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Noncommutative Differential Forms on the kappa-deformed Space

We construct a differential algebra of forms on the kappa-deformed space. For a given realization of the noncommutative coordinates as formal power series in the Weyl algebra we find an infinite family of one-forms and nilpotent exterior derivatives. We derive explicit expressions for the exterior derivative and one-forms in covariant and noncovariant realizations. We also introduce higher-order forms and show that the exterior derivative satisfies the graded Leibniz rule. The differential forms are generally not graded-commutative, but they satisfy the graded Jacobi identity. We also consider the star-product of classical differential forms. The star-product is well-defined if the commutator between the noncommutative coordinates and one-forms is closed in the space of one-forms alone. In addition, we show that in certain realizations the exterior derivative acting on the star-product satisfies the undeformed Leibniz rule.


Introduction
Recent years have witnessed a growing interest in the formulation of physical theories on noncommutative (NC) spaces. The structure of NC spaces and their physical implications were studied in [1]- [7]. Such spaces have roots in quantum mechanics where the canonical phase space becomes noncommutative (see [8] for a historical treatment and the references therein). Classification of the NC spaces and investigation of their properties, in particular the development of a general theory suitable for physical applications, is an important problem. In this note we investigate differential calculus in the Euclidean kappa-deformed space. The kappa-space is a mild deformation of the Euclidean space whose coordinatesx µ , µ = 1, 2, . . . , n, satisfy a Lie algebra type commutation relations.
The commutation relations forx µ depend on a deformation vector a ∈ R n which is on a very small length scale and yields the undeformed space when a → 0.
The kappa-space was studied by different groups, from both the mathematical and physical points of view [9]- [33]. It provides a framework for doubly special relativity [18], [19], and it has applications in quantum gravity [34] and quantum field theory [35], [36] A crucial tool in the development of a physical theory is differential calculus. There have been several attempts to develop differential calculus in the kappa-deformed space [14], [25]. For a general associative algebra Landi gave a construction of a differential algebra of forms in [37]. In this work we present a construction of differential forms and exterior derivative in the kappa-deformed space using realizations of the NC coordinatesx µ as formal power series in the Weyl algebra. Our approach is based on the methods developed for algebras of deformed oscillators and the corresponding creation and annihilation operators [38]- [47]. The realizations of the NC coordinatesx µ in various orderings have been found in [26] and [28]. The realization of a general Lie algebra type NC space in the symmetric Weyl ordering has been given in [48].
The outline of the paper is as follows. In section 2 we present a novel construction of a differential algebra of forms on the kappa-deformed space. The exterior derivatived and one-forms ξ µ are defined as formal power series in the Lie superalgebra generated by commutative coordinates x µ , derivatives ∂ µ and ordinary one-forms dx µ . The number of one-forms ξ µ is the same as the number of NC coordinatesx µ , and the results are valid for a general deformation vector a ∈ R n . In the present work we do not require compatibility of the differential structure with a kappa-deformed symmetry. This distinguishes our approach from [14] where compatibility of the differential calculus with the kappa-deformed symmetry group was considered. This compatibility requires that in addition to ξ µ there is an extra one-form φ. The realizations ofd and ξ µ are related to realizations ofx µ through a system of partial differential equations. We also define higher-order forms and show thatd is a nilpotent operator which satisfies the graded Leibniz rule. However, the differential forms are generally not graded commutative. In the smooth limit when a → 0 our theory reduces to classical results. In section 3 we analyze the exterior derivative and one-forms in covariant realizations of the kappa-deformed space. We show that the algebra generated byx µ and ξ µ generally does not close under the commutator bracket since [ξ µ ,x ν ] may involve an infinite series in derivatives ∂ µ . We have derived a condition for the commutator [ξ µ ,x ν ] to be closed and found realizations in which the condition holds. A similar analysis was carried out by Dimitrijević et. al. in [25], but our results are more general and in certain aspects different. Section 4 deals with the differential algebra of forms in noncovariant realizations. We introduce a general Ansatz for the exterior derivative and find the corresponding one-forms in the left, right and symmetric left-right realization. In these realizations the commutator [ξ µ ,x ν ] is always closed in the space of one-forms ξ µ alone. In section 5 we present a novel construction of the star-product of (classical) differential forms.
The star-product depends on realizations ofx µ and is well-defined if the commutator [ξ µ ,x ν ] is closed in the space of one-forms ξ µ alone. We show that for differential forms with constant coefficients the star-product is undeformed and graded-commutative. However, this property does not hold for arbitrary forms.
