Differential forms and k-Minkowski spacetime from extended twist
aa r X i v : . [ h e p - t h ] J u l Di ff erential forms and κ -Minkowski spacetime from extended twist Tajron Juri´c Rudjer Boškovi´c Institute, Bijeniˇcka c.54, HR-10002 Zagreb, CroatiaStjepan Meljanac ,Rudjer Boškovi´c Institute, Bijeniˇcka c.54, HR-10002 Zagreb, CroatiaRina Štrajn ,Jacobs University Bremen, 28759 Bremen, GermanyWe analyze bicovariant di ff erential calculus on κ -Minkowski spacetime. It is shown that correspondingLorentz generators and noncommutative coordinates compatible with bicovariant calculus cannot be real-ized in terms of commutative coordinates and momenta. Furthermore, κ -Minkowski space and NC formsare constructed by twist related to a bicrossproduct basis. It is pointed out that the consistency condition isnot satisfied. We present the construction of κ -deformed coordinates and forms (super-Heisenberg algebra)using extended twist. It is compatible with bicovariant di ff erential calculus with κ -deformed igl (4)-Hopfalgebra. The extended twist leading to κ -Poincaré-Hopf algebra is also discussed. Keywords: noncommutative space, κ -Minkowski spacetime, di ff erential forms, super-Heisenberg alge-bra, realizations, twist. I. INTRODUCTION
The structure of spacetime at very high energies (Planck scale lengths) is still unknown and it is believedthat, at these energies, gravity e ff ects become significant and we need to abandon the notion of smooth andcontinuous spacetime. Among many attempts to find a suitable model for unifying quantum field theoryand gravity, one of the ideas that emerged is that of noncommutative spaces [1]-[5]. Authors inclined to thisidea have followed di ff erent approaches and considered di ff erent types of noncommutative (NC) spaces,where the concept of invoking twisted Poincaré symmetry of the algebra of functions on a Minkowskispacetime using twist operator is the most elaborated one [6]. In formulating field theories on NC spaces,di ff erential calculus plays an essential role. The requirement that this di ff erential calculus is bicovariant andalso covariant under the expected group of symmetries leads to some problems. e-mail:e-mail: [email protected] e-mail: [email protected] e-mail: [email protected] κ -Minkowski space [7]-[48]. This space is a Lie algebra type of deformation of the Minkowski spacetimeand here the deformation parameter κ is usually interpreted as the Planck mass or the quantum gravityscale. κ -Minkowski space is also related to doubly special relativity [28]-[31]. For each NC space there is acorresponding symmetry algebra. In the case of κ -Minkowski space, the symmetry algebra is a deformationof the Poincaré algebra, known as the κ -Poincaré algebra. The κ -Poincaré algebra is also an example of aHopf algebra. Some of the results of pursuing this line of research are, e.g., the construction of quantumfield theories [32]-[39], electrodynamics [40]-[42], considerations of quantum gravity e ff ects [43]-[45] andthe modification of particle statistics [46]-[48] on κ -Minkowski space.Regarding the problem of di ff erential calculus on κ -Minkowski space, Sitarz has shown [11] that in or-der to obtain bicovariant di ff erential calculus, which is also