict to the case when ⁸ g = V 0 ⊕ V 1 with V 0 , V 0 ⊂ V 0 , V 0 , V 1 ⊂ V 1 , V 1 , V 1 ⊂ V 0 . (2.14)

(We use different letters here for the graded pieces in order to avoid confusion with the contracted algebra g 0 studied above.) A resonant expansion of g with S E ( N ) is then given by the space ⊕ i = 0 N V i mod 2 ⊗ λ i (2.15)

with the obvious Lie brackets. It is this refined version of a Lie algebra expansion that makes direct contact with Eq. 2.7. A simple example of the refined expansion would be to take the Poincaré algebra Eq. 2.1 and write it as i s o 1 , d = 〈 M i j , P 0 〉 ︸ V 0 ⊕ 〈 M 0 i , P i 〉 ︸ V 1 . (2.16)

The expansion with S E ( 2 ) then would have the basis elements M i j ⊗ λ 0 , P 0 ⊗ λ 0 , M 0 i ⊗ λ 1 , P i ⊗ λ 1 , M i j ⊗ λ 2 , P 0 ⊗ λ 2 (2.17)

with new non-trivial commutators M 0 i ⊗ λ 1 , M 0 j ⊗ λ 1 = M i j ⊗ λ 2 , M 0 i ⊗ λ 1 , P j ⊗ λ 1 = δ i j P 0 ⊗ λ 2 , P i ⊗ λ 1 , P j ⊗ λ 1 = 0 (2.18)

in the expanded algebra.

The algebra obtained by expanding with S E ( 1 ) gives the Galilei algebra Eq. 2.4. The algebra above is a quotient of Eq. 2.9. A more general discussion of the expansion method can be found in Ref. 20. We shall apply this method to several more cases in this paper.

We now turn to the discussion of free Lie algebras. General references for this are Refs. 54 and 55 and we follow the exposition in Refs. 16 and 29.

A free Lie algebra on a (finite) set of D = d + 1 generators f 1 = P a | a = 0,1 , … d (2.19)

is obtained by considering all possible multi-commutators of the generators P _a only subject to anti-symmetry and the Jacobi identity. There is a natural grading of the free Lie algebra by the number of times the generators P _a appear in the multi-commutator. ⁹ The infinite-dimensional free Lie algebra f is therefore f = ⊕ ℓ = 1 ∞ f ℓ (2.20)

with for example f 2 = P a , P b | a , b = 0,1 , … , d = ∧ 2 f 1 (2.21)

being of dimension D ( D − 1 ) 2 because of the anti-symmetry of the commutator. We use the symbol ∧ ^k V to denote the kth anti-symmetric tensor power of a vector space V. The element [P _a , P _b ] is an independent element in the free Lie algebra.

The full structure of f can be summarised elegantly by a generating series in a formal parameter t as [56, 57] ⊕ ℓ = 1 ∞ ⊕ ℓ = 1 ∞ − 1 k t k ℓ ∧ k f ℓ = 1 − t f 1 (2.22)

that leads for example to f 2 = ∧ 2 f 1 , f 3 = f 1 ⊗ f 2 ⊖ ∧ 3 f 1 , f 3 = f 1 ⊗ f 3 ⊕ ∧ 2 f 2 ⊖ f 2 ⊗ ∧ 2 f 1 ⊕ ∧ 4 f 2 . (2.23) With ⊖ we mean the removal of a vector space from the tensor product, so that for f 3 the formula states that one takes all commutators of P _a from f 1 with the anti-symmetric [P _b , P _c ] from f 2 but has to remove the completely anti-symmetric Jacobi identity in all three elements.