Also, we consider the induced exterior derivative acting on the star-product of differential forms. A short conclusion is given in section 6.

Differential Forms
In this section we present a general construction of a differential algebra of forms in the Euclidean kappa-deformed space. This construction is based on realizations of the NC coordinatesx µ as formal power series in the Weyl algebra introduced in [26] and [28]. We find that for a given realization ofx µ there is an infinite family of exterior derivativesd and one-forms ξ µ where ξ µ are obtained by the action ofd onx µ . This infinite family includes two canonical types ofd and ξ µ whose realizations are studied in detail in the following sections.
The n-dimensional kappa-deformed space is a noncommutative space of Lie algebra type with generatorsx 1 ,x 2 , . . . ,x n satisfying the commutation relations The vector a ∈ R n describes the deformation of the n-dimensional Euclidean space. The Lie algebra satisfying (1) will be denoted by g. The structure constants of g are given by Our construction of the differential calculus uses realizations ofx µ as formal power series in the deformation parameter a with coefficients in the Weyl algebra. The Weyl algebra is generated by the operators x µ and ∂ µ , µ = 1, 2, . . . , n, satisfying It has been shown in [26] and [28] that there exist infinitely many realizations ofx µ of the form where φ αµ is a formal power series We denote ∂ k = ∂ k 1 1 ∂ k 2 2 . . . ∂ kn n where k is a multi-index of length |k| = µ k µ . In the limit as a → 0 we have φ αµ → δ αµ , whencex µ become the commutative coordinates x µ . A representation (3) of the NC coordinatesx µ will be called a φ-realization. The NC coordinatesx µ and derivatives ∂ µ generate a deformed Heisenberg algebra satisfying We will assume that the matrix [φ µν ] is invertible, allowing us to express x µ as where φ −1 αµ (∂) is also a formal power series of the type (4). The existence of φ −1 µν implies that threre is a vector space isomorphism between the symmetric algebra generated by x µ , µ = 1, 2, . . . , n, and the enveloping algebra of g. This isomorphism will be important in defining the star-product discussed in section 5. With regard to the action of the rotation algebra so(n) the realizations of the kappa-space can be divided into covariant [28] and noncovariant [26]. Both types of realizations will be used in the construction of differential forms in sections 3 and 4.
It is useful to introduce a unital associative algebra A over C generated by x µ , ∂ µ and ordinary one-forms dx µ , 1 ≤ µ ≤ n, satisfying the additional relations where α i , β i ∈ N 0 and 1 ≤ σ 1 < σ 2 . . . < σ p ≤ n for p = 1, 2, . . . , n. We define a Z 2 -gradation of A by A = A 0 ⊕ A 1 where A 0 and A 1 are spanned by the monomials (7) with p even and odd, respectively. The algebra A is equipped with the graded commutator defined on homogeneous elements by where |u| denotes the degree of u, (|u| = 0 or |u| = 1). The commutator (8) makes A into a Lie superalgebra, and it satisfies the graded Jacobi identity [[u, v]] ]] = 0. (9) Recall that in the ordinary Euclidean space the exterior derivative is given by d = α dx α ∂ α . It is a nilpotent operator, d 2 = 0, satisfying the commutation relation [d, x µ ] = dx µ . Our goal is to construct smooth deformations of d and dx µ , denotedd and ξ µ , µ = 1, 2, . . . , n, which preserve the basic relation Let us assume thatd and ξ µ are represented by where h µν and k µν are formal power series of the type (4). The boundary conditions lim a→0 h µν = δ µν and lim a→0 k µν = δ µν ensure that in the smooth limit ξ µ → dx µ andd → d as a → 0. As in the classical case, the deformed one-forms anticommute and the exterior derivative is nilpotent. Indeed, since {dx α , dx β } = 0. We assume that the matrix [h µν ] is invertible so that we may express dx µ in terms of ξ µ . Using representation (11) one finds that the commutation relation (10) is equivalent to a system of partial differential equations for the unknown functions h µν and k µν : This is an underdetermined system of n 2 equations for 2n 2 unknown functions.
Taking the commutator ofd with both sides of the commutation relations (1), and applying the Jacobi identity to the commutator [d, [x µ ,x ν ]], we find thatx µ and ξ ν satisfy the compatibility condition Hence, every solution of Eq. (14) must be compatible with the differential equation implicit in (15). We note that Eq. (15) implies that, since a = 0, not all commutators [x µ , ξ ν ] can be simultaneously zero.