Lorentz covariant, one has to introduce an extracotangent direction. While Sitarz considered 3 + n dimensions in Ref. [12]. Another attempt to dealwith this issue was made in [20] by the Abelian twist deformation of U[ igl (4 , R )]. Bu et al. in [21] extendedthe Poincaré algebra with the dilatation operator and constructed a four dimensional di ff erential algebraon the κ -Minkowski space using a Jordanian twist of the Weyl algebra. Di ff erential algebras of classicaldimensions were also constructed in [18] and [19], from the action of a deformed exterior derivative.In [26] the authors have constructed two families of di ff erential algebras of classical dimensions on the κ -Minkowski space, using realizations of the generators as formal power series in a Weyl superalgebra. Inthis approach, the realization of the Lorentz algebra is also modified, with the addition of Grassmann-typevariables. As a consequence, generators of the Lorentz algebra act covariantly on one-forms, without theneed to introduce an extra cotangent direction. The action is also covariant if restricted to the κ -Minkowskispace. However, one loses Lorentz covariance when considering forms of order higher than one.Our motivation in this letter is to unify κ -Minkowski spacetime, κ -Poincare algebra and di ff erentialforms. We embed them into κ -deformed super-Heisenberg algebra related to bicrossproduct basis. Usingextended twist, we construct a smooth mapping between κ -deformed super-Heisenberg algebra and super-Heisenberg algebra. We present an extended realization for κ -deformed coordinates, Lorentz generators,and exterior derivative compatible with Lorentz covariance condition.In section II, super-Heisenberg algebra is described. In section III, realization of κ -Minkowski spaceand κ -Poincare algebra related to bicrossproduct basis is given. In section IV, bicovariant di ff erential cal-culus is analyzed. It is pointed out that there does not exist a realization of Lorentz generators and NCcoordinates compatible with bicovariant calculus in terms of commutative coordinates and momenta. Insection V, κ -Minkowski space and NC forms are constructed by twist related to bicrossproduct basis. It is2hown that the consistency condition is not satisfied. In section VI, we present our main construction of κ -deformed super-Heisenberg algebra using extended twist. Extended realizations for Lorentz generators andexterior derivative invariant under igl (4)-Hopf algebra are presented. Finally, in section VII we outline theconstruction of Lorentz generators, exterior derivative and one-forms for bicovariant calculus compatiblewith κ -Poincaré-Hopf algebra. II. SUPER-HEISENBERG ALGEBRA
In the undeformed case we consider spacetime coordinates x µ , derivatives ∂ µ ≡ ∂∂ x µ , one forms d x µ ≡ ξ µ ,and Grassmann derivatives q µ ≡ ∂∂ξ µ satisfying the following (anti)commutation relations:[ x µ , x ν ] = [ ∂ µ , ∂ ν ] = , [ ∂ µ , x ν ] = η µν , { ξ µ , ξ ν } = { q µ , q ν } = , { ξ µ , q ν } = η µν , [ x µ , ξ ν ] = [ x µ , q ν ] = [ ∂ µ , ξ ν ] = [ ∂ µ , q ν ] = , (1)where µ = { , , , } and η µν = diag( − , , , SH ( x , ∂, ξ, q ) i.e. superphase space. The exterior derivative is defined as d = ξ α ∂ α , so ξ µ = [d , x µ ].We define the action ⊲ : SH ( x , ∂, ξ, q )