Free Lie algebras as defined above are graded consistently with Eq. 2.20, i.e., they satisfy f ℓ , f m ⊂ f ℓ + m . (2.24)

The elements in f ℓ can be represented by Young diagrams with ℓ boxes that represent the irreducible action of the symmetric group S D on the elements in a set of multicommutator. In this way we write

The free Lie algebra can also be viewed as a successive extension of the real commutative Lie algebra f 1 using the method of Chevalley–Eilenberg Lie algebra cohomology [29]. The second cohomology of f 1 with values in R is non-trivial and of dimension D ( D − 1 ) 2 and therefore the Lie algebra f 1 can be extended by introducing anti-symmetric generators Z _ab = Z _[ab] with the new commutator P a , P b = Z a b , (2.26) but the Z _ab are central in this extended algebra. Thus one has obtained a graded Lie algebra f 1 ⊕ f 2 by considering the cohomology of f 1 . The process can now be repeated by studying the cohomology of f 1 ⊕ f 2 which leads to f 1 ⊕ f 2 ⊕ f 3 and so on. In this way, the free Lie algebra f is the maximal cohomological extension of f 1 .

As suggested by the notation (Eq. 2.19), we wish to think of the elements of f 1 for instance as the translation generators of some kinematic algebra. Typically, there is also a set of rotation generators, such as the Lorentz generators M _ab , under which the translation generators form a module. We call the space of the rotation generators f 0 as then we have a graded structure f 0 ⊕ f 1 (2.27)

to begin with. As a Lie algebra this is a semi-direct sum since f 0 acts on its representation space f 1 . The free Lie algebra based on f 1 then inherits an action of f 0 on each f ℓ and the expressions in Eq. 2.23 can be viewed as products and sums of f 0 modules. Extensions to super-algebras are discussed for example in Refs. 56, 57.

Free Lie algebras f admit many different quotients. We list a few important and representative examples and consider the case when there are also rotations f 0 acting on the algebra, see also Ref. 29.

1. Level truncation: Due to the grading (Eq. 2.24), the space

The corresponding quotient d = f / u (2.32) then consists only of those generators of f whose Young diagrams have the shape

with an arbitrary number of boxes in the first row. Why we refer to this quotient as the derivative truncation will become clear in Section 4.6 below.

Yet another common quotient is described by Serre relations and this arises for Kac–Moody algebras [25, 58] as we shall review in Section 2.3 below.

In this final section on algebraic constructions we would like to make a brief comment on the relation to (affine) Kac–Moody algebras. For any finite-dimensional Lie algebra g it is well-known that one can construct the (untwisted) loop algebra by letting g ̂ = g λ , λ − 1 (2.42) of Laurent polynomials in λ with values in g . It is also possible to add a central term and a derivation element to this construction to obtain a proper Kac–Moody algebra [25].

The relation to the constructions above becomes transparent by restricting to the parabolic subalgebra of polynomials g [ λ ] whose elements can be written in terms of the basis t _α ⊗ λ ⁿ for n = 0, 1, …. Clearly, this can be seen as a version of the method (Eq. 2.7) when setting all offsets n ₀(α) to zero. Setting some of the offsets to a non-zero value can result in twisted in affine algebras, see Ref. 16 for examples.

The parabolic subalgebra g [ λ ] is also closely related to free Lie algebras. Indeed it is known that the Borel subalgebra of a Kac–Moody algebra can be described as the quotient of a free Lie algebra on the simple Chevalley generators (often denoted e _i ) subject to the Serre relations that are encoded in the generalised Cartan matrix of the Kac–Moody algebra [58]. This is in particular true for affine algebras. Since we are dealing with a parabolic subalgebra rather than a Borel subalgebra in that the starting finite-dimensional Lie algebra g is not the abelian Cartan subalgebra, the Serre relations have to be adapted slightly but one can still describe g [ λ ] as a quotient of the free Lie algebra with f 1 = g that is acted upon by f 0 = g . Again, refinements of this constructions are available when g is decomposed already into V ₀ ⊕ V ₁, see Ref. 16 for examples. ¹⁰ This shows that all the various algebraic constructions in this section are interrelated.

For the Poincaré algebra (Eq. 2.1) the space M is Minkowski space in D dimensions with coordinates x ^a of dimension L = length. The faithful action of the Poincaré algebra can described as follows. Let 1 2 ω a b M a b + α a P a (3.3) be an arbitrary element of the Poincaré algebra. Its action on the coordinate x ^a is given by δ x a = − ω a b η b c x c + α a . (3.4) For the Lorentz part s o ( 1 , d ) this is nothing but the fundamental representation on which the P _a act as translations.