The condition (15) places constraints on the choice of k µν and h µν . For a given function k µν satisfying lim a→0 k µν = δ µν , Eq. (14) uniquely determines h µν .
LetĀ denote the formal completion of A. We associate to the exterior derivatived a linear map or actiond :Ā →Ā defined bŷ It follows from Eq. (10) thatd·x µ = ξ µ , hence the action ofd on the coordinatex µ yields the one-form ξ µ . The action ofd on the product of homogeneous elements u, v, ∈Ā satisfies the graded Leibniz rulê For zero-formsf =f(x) andĝ =ĝ(x) this reduces to the undeformed Leibniz It turns out that it is quite natural to consider the following canonical representation ofd and ξ µ : Type IId The first type is obtained by choosing k µν = δ µν , in which case Eq. (14) yields h µν = φ µν . This provides the simplest possible realization of the one-form ξ µ .
The second type is obtained by demanding that k µν = h µν . Then the functions h µν satisfy the system of partial differential equations subject to the boundary conditions lim a→0 h µν = δ µν . In this case both the exterior derivatived and one-forms ξ µ depend in a very nontrivial manner on the given φ-realization. In the following sections sections we shall analyzed and ξ µ in covariant and noncovariant realizations found in [26] and [28]. Note that the generatorsx µ , ∂ µ , ξ µ , 1 ≤ µ ≤ n, form an associative superalgebra which inherits the grading from the superalgebra A. The subalgebra generated byx µ , ∂ µ , 1 ≤ µ ≤ n is the deformed Heisenberg algebra (5).
So far we have defined the exterior derivatived and one-forms ξ µ such that d ·x µ = ξ µ . We would like to extend the above construction to higher-order forms so that the action ofd on k-forms yields (k + 1)-forms. First, we need to define what is meant by a k-form for k ≥ 1. A k-form is a finite linear combination of monomials inx 1 ,x 2 , . . . ,x n and ξ 1 , ξ 2 , . . . , ξ n such that there are precisely k oneforms ξ µ in each monomial. The one-forms ξ µ may be placed in any order in a given monomial. For example, bothω 1 =x µxν ξ ρ andη 1 =x µ ξ ρxν are one-forms, albeit different. LetΩ k denote the space of k-forms and letΩ = k≥0Ω k . The multiplication inΩ is simply given by juxtaposition of the elements. This defines a grading onΩ sinceΩ kΩl ⊆Ω k+l . We note that the product of differential forms is not graded-commutative in general, The product is graded-commutative only for constant formsω k = ξ µ 1 ξ µ 2 . . . ξ µ k since ξ µ i and ξ µ j anticommute.
Next we show that the exterior derivatived mapsΩ k intoΩ k+1 for k ≥ 0.
First, using the Leibniz rule (17) it is easily seen that Furthermore, using Eq. (11) we find The relations (23) and (25) together with the Leibniz rule (17) imply thatd maps k-forms to (k + 1)-forms. For example, The exterior derivative satisfies the graded Leibniz rulê Hence, the algebraΩ together with the linear mapd :Ω k →Ω k+1 is a differential algebra. Our approach is essentially the same as the construction of the differential algebra of forms discussed in [37]. In our case the algebra of zeroforms has the additional structure of the universal enveloping algebra satisfying relations (1). We note that in general one cannot rewrite a given k-form such that ξ µ 1 , ξ µ 2 , . . . , ξ µ k are placed to the far right. This is possible only in special realizations in which the commutator [ξ µ ,x ν ] closes in the space of one-forms ξ µ alone.

Covariant realizations
In this section we shall investigate the differential algebra of forms in covariant realizations of the kappa-deformed space introduced in [28]. These realizations are covariant under the action of the rotation aglebra so(n). Of particular interest is a class of simple realizations obtained for the following choice of φ µν in the representation (3): Left realization: Right realization: Natural realization: Symmetric realization: Here A and B are commuting operators defined by A = ia∂ and B = a 2 ∂ 2 where we use the convention a∂ = α a α ∂ α , ∂ 2 = α ∂ 2 α , etc. The symmetric realization corresponds to the Weyl symmetric ordering of the monomials inx µ .
We remark that for a general Lie algebra type NC space there is a universal formula for φ µν in Weyl symmetric ordering given in [48] as follows. Supposê x 1 ,x 2 , . . . ,x n are generators of a Lie algebra with structure constants θ µνα : Let M = [M µν ] denote the n × n matrix of differential operators with elements Then the Weyl symmetric realization of the Lie algebra (32) is given by is the generating function for the Bernoulli numbers (see also [49]). In principle the exterior derivative and one-forms may be constructed using any of the above realizations. Here we shall consider the left, right and natural realization.