7→ SA ( x , ξ ), where SA ( x , ξ ) ⊂ SH ( x , ∂, ξ, q ). The super-Heisenberg algebra SH ( x , ∂, ξ, q ) can be written as SH = SA ST , where ST ( ∂, q ) ⊂ SH ( x , ∂, ξ, q ). Forany element f ( x , ξ ) ∈ SA ( x , ξ ) we have x µ ⊲ f ( x , ξ ) = x µ f ( x , ξ ) , ξ µ ⊲ f ( x , ξ ) = ξ µ f ( x , ξ ) ,∂ µ ⊲ f ( x , ξ ) = ∂ f ∂ x µ , q µ ⊲ f ( x , ξ ) = ∂ f ∂ξ µ . (2)The coalgebra structure of ST ( ∂, q ) is defined by undeformed coproducts: ∆ ∂ µ = ∂ µ ⊗ + ⊗ ∂ µ , ∆ q µ = q µ ⊗ + ( − ) deg ⊗ q µ , deg = ξ α q α (mod2) . (3)The coalgebra structure with antipode and counit is (undeformed) super-Hopf algebra. Let us mention thatsuper-Heisenberg algebra SH has also super-Hopf-algebroid structure, which will be elaborated separately.The Hopf-algebroid structure of Heisenberg algebra was discussed in [52] .Now we introduce Lorentz generators M µν :[ M µν , M λρ ] = η νλ M µρ − η µλ M νρ − η νρ M µλ + η µρ M νλ , (4) for Hopf-algebroid structure also see [49], [50] and [27] ∆ M µν = M µν ⊗ + ⊗ M µν (5)and action ⊲ : M µν ⊲ x λ = η νλ x µ − η µλ x ν , M µν ⊲ ξ λ = η νλ ξ µ − η µλ ξ ν . M µν ⊲ = M µν , x λ ] = η νλ x µ − η µλ x ν , [ M µν , ξ λ ] = η νλ ξ µ − η µλ ξ ν , (7)so that x µ and ξ µ transform as vectors (the same holds for ∂ µ and q µ ).The Lorentz covariance condition M µν ⊲ f ( x , ξ ) g ( x , ξ ) = m (cid:0) ∆ M µν ⊲ f ⊗ g (cid:1) M µν ⊲ d f ( x , ξ ) = d( M µν ⊲ f ( x , ξ )) (8)(where m is the multiplication map) implies [ M µν , d] = , (9)where d f ( x , ξ ) = d ⊲ f ( x , ξ ) = [d , f ( x , ξ )] ⊲ ⊲ =
0. Note that the action M µν ⊲ f ( x , ξ ) in Eq. (8) iscompatible with (6), (7) and M µν ⊲ f ( x , ξ ) g ( x , ξ ) = M µν f ( x , ξ ) g ( x , ξ ) ⊲ M µν in SH ( x , ∂, ξ, q ) is M µν = x µ ∂ ν − x ν ∂ µ + ξ µ q ν − ξ ν q µ (11)and now it is easy to verify Eqs.(4) - (7). Note that the Lorentz generators without the Grassmann-part( ξ µ q ν − ξ ν q µ ) cannot satisfy the condition (9). Usually in di ff erential geometry vector field v = v µ ∂ µ actson a one-form ξ β = d x β as a Lie derivative L v ξ β = d L v x β = dv β . In our approach the action through Liederivative is equivalent to the action of (v µ ∂ µ + dv µ q µ ) ⊲ d x β = dv β and [v µ ∂ µ + dv µ q µ , d] =
0. Our approachis more suitable for studying NC case (see section VI.).Hidden supersymmetry proposed in [53] could be interpreted as having origin in additional vector-like Grassmann coordinates. The action of superspace realization of Lorentz generators (11) on physicalsuperfields and possible physical consequences are still under consideration and will be presented elsewhere.4
II. κ -MINKOWSKI SPACE IN BICROSSPRODUCT BASIS In κ -Minkowski space with deformed coordinates { ˆ x µ } we have[ ˆ x i , ˆ x j ] = , [ ˆ x , ˆ x i ] = ia ˆ x i , (12)where a is the deformation parameter. The deformed coproducts ∆ for momentum generators p µ andLorentz generators ˆ M µν in bicrossproduct basis [10] are ∆ p = p ⊗ + ⊗ p , ∆ p i = p µ ⊗ + e a p ⊗ p i , ∆ ˆ M i = ˆ M i ⊗ + e a p ⊗ ˆ M i − a p j ⊗ ˆ M i j , ∆ ˆ M i j = ˆ M i j ⊗ + ⊗ ˆ M i j . (13)The algebra generated by ˆ M µν and p µ is called κ -Poincaré algebra where ˆ M µν generate undeformed Lorentzalgebra, p µ satisfy [ p µ , p ν ] = M µν , p λ ] are given in [10]. Equations in(13) describe the coalgebra structure of the κ -Poincaré algebra and together with antipode and counit makethe κ -Poincaré-Hopf algebra. We have the action (for more details see [24]) ◮ : ˆ H ( ˆ x , p ) ˆ A ( ˆ x ), whereˆ H ( ˆ x , p ) is the algebra generated by ˆ x µ and p µ and ˆ A ( ˆ x ) is a subalgebra of ˆ H ( ˆ x , p ) generated by ˆ x µ :ˆ x µ ◮ ˆ g ( ˆ x ) = ˆ x µ ˆ g ( ˆ x ) , p µ ◮ = , ˆ M µν ◮ = p µ ◮ ˆ x ν = − i η µν , ˆ M µν ◮ ˆ x λ = η νλ ˆ x µ − η µλ ˆ x ν . (14)Namely, using coproducts (13) and action (14) one can extract the following commutation relations betweenˆ M µν , p µ , and ˆ x µ : [ p , ˆ x µ ] = − i η µ , [ p k , ˆ x µ ] = − i η k µ + ia µ p k , (15)[ ˆ M µν , ˆ x λ ] = η νλ ˆ x µ − η µλ ˆ x ν − ia µ ˆ M νλ + ia ν ˆ M µλ . (16)The realization corresponding to bicrossproduct basis for ˆ M µν , p µ , and ˆ x µ in terms of undeformed x µ and ∂ µ is : ˆ x ( o ) i = x i , ˆ x ( o )0 = x + ia x k ∂ k , p µ = − i ∂ µ ˆ M ( o ) i = x i − Zia + ia ∂ k − ia sh (cid:0) A (cid:1) Z ! − (cid:0) x + ia x k ∂ k (cid:1) ∂ i , ˆ M ( o ) i j = x i ∂ j − x j ∂ i , (17)where A = − ia ∂ and Z = e A (for more details see [16] and [19]). Greek indices ( µ, ν, ... ) are from 0 to 3, and Latin indices ( i , j , ... ) from 1 to 3. Summation over repeated indices is assumed. Here the superscript ( o ) denotes that the Lorentz generators and NC coordinates ˆ x are realized only in terms of undeformed x µ and ∂ µ . V. BICOVARIANT DIFFERENTIAL CALCULUS
In the paper by Sitarz [11] there is a construction of a bicovariant di ff erential calculus [51] on κ -Minkowski space compatible with Lorentz covariance condition (20), but with an extra one-form φ , whichtransforms as a singlet under the Lorentz generators. The algebra generated by ˆ x µ and one-forms ˆ ξ µ , φ isclosed in one-forms.The deformed exterior derivative is defined by ˆd =
0, [ˆd , ˆ x µ ] = ˆ ξ µ and satisfies ordinary Leibniz rule.In the bicovariant calculus it is also assumed that the coproduct for Lorentz generator ˆ M µν and momentumgenerator p µ is in the bicrossproduct basis (13), the action ◮ is defined in (14) and it is extended to one-forms by ˆ ξ µ ◮ = ˆ ξ µ , φ ◮ = φ p µ ◮ ˆ ξ ν = p µ ◮ φ = ˆ M µν ◮ φ = , ˆ M µν ◮ ˆ ξ λ = η νλ ˆ ξ µ − η µλ ˆ ξ ν . (18)From coproducts (13) and eq.(18) we can find commutation relations [ ˆ M µν , ˆ ξ λ ] and [ p µ , ˆ ξ ν ]. In addition toEqs. (15) and (16) we have [ p µ , ˆ ξ ν ] = [ p µ , φ ] = [ ˆ M µν , φ ] = , [ ˆ M µν , ˆ ξ λ ] = η νλ ˆ ξ µ − η µλ ˆ ξ ν . (19)The Lorentz covariance condition ˆ M µν ◮ ˆ f ( ˆ x , ˆ ξ )ˆ g ( ˆ x , ˆ ξ ) = m (cid:0) ∆ ˆ M µν ◮ ˆ f ⊗ ˆ g (cid:1) ˆ M µν ◮ ˆd ˆ f = ˆd( ˆ M µν ◮ ˆ f ) (20)implies [ ˆ M µν , ˆd] = , (21)where ˆd ˆ f = ˆd ◮ ˆ f = [ˆd , ˆ f ] ◮ ◮ = between one-forms ˆ ξ µ , φ and NC coordinate ˆ x µ that is compatible with(20) - (21) is given by [ ˆ x µ , φ ] = ˆ ξ µ , [ ˆ x , ˆ ξ ] = − a φ, [ ˆ x i , ˆ ξ j ] = − ia δ i j (cid:0) ˆ ξ + ia φ (cid:1) , [ ˆ x , ˆ ξ i ] = , [ ˆ x i , ˆ ξ ] = − ia ˆ ξ i . (22) Sitarz denotes this action with ⊲ . The correspondence between algebra in [11] and (22) is κ = − ia , x µ = ˆ x µ , d x µ = ˆ ξ µ , φ = φ , N i = ˆ M i , and M i = ǫ ijk ˆ M jk . x µ is given in (17) and the realization for one-forms ˆ ξ µ and φ that satisfies (22) canbe given in terms of undeformed x µ , ∂ µ and ξ µ . The realizations for one-forms and exterior derivative areˆ ξ = ξ (cid:16) + a (cid:3) (cid:17) + ia ξ k ∂ k , ˆ ξ k = ξ k − ia ξ ∂ k Z − ,φ = − ˆd s = ξ (cid:16) Z − − ia + ia (cid:3) (cid:17) − ξ k ∂ k , (cid:3) = ∂ i Z − − a sh (cid:0) A (cid:1) , (23)where we have denoted the exterior derivative for Sitarz’s case with ˆd s .Relations (22), (23) and (2) imply φ = − ˆd s and φ ⊲ =