For the Galilean contraction (Eq. 2.2) we split space and time a = (0, i) and let t n = x 0 ⊗ λ − n , x n i = x i ⊗ λ − n − 1 / 2 . (3.5) Here, the dimensions of t _(n) and x ( n ) i are fixed by the dimension of x ⁰ and x ⁱ (that we always maintain at dimension length) and that of λ which for the Galilean case follows from λ = c ⁻². In particular, even though we use the notation t _(m), the lowest element t ₍₀₎ does not have the dimension of time (T) but of length (L) and x ( 0 ) i does not have the dimension of length but of L ²/T. We note also that in our conventions the Poincaré generators P _a in Eq. 2.1 have dimension of L ⁻¹ while the M _ab are dimensionless.

The action of an element ∑ m ≥ 0 1 2 ω m i j J i j m + v m i B i m + α m i T i m + ϵ m H m (3.6) of the algebra (Eq. 2.9) on a (dimensionless) coordinate element ∑ n ≥ 0 x n i T i n + t n H n (3.7) is then given by the commutator of the two elements, leading to δ t n = ϵ n + ∑ m = 0 n − 1 δ i j v m i x n − 1 − m j , (3.8) δ x n i = α n i + ∑ m = 0 n − ω m i j δ j k x n − m k + v m i t n − m . (3.9)

We see that restricting to only level 0 this becomes the usual action on the Galilean coordinates (t, z ⁱ ) with t = t ₍₀₎ and z i = x ( 0 ) i . In particular, the Galilean boost with only v ( 0 ) i ≠ 0 yields δ t = 0 , δ z i = v 0 i t , (3.10) which is the lowest order term of the Lorentz boost. Note that in our conventions the parameter v ₍₀₎ has dimension of L/T as a velocity, but we recall that [ x ( 0 ) i ] = L 2 / T and [t ₍₀₎] = L.

In order to see the systematic higher order expansion of the Lorentz boost encoded in Eq. 3.5, we follow [48] and define collective coordinates formally by X 0 = ∑ n ≥ 0 t n λ n , X i = ∑ n ≥ 0 x n i λ n + 1 / 2 , (3.11) as well as the collective boost parameter Θ i = ∑ n ≥ 0 v n i λ n + 1 / 2 . (3.12)

The transformation of the collective coordinates (Eq. 3.11) under such a collective boost then works out as δ X 0 = δ i j Θ i X j , δ X i = Θ i X 0 , (3.13)

the usual expression for an infinitesimal relativistic Lorentz boost with rapidity Θ ⁱ . However, the difference is that now the boost parameter and the coordinate are collective.

If one imposes that v n i = 1 2 n + 1 v 2 n + 1 n i (3.14) for some scalar v and spatial unit vector n ⁱ , i.e., δ _ij n ⁱ n ^j = 1, then the transformations (Eq. 3.13) become for λ = c ⁻² δ X 0 = ∑ n ≥ 0 1 2 n + 1 v c 2 n + 1 δ i j n i X j , δ X i = ∑ n ≥ 0 1 2 n + 1 v c 2 n + 1 n i X 0 , (3.15) which are the expansions of the infinitesimal Lorentz boost with parameter θ ⁱ = θn ⁱ , where tanh θ = v/c for v/c ≪ 1.

At this point we should comment on the geometrical meaning of the collective coordinates (Eq. 3.11). These define a hyperspace of co-dimension D within the infinite-dimensional generalised Minkowski space with coordinates (Eq. 3.5). Since the sums are infinite and we are not making any assumptions about convergence here, the expressions are formal but the formal expansion parameter λ is introduced in such a way as to render meaningful expressions at any finite order in the expansion. What the transformation (Eq. 3.13) then describes is a transformation from one hyperspace to another one, so we obtain a description of ordinary Minkowski space as a family of hyperspaces inside generalised Minkowski space. We shall see that a similar picture applies to all other expansions considered in this paper.