Covariant realizations of type I
Let us consider realizations of type I where the exterior derivative is undeformed, d = α dx α ∂ α , and one-forms are given by ξ µ = α dx α φ αµ (∂). We investigate the conditions under which the commutator [ξ µ ,x ν ] is closed in the space of oneforms ξ µ . The closedness of the commuatator is important when considering the extended star-product of (classical) forms in section 5.
The matrix [φ µν ] is invertible, hence we may express dx µ in terms of ξ µ to obtain Clearly, the commutator (36) is closed in the space of one-forms ξ µ only if the coefficients C µνσ are constant. This condition is satisfied in the left and right realizations, as shown in the following. In the left realization we havê which yields Similarly, in the right realization we havê which leads to On the other hand, in the natural and symmetric realizations the coefficients C µνσ involve partial derivatives so the commutators between ξ µ andx ν are not closed.

Covariant realizations of type II
Consider now realizations of type II where the exterior derivative and one-forms are given byd = α ξ α ∂ α and ξ µ = α dx α h αµ (∂), and h αµ is a solution of Eq. (21). In this section we shall constructd and ξ µ using the natural realization (30). The construction of NC forms in type II realization was considered in [25], but not in a proper and complete way. Our motivation for using the natural realization is to present a proper analysis of this problem.
Let us write Eq. (21) in a more compact form where Λ α (∂) = β h αβ (∂)∂ β . The idea is to solve an auxiliary problem for Λ α and then calculate h µν from Eq. (42). Multiplying Eq. (42) by ∂ µ and summing we obtain the following boundary value problem for Λ α : where Ψ ρ (∂) = µ φ ρµ (∂)∂ µ . In the natural realization (30) we find Let us denote This is the inverse shift operator introduced in [28]. The index structure of Ψ ρ and Eq. (43) suggest that we should look for for unknown functions H 1 and H 2 . From Eqs. (44) and (45) we obtain Substituting the above result into Eq. (43) we find that H 1 and H 2 satisfy the following system of differential equations: Since Λ α (∂) → ∂ a as a → 0, H 1 and H 2 are subject to the boundary conditions It is shown in Appendix A that the above system has a unique solution Inserting the expressions for H 1 and H 2 into Eq. (45) we find Since the exterior derivative is given byd Thus, we find from Eq. (52) that Keeping only the first-order terms in a ∈ R n we obtain the approximation where d = ∂ dx is the undeformed exterior derivative.
(42) we find after some manipulation that where Therefore, in the natural realization of type II the one-form ξ µ is given by Although the above realization of ξ µ is rather complicated, the first-order approximation has a particularly nice form Let us now investigate the commutation relations for ξ µ andx ν . The NC coordinates in the natural realization (30) are given bŷ The explicit form of the commutator [ξ µ ,x ν ] is fairly complicated and a complete derivation is given in Appendix B. Here we only state that it can be expressed as µν are certain combinations of the functions L 1 , L 2 , . . . , L 5 and their partial derivatives. We note that the commutator (65) is not closed since the right-hand side involves derivatives ∂ µ . To gain an insight into the form of the commutator it is instructive to find a first-order approximation in the parameter a. To first order in a the natural realization ofx µ is given bŷ Using the approximations (63) and (66) we obtain As a special case suppose that the vector a ∈ R n has only one non-zero component, a µ = aδ µn for µ = 1, 2, . . . , n. Then The above result agrees to first order in a with the commutator [ξ µ ,x ν ] for vectorlike transforming one-forms considered in [25]. We emphasize, however, that the exact expression (65) does not agree with this commutator for higher orders in a.

Noncovariant realizations
In this section we consider the exterior derivative and one-forms in noncovariant realizations of the kappa-space introduced in [26]. We assume that the components of the deformation vector a ∈ R n are given by a k = 0 for k = 1, 2, . . . , n − 1 and a n = a. Then the commutation relations (1) yield We use the Latin alphabet for the indices 1, 2, . . . , n − 1 and the Greek alphabet for the full set 1, 2, . . . , n. It was shown in [26] that the NC coordinatesx µ have infinitely many realizations of the form where respectively. We remark that only the symmetric Weyl realization is covariant under the full algebra so(n).