0, so that the algebra ˆ SA is not isomorphic to SA ⋆ . Also the problem with this construction is that the Lorentz generators ˆ M µν cannot be realized in termsof x µ , ∂ µ , ξ µ and q µ in order to satisfy Lorentz covariance condition (20) which implies (21). Namely, ifwe take the realization (23) for ˆd s and just want to find realization for ˆ M µν so that [ ˆ M µν , ˆd s ] = M µν do not satisfy the Lorentz algebra (4).Furthermore, in [26] di ff erential algebras D and D of classical dimension were constructed (avoidingthe extra form φ ), where all conditions were satisfied, except (16) and ˆ M µν does not commute with exteriorderivative. All these arguments lead to a conclusion that for the fixed realization (17) for ˆ x µ , there is norealization for ˆ M µν that satisfies κ -Poincaré-Hopf algebra (13) and Lorentz covariance condition (20), (21). V. κ -MINKOWSKI SPACETIME FROM TWIST AND NC ONE-FORMS In this section we will construct the noncommutative coordinates ˆ x µ , coproducts, and NC one-formsusing the twist operator. A. κ -Minkowski spacetime from twist We start with an Abelian twist (see [47], [20], [36] and [22]) F = exp (cid:0) − A ⊗ x k ∂ k (cid:1) , (24)where A = ia ∂ = − ia ∂ . The bidi ff erential operator (24) satisfies all the properties of a twist (2-cocyclecondition and normalization) and leads to noncommutative coordinatesˆ x µ = m (cid:16) F − ⊲ ( x µ ⊗ id ) (cid:17) . (25) The algebra of undeformed operators is defined in Section II. and for ◮ action we have x µ ◮ = ˆ x µ , ξ µ ◮ = ˆ ξ µ , ∂ µ ◮ = q µ ◮ = For more details see [26] The algebra ˆ SA is generated by ˆ x µ , ˆ ξ µ and φ , and the algebra SA ⋆ is generated by x µ and ξ µ but with ⋆ -multiplication. Thestar-product is defined by f ( x , ξ ) ⋆ g ( x , ξ ) = ˆ f ( ˆ x , ˆ ξ )ˆ g ( ˆ x , ˆ ξ ) ⊲
7t follows that for this twist we get a realization for ˆ x µ exactly as in (17). The twist given by Eq. (24) alsoleads to an associative star product f ( x ) ⋆ g ( x ) = m (cid:16) F − ⊲ ( f ⊗ g ) (cid:17) . (26)If we define the operators M µν as M µν = x µ ∂ ν − x ν ∂ µ , then M µν generate the undeformed Lorentz algebra,but their coproducts, obtained from the twist (24) do not close in the Poincaré-Hopf algebra. For this reasonwe consider the algebra igl (4), generated by ∂ µ and L µν = x µ ∂ ν , which also has a Hopf algebra structure[25]. The coproducts of ∂ µ and L µν calculated as ∆ ∂ µ = F ∆ ∂ µ F − , and analogously for L µν , are ∆ ∂ = ∆ ∂ , ∆ ∂ i = ∂ i ⊗ + e A ⊗ ∂ i (27) ∆ L i j = ∆ L i j , ∆ L = ∆ L + A ⊗ L kk (28) ∆ L i = L i ⊗ + e − A ⊗ L i , ∆ L i = L i ⊗ + e A ⊗ L i − ia ∂ i ⊗ L kk . (29)Coproducts of the momenta p µ , obtained from (27) by expressing p µ in terms of ∂ µ ( p µ = − i ∂ µ ), coincidewith the coproducts of momenta in