For the case of the Carroll algebra, we use “Carroll time” s = Cx ⁰ introduced in Refs. 12, 39. The contraction limit in these variables is C → ∞. Morally, we can think of C as being related to the inverse of the speed of light, so that the speed of light goes to zero. However, the dimension of C is that of a velocity. The expansion parameter λ = C ⁻², so that s ₍₀₎ = x ⁰ ⊗ C is the Carroll time of Refs. 12, 39.

For the Carrollian contraction (Eq. 2.5) we proceed analogously to the generalised Galilei space-time and define s n = x 0 ⊗ λ − n − 1 / 2 , x n i = x i ⊗ λ − n . (3.16) where the difference to Eq. 3.5 that the constant shift has moved from the space to the time translations. The dimensions of the coordinates implied by these definitions are [s _(n)] = L ²ⁿ⁺²/T ²ⁿ⁺¹ and [ x ( n ) i ] = L 2 n + 1 / T 2 n .

A dimensionless element ∑ m ≥ 0 1 2 ω m i j J i j m + v m i K i m + α m i T i m + ϵ m K m (3.17) of the expanded Carroll algebra (Eq. 2.11) then acts on the coordinates by δ s n = ϵ n + ∑ m = 0 n δ i j v m i x n − m j , (3.18) δ x n i = α n i − ∑ m = 0 n ω m i j δ j k x n − m k + ∑ m = 0 n − 1 v m i s n − 1 − m . (3.19)

Especially, restricting to lowest order we obtain for the Carrollian time coordinate s = s ₍₀₎ and z i = x ( 0 ) i that the Carrollian boost (only v ( 0 ) i ≠ 0 ) acts by δ s = δ i j v 0 i z j , δ z i = 0 , (3.20) the well-known expression for this boost, see, e.g., Refs. 12, 39. In particular, an ordinary particle at rest cannot be Carroll boosted to one in motion: it is effectively stationary in any frame.

We now turn to corrections to this classical statement as contained in the infinite-dimensional algebra (Eq. 2.11). We introduce the collective coordinates X 0 = ∑ n ≥ 0 s n λ n + 1 / 2 , X i = ∑ n ≥ 0 x n i λ n (3.21) as well as the collective boost parameter Θ i = ∑ n ≥ 0 v n i λ n + 1 / 2 . (3.22) The transformation then becomes δ X 0 = δ i j Θ i X j , δ X i = Θ i X 0 (3.23) just as in Eq. 3.13 and thus formally resembles the usual infinitesimal Lorentz boost with parameter Θ ⁱ . With λ = C ⁻² we can now specialise to v n i = 1 2 n + 1 b 2 n + 1 n i (3.24) to arrive at δ X 0 = ∑ n ≥ 0 1 2 n + 1 b C 2 n + 1 δ i j n i X j , δ X i = ∑ n ≥ 0 1 2 n + 1 b C 2 n + 1 n i X 0 . (3.25) This is the correct expansion of a Lorentz boost in Carroll parametrisation where b = C v c is fixed in the limit C → ∞ [39].

The relativistic conformal algebra in D > 2 dimensions is s o ( D , 2 ) that contains, besides the Poincaré generators (Eq. 2.1), the special conformal generators S _a and the dilatation generator D with additional commutators M a b , S c = η b c S a − η a c S b , D , P a = P a , D , S a = − S a , S a , P b = 2 M a b − 2 η a b D . (3.26) A non-relativistic Galilean version of this algebra can be obtained by considering the contraction (λ = c ⁻²) J i j = M i j , D = D , H = P 0 , S = S 0 , B i = λ 1 / 2 M i 0 , T i = λ 1 / 2 P i , G i = λ 1 / 2 S i (3.27) that extends the Galilean contraction (Eq. 2.2) from Poincaré to the conformal algebra. The resulting contracted algebra is known as the Galilei conformal algebra and has been studied for example in Refs. 61–65, see also Refs. 66, 67 for a recent extension to higher spin algebras.