For a given parameter function ϕ and an arbitrary choice of N 1 , N 2 and G one can find the one-forms ξ k satisfying [d,x µ ] = ξ µ . As in the case of the covariant realizations one can express the commutator [ξ µ ,x ν ] in terms of the one-forms ξ µ and partial derivatives ∂ µ , but the general expressions are fairly complicated.
In the following we will focus our attention to a subfamily of the noncovariant realizations which lead to some interesting results. These realizations are parameterized by ϕ(A) = e −cA , c ∈ R: x k = x k e −cA , k = 1, 2, . . . , n − 1, The algebra generated byx µ and ξ µ satisfies the commutation relations [ξ n ,x l ] = 0, [ξ n ,x n ] = 0.
This algebra satisfies the graded Jacobi relations (9). We note that the relations (80)-(81) correspond to the algebra found by Kim et. al. [50] where the commutators are defined in terms of the star-product, except that in our work ξ µ and ξ ν anticommute. In particular, for c = 0 the exterior derivative becomeŝ which is the type II realization ofd. In addition to the examples in section 3 the commutators (80)-(81) also close in the space of one-forms ξ µ alone. Moreover, the right realization (c = 0) is an example of a type II realization with closed commutator.
The above construction can be extended to any parameter function ϕ. It can be shown that for a given ϕ one can find N 1 , N 2 and G such thatd = α ξ α ∂ α and [d,x µ ] = ξ µ . However, this may be very complicated as already seen in the natural realization in section 3.

Extended star-product
Regarding functions as zero-forms we want to extend the star-product to differential forms of arbitrary degree. The star-product of differential forms in the context of deformation quantization has been investigated recently in [51]. The construction of the star-product presented here is valid for a general Lie algebra type noncommutative space. We recall that the realization of NC coordinatesx µ in terms of x µ and ∂ µ is given by Eq. (3). Also, since the matrix [φ µν ] is invertible the commutative coordinates x µ admit realization in terms ofx µ and ∂ µ via Eq. (6). The duality betweenx µ and x µ induces a vector space isomorphism Ω φ : U(g) → S between the enveloping algebra U(g) of the Lie algebra (1) and the symmetric algebra S generated by x µ , µ = 1, 2, . . . , n. The isomorphism Ω φ depends on the realization φ, and is given as follows. Let 1 denote the unit in S (S is isomorphic to the Fock space built on the vacuum vector |0 ≡ 1). Then x µ and ∂ µ act on f ∈ S in a natural way by x µ · f = x µ f and ∂ µ · f = ∂f ∂xµ . In particular, For a monomialf (x) ∈ U(g) we define and extend Ω φ linearly to U(g). The map Ω φ is evaluated atf (x) by using the realization (3) and action (83). For example, Similarly, for monomials of order two we have where ∂φαµ ∂∂ν · 1 is a first-order coefficient in the Taylor expansion of φ αµ (∂). In general, Ω φ (x µ 1x µ 2 . . .x µm ) is a polynomial in the variables x µ 1 , x µ 2 , . . . , x µm whose coefficients are given by the Taylor expansion of φ µν . The computation of Ω φ (x µ 1x µ 2 . . .x µm ) can be done using a recursive formula. Suppose that Then The commutator in the above expression is calculated according to The inverse map Ω −1 φ is defined analogously. Let1 be the unit in U(g). Define the action ofx µ on a monomialf(x) ∈ U(g) byx µ ·f (x) =x µf (x). The action is expressed using the commutation relations [∂ µ ,x ν ] = φ µν (∂). For the lowest order vector we havex Then Ω −1 φ is given by where f (x) ·1 is calculated using the realization (6) and relations (90). For example, , and for monomials of order two we have One can show that the right hand side of Eq. (93) is invariant under the transposition of indices µ ↔ ν, hence Ω −1 φ (x µ x ν ) is well-defined. Clearly, Ω φ and Ω −1 φ can be readily extended to U(g) and S, the formal completions of U(g) and S.
The star-product of f, g ∈ S is defined by (g(x)). In the limit as the deformation parameter a → 0 the star-product reduces to ordinary product of functions (c.f. Eq. (4)). The star-product on the kappa-deformed space was discussed in [26], [28], [29]; see also [52]. Equation (94) defines the star-product of zero-forms. Following the ideas outlined above we want to extend the star-product to differential forms of arbitrary degree. Our strategy is to associate to ω k a noncommutative formω k such that ω k · 1 = ω k , and define the star-product by It turns out that the star-product (95) is well-defined provided the commutator [ξ µ ,x ν ] is closed in the space of one-forms ξ µ alone. It depends only on the realizations of the coordinatesx µ , hence we also denote it by ⋆ φ .