the bicrossproduct basis (13) (see also [10]). On the other hand, co-products of the Lorentz generators M µν , calculated from Eqs. (28) and (29) as ∆ M µν = ∆ L µν − ∆ L νµ , aredi ff erent from the ones in the bicrossproduct basis (13) (more precisely, ∆ M i is di ff erent), [25]. In this waywe have constructed the igl (4)-Hopf algebra structure using twist F .We point out that in [27, 52] it is shown that for Lorentz generators in bicrossproduct basis (17) the twist F gives the correct Hopf algebra structure (13) of κ -Poincaré algebra generated by ˆ M µν and p µ . B. Noncommutative one-forms from twist
Our aim is to construct an exterior derivative ˆd and noncommutative one-forms ˆ ξ µ with the followingproperties: ˆd = , [ˆd , ˆ x µ ] = ˆ ξ µ (30) { ˆ ξ µ , ˆ ξ ν } = , [ ˆ ξ µ , ˆ x ν ] = K λµν ˆ ξ λ , K λµν ∈ C (31)[ ˆ ξ µ , ˆ x ν ] − [ ˆ ξ ν , ˆ x µ ] = ia µ ˆ ξ ν − ia ν ˆ ξ µ (consistency condition) , (32)where we have introduced a µ = ( a , ~
0) so that (12) can be written in a unified way as[ ˆ x µ , ˆ x ν ] = i ( a µ ˆ x ν − a ν ˆ x µ ) . (33)We want to find a realization of NC one-forms in terms of the undeformed algebra SH ( x , ∂, ξ, q ). If wenow calculate ˆ ξ µ , by analogy to Eq. (25), as ˆ ξ µ = m (cid:16) F − ⊲ ( ξ µ ⊗ id ) (cid:17) , we get ˆ ξ µ = ξ µ , so that the LHS of832) equals 0, while the RHS gives ia µ ˆ ξ ν − ia ν ˆ ξ µ and the consistency condition is not fulfilled. Obviouslywe need to extend the twist defined in (24).We have shown that the bicovariant di ff erential calculus á la Sitarz [11] could not be realized in terms ofHeisenberg or super-Heisenberg algebra. In the next section we will propose a new version of bicovariantcalculus compatible with igl (4)-Hopf algebra. VI. EXTENDED TWIST
Our main goal is to construct a twist so that our bicovariant calculus satisfies the following properties:1. The bicovariant calculus has classical dimension, i.e. there is no extra form like φ .2. The algebra between ˆ ξ µ and ˆ x µ is closed in one-forms.3. Generators M µν satisfy the Lorentz algebra.4. The condition [ M µν , ˆd] =
0, which is su ffi cient condition for (20), (21).In order to satisfy all the requirements for ˆ ξ µ and ˆd we define the extended twist F ext = exp (cid:16) − A ⊗ ( x k ∂ k + ξ k q k ) (cid:17) . (34)This twist leads toˆ x i = m (cid:16) F − ext ⊲ ( x i ⊗ id ) (cid:17) = x i , ˆ x = m (cid:16) F − ext ⊲ ( x ⊗ id ) (cid:17) = x + ia ( x k ∂ k + ξ k q k ) (35)ˆ ξ µ = m (cid:16) F − ext ⊲ ( ξ µ ⊗ id ) (cid:17) = ξ µ . (36)Although the realization of ˆ x is changed with the addition of a term containing Grassmann variables, ˆ x µ still satisfy the same commutation relations Eq. (12), but the commutation relations between ˆ x µ and ˆ ξ µ areno longer all equal to 0 [ ˆ ξ µ , ˆ x i ] = , [ ˆ ξ , ˆ x ] = , [ ˆ ξ i , ˆ x ] = − ia ˆ ξ i . (37)Inserting (37) into (32) shows that in this case the consistency condition and the requirement from (31) aresatisfied. Note that ˆ x µ , ˆ ξ µ , ∂ µ and q µ generate the deformed super-Heisenberg algebra ˆ SH ,which also hassuper-Hopf-algebroid structure.We now want to introduce an exterior derivative ˆd such that (30) is also fulfilled and gives rise to thesame expression for ˆ ξ µ as (36). It is easily shown that this is achieved withˆd = ξ α ∂ α = d . (38)9or the exterior derivative d in (38), we wanted to constructed an operator ˆ M µν = ˆ M (0) µν + Grassmann partby extending the realization in SH with the property that it commutes with exterior derivative, i.e.[ ˆ M µν , d] =