The two lines of Eq. 3.27 also define spaces V ₀ and V ₁ satisfying (Eq. 2.14), so that an infinite expanded algebra undoing the contraction can be defined, exactly in the same way as for the previous cases. An infinite generalised space-time on which the infinite expanded Galilei conformal algebra can act is then defined by introducing coordinates t n = x 0 ⊗ λ − n , x n i = x i ⊗ λ − n − 1 / 2 . (3.28) The action of the algebra on these coordinates can be worked out in the same way as in the previous cases, with the additional feature that the action of the special conformal transformation is non-linear in the coordinates due to the relativistic expressions δ σ D x a = σ x a , δ β b S b x a = 2 x ⋅ β x a − x ⋅ x β a , (3.29) extending the Poincaré transformations (Eq. 3.4).

Collectives coordinates are defined by X i = ∑ n ≥ 0 x n i λ n + 1 / 2 , X 0 = ∑ n ≥ 0 t n λ n (3.30) exactly as for the non-conformal Galilei case (Eq. 3.11). Under special conformal transformations (Eq. 3.29), the lowest order coordinates transform as δ t 0 = − b 0 t 0 2 , δ t 1 = − 2 b 0 t 0 t 1 − b 1 t 0 2 + 2 δ i j β 0 i x 0 j t 0 − δ i j x 0 i x 0 j b 0 , δ x 0 i = − 2 b 0 t 0 x 0 i + t 0 2 β 0 i , (3.31) where we have expanded the parameter of the transformation as β 0 = ∑ n ≥ 0 b n λ n , β i = ∑ n ≥ 0 β n i λ n + 1 / 2 . (3.32)

The small parameter can also be taken to be the curvature of space-time in appropriate dimensions. This was considered in Ref. 68 and leads to corrections to Minkowski space-time towards (Anti-)de Sitter space when the starting point is the (A)dS algebra that differs from the Poincaré algebra (Eq. 2.1) by the non-trivial commutator P a , P b = σ M a b (3.33) among the translations. The sign σ = +1 is the AdS algebra s o ( D − 1,2 ) and σ = −1 is the dS algebra s o ( D , 1 ) . ¹² Here, and in contrast to the Poincaré algebra (Eq. 2.1), we have rescaled all generators to be dimensionless. ¹³

Following our usual expansion method we define the generators M a b n = M a b ⊗ λ n , P a n = P a ⊗ λ 1 / 2 + n , (3.34) where now λ = R ⁻² is to be thought of as the curvature scale of the (A)dS space-time. For R → ∞ we obtain the Poincaré algebra (Eq. 2.1) as a contraction of the (A)dS algebra similar to the non-relativistic cases in Section 2.1.

One can now similarly consider an extended space-time with coordinates x n a = x a ⊗ λ − 1 / 2 − n . (3.35)

The transformations formula for these coordinates is now more complicated since the underlying translations no longer commute due to Eq. 3.33. Since we do not rely on them in the following, we refer the reader to Ref. 68. In Section 4.5 we shall study a particle model based on this generalised space-time.

In the case of the Maxwell extension of Poincaré we also deal with non-commuting translations P _a , the basic commutator is Eq. 2.26, where Z _ab is a new generator unlike in the case of the (A)dS algebra.

The most general algebra that we can construct when starting from the Poincaré algebra is the Maxwell free Lie algebra that was introduced in Section 2.2.1. An associated generalised space-time can be defined by considering the P _a and all their free commutators as translation generators. This means that one has coordinates for each of them [28, 29] P a ↔ x a , Z a b ↔ θ a b , Y a b , c ↔ ξ a b , c etc. (3.36) The generalised space-time defined by these coordinates has non-abelian translations δ x a = ϵ a , δ θ a = ϵ a b − 1 2 x a ϵ b − x b ϵ a , δ ξ a b , c = ϵ a b , c + 1 3 2 ϵ a b x c − ϵ b c x a − ϵ c a x b + 1 3 ϵ a x b x c − ϵ b x a x c , (3.37) where the higher-level coordinates are also affected by the translations of all lower levels.