Finally, let us consider the exterior derivative acting on the star-product of forms. In the realization of type I the exterior derivative is undeformed,d = d ≡ α dx α ∂ α . Then one can show that whereω · 1 = ω. Using the star-product (95) and Leibniz rule (17) one finds Hence, in type I realization the Leibniz rule for the extended star-product is undeformed. It would be interesting to invstigate the action of the induced exterior derivative on the star-product of forms in other realizations whend is given by a general expression (11).

Concluding remarks
In this paper we have investigated the differential algebra of forms on the kappadeformed space. Our construction of the exterior derivatived and one-forms ξ µ is based on the realizations of NC coordinatesx µ in terms of formal power series in the Weyl algebra. We have shown that for each realization ofx µ there is an infinite family of the exterior derivativesd which uniquely determine the one-forms ξ µ . The exterior derivative is a nilpotent operator and it satisfies the undeformed Leibniz rule. The NC coordinatesx µ , derivatives ∂ µ and one-forms ξ µ generate a Z 2 -graded algebra. The subalgebra generated byx µ and ∂ µ is a deformed Heisenberg algebra. The algebra generated byx µ and ξ µ is generally We have also extended the star-product from zero-forms to differential forms of arbitrary degree. The star-product can be defined for realizations in which [ξ µ ,x ν ] is closed in the space of one-forms ξ µ . It depends only on the realizations of both the NC coordinatesx µ . For diffential forms with constant coefficients the star-product is undeformed and graded-commutative, but for arbitrary forms this is no longer true. It was shown the the exterior derivative acting on the extended star-product satisfies the undeformed Leibniz rule in type I realization.
It would be interesting to investigate possible relations between our approach to the star-product of differential forms and the recent work presented in [51].
Finally, the notion of the twist operator is very important in the construction of the star-product from both the mathematical ( [53], [54]) and physical ( [55], [56], [57], [58], [59]) points of view. The twist operator for zero-forms on the kappa-deformed space was constructed in [30] and [59], and was also considered in [27]. However, it remains an open problem to see if there exisits a twist operator that leads to the star-product of differential forms defined in this work.

Appendix A
In this appendix we find the solution of the system of equations (47)- (48). Let us write Eq. (47) in equivalent forms as We assume that H 1 can be factored as H 1 (A, B) = ZF 1 (B) which leads to the following differential equation for F 1 , The boundary condition for H 1 implies that lim a→0 F 1 (B) = 1. Now the solution of Eq. (108) is readily found to be hence Next, let us consider Eq. (48) which we write equivalently as We apply a similar method of "separation of variables" assuming that H 2 (A, B) = ZF 2 (B)+F 3 (B). Inserting the Ansatz for H 1 and H 2 into Eq. (111), and grouping the terms depending only on B on the right-hand side, we obtain Let us define the function Then the variables in Eq. (112) can be separated as We conclude that both sides of the equation must be zero which implies that F 2 and F 3 satisfy the following system of differential equations: Using the boundary condition for H 2 we find that in the limit a → 0 both F 2 (B) and F 3 (B) must be finite. Taking this into account, integration of the system (115)-(116) yields Therefore, 8 Appendix B In this Appendix we give a brief derivation of the result (65). We shall do this in two steps. First we calculate the commutator [ξ µ , andx ν is given in the natural realization (30). We have Expressing h αµ by Eq. (56) and making use of after some manipulation we find where we have defined A similar computation yields α [h αµ , iax] dx α = a 2 E 1 dx µ + (iE 2 a µ + a 2 E 3 ∂ µ )(iadx) + a 2 (iE 4 a µ + a 2 E 5 ∂ µ )(∂dx) (124) where the functions E i are defined by Combining equations (121) and (124) we obtain where the functions P (i) µ and Q (i) µν are given by In the second step we wish to express the commutator (130) in terms of the one-forms ξ µ and derivatives ∂ µ . In order to replace dx µ by ξ µ we write dx µ = α h −1 αµ (∂)ξ µ where h −1 αµ is the inverse of the matrix h αµ . The inverse matrix should have the same index structure as h αµ , hence we look for h −1 αµ in the form h −1 αµ (∂) = G 1 δ αµ + iG 2 a α ∂ µ + iG 3 a µ ∂ α + a 2 G 4 ∂ α ∂ µ − ∂ 2 G 5 a α a µ .
Aknowledgements. This work is supported by the Croatian Ministry of Science, Education and Sports grants no. 098-0000000-2865 and 177-0372794-2816.