0, but in doing so, we find that this operator does not satisfy the Lorentz algebra (4). Hence,exterior derivative (38) is not be compatible with κ -Poincaré-Hopf algebra in the bicrossproduct basis evenif we consider the realizations in SH .Since ˆ ξ µ are undeformed, their ⊲ and ◮ actions are the same as for ξ µ . Our construction can be extendedto forms of higher order in a natural way. E.g., the space of two-forms can be defined as the space generatedby ˆ ξ µ ∧ ˆ ξ ν . These two-forms then automatically satisfyˆ ξ µ ∧ ˆ ξ ν = ξ µ ∧ ξ ν = − ξ ν ∧ ξ µ = − ˆ ξ ν ∧ ˆ ξ µ . (39)Now we define the extended ⋆ -product with f ( x , ξ ) ⋆ g ( x , ξ ) = m (cid:16) F − ext ⊲ f ⊗ g (cid:17) . (40)For f ( x ) and g ( x ), functions of x only, (40) coincides with (26), and if f ( ξ ) and g ( ξ ) are functions of ξ only,their extended ⋆ -product is just the ordinary multiplication, f ( ξ ) ⋆ g ( ξ ) = f ( ξ ) g ( ξ ). As before (see (26)),the extended ⋆ -product can be equivalently defined with the ⊲ action: f ( x , ξ ) ⋆ g ( x , ξ ) = ˆ f ( ˆ x , ˆ ξ )ˆ g ( ˆ x , ˆ ξ ) ⊲ igl (4 , R ) covariant action, generators of gl (4) also need to be extended. L ext µν are defined by L ext µν = x µ ∂ ν + ξ µ q ν . It can be easily checked that L ext µν , defined in this way, still satisfy the igl (4) algebra, and furthermore, that they commute with d, [ L ext µν , d] =
0, so that L ext µν ◮ ˆ ξ λ = [ L ext µν , ˆ ξ λ ] ◮ = d[ L ext µν , ˆ x λ ] ◮ = d( L ext µν ◮ ˆ x λ ) = ˆ ξ µ η νλ . (41)The results for the coproducts, obtained from the extended twist, are ∆ q = ∆ q , ∆ q i = q i ⊗ + ( − ) deg e A ⊗ q i (42) ∆ L exti j = ∆ L exti j , ∆ L ext = ∆ L ext + A ⊗ L extkk (43) ∆ L exti = L exti ⊗ + e − A ⊗ L exti , ∆ L ext i = L ext i ⊗ + e A ⊗ L ext i − ia ∂ i ⊗ L extkk . (44)The coproducts of ∂ and ∂ i , calculated in the same way, are again given by Eq. (27).The action of L ext µν on the product of ˆ x ρ and ˆ ξ σ , calculated in three di ff erent ways:(i) L ext µν ◮ ˆ x ρ ˆ ξ σ = [ L ext µν , ˆ x ρ ] ◮ ˆ ξ σ + ˆ x ρ ◮ ([ L ext µν , ˆ ξ σ ] ◮ L ext µν ◮ ˆ x ρ ˆ ξ σ = (cid:0) L ext µν (1) ◮ ˆ x ρ (cid:1)(cid:0) L ext µν (2) ◮ ˆ ξ σ (cid:1) (iii) L ext µν ◮ ˆ x ρ ˆ ξ σ = (cid:0) L ext µν (1) ◮ ˆ x ρ (cid:1) d (cid:0) L ext µν (2) ◮ ˆ x σ (cid:1) ,gives the same result, i.e., the action is in accordance with bicovariant calculus. One can easily showthat neither the last equality, (iii), nor (41) would be satisfied had we used the ordinary definition of L µν L µν = x µ ∂ ν ). Similar expressions can be written in terms of the ⊲ action and the extended ⋆ product.Hence, we have constructed igl (4)-Hopf algebra structure using the extended twist F ext that satisfies all therequirements for bicovariant calculus (listed in the beginning of this section 1-4). Note that the coprodutsof M µν = L ext µν − L ext νµ calculated in this way di ff er from the one in bicrossproduct basis.The action (41) could also be obtained using the ordinary definition of L µν and promoting these genera-tors to Lie derivatives. Using Cartan’s identity, one would get L x µ ∂ ν ⊲ ξ λ = d( L x µ ∂ ν ⊲ x λ ) = d( x µ η νλ ) = ξ ν η νλ . (45)However, there is a problem in this approach. Namely, promoting L µν to Lie derivatives would again givethe realization (17) for ˆ x µ and in the case of deformed one-forms it would give ˆ ξ µ = ξ µ , i.e., the consistencycondition would not be fulfilled. VII. OUTLOOK AND DISCUSSION
We have shown that if the NC coordinates (12) are given only in terms of Heisenberg algebra H ( x , ∂ ),then there is no realization of Lorentz generators compatible with all the requirements of bicovariant calcu-lus [11]. Hence, if one wants to unify κ -Minkowski spacetime, κ -Poincaré algebra and di ff erential forms itis crucial to embed them into κ -deformed super-Heisenberg algebra ˆ SH ( ˆ x , ˆ ξ, ∂, q ). This is explicitly donefor κ -deformed igl (4)-Hopf algebra using the extended twist. We choose a bicrossproduct basis just as oneexample, but similar constructions for other bases are also possible.In section VI we have constructed a bicovariant di ff erential calculus compatible with κ -deformed igl (4)-Hopf algebra. The question is whether it is possible to construct bicovariant calculus compatible with κ -Poincaré-Hopf algebra. One has to develop the notion of superphase space and its super-Hopf-algebroidstructure. The idea is to generalize the method developed in [52] where we have analyzed the quantumphase space, its deformation and Hopf-algebroid structure. In [52] we have also constructed κ -Poincaré-Hopf algebra from twist.We present the realization of κ -Poincaré algebra compatible with bicovariant di ff erential calculus in thebicrossproduct basis. Starting with the extended realization of ˆ x µ ,ˆ x i = x i , ˆ x = x + ia ( x k ∂ k + ξ k q k ) (46)and demanding the Lorentz algebra (4), (13), and (16) we find the realization for the Lorentz generators:ˆ M i j = x i ∂ j − x j ∂ i + ξ i q j − ξ j q i ˆ M i = ˆ M (0) i + ξ i ( q Z + ia q k ∂ k ) − [ ξ q i + ia ( ξ k q k ∂ i + ξ k ∂ k q i )] (47)11here ˆ M (0) i is given in (17). The requirement that the Lorentz generators ˆ M µν commutes with exteriorderivatives ˜d, i.e. [˜d , ˆ M µν ] = = + a (cid:3) " ξ sh Aia + ia ∂ i Z − ! + ξ j ∂ j Z (48)The corresponding one-forms are defined as ˜ ξ µ = [˜d , ˆ x µ ]. We are working on generalizing the results in [52]in order to construct the extended twist operator within the superphase space which will provide the correct κ -Poincaré-Hopf algebra compatible with bicovariant calculus. All the details of this construction will bepresented elsewhere.Our main motivation for studying these problems is related to the fact that the general theory of relativitytogether with uncertainty principle leads to NC spacetime. In this setting, the notion of smooth spacetimegeometry and its symmetries are generalized using the Hopf algebraic approach. Further development ofthe approach presented in this paper will lead to possible application to the construction of NC quantumfield theories (especially electrodynamics and gauge theories), quantum gravity models, particle statistics,and modified dispersion relation. Acknowledgment
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