Lie-deformed quantum Minkowski spaces from twists: Hopf-algebraic versus Hopf-algebroid approach

We consider new Abelian twists of Poincare algebra describing non-symmetric generalization of the ones given in [1], which lead to the class of Lie-deformed quantum Minkowski spaces. We apply corresponding twist quantization in two ways: as generating quantum Poincare-Hopf algebra providing quantum Poincare symmetries, and by considering the quantization which provides Hopf algebroid describing the class of quantum relativistic phase spaces with built-in quantum Poincare covariance. If we assume that Lorentz generators are orbital i.e.do not describe spin degrees of freedom, one can embed the considered generalized phase spaces into the ones describing the quantum-deformed Heisenberg algebras.


Introduction
Due to quantum gravity (see [2]- [6]) as well as quantized strings effects (see e.g. [7,8]) at Planck distances the notion of classical space-time can not be maintained. The quantum-mechanical space-time localization in the presence of gravitational interactions are constrained by new type of bounds, extending Heisenberg uncertainty relations to the measurements of pairs of space-time coordinates (see e.g. [4]) 1 . Algebraically DSR uncertainty relations can be derived In such a way we obtain consistent classical action of Poincare algebra P on the Minkowski space M, describing the cross product P#M 4 . Important class of quantum Poincare algebras are described by twist quantizations of classical Poincare algebra, with all deformation located only in coalgebraic sector of Poincare-Hopf algebra H. In such a case the twist F ≡ F (1) ⊗F (2) ∈ U(g) ⊗ U(g) depends on classical Poincare generators, and the formula for coproducts can be calculated using only the classical Poincare algebra commutators. The quantum Minkowski space M( x µ ∈ M) in the case of twist quantization is again fully specified if it is given as the irreducible four-dimensional module of the respective twisted Poincare-Hopf algebra H F H F = (U(P), m, ∆ F , S F , ǫ), where S F denotes twisted antipode (coinverse) and ǫ is a counit. The standard Hopf-algebraic way of introducing the quantum Minkowski coordinates x µ is to consider quantum Poincare group and use the Hopf-algebraic duality between the coproducts of p µ and the coordinates x µ and introduce the notion of Heisenberg double [10,18]. However, the twist F provides directly the formula linking x µ ∈ M with x µ ∈ M, by means of the relation which is a special case of derived in Sect.2 star-product relation (34) with the action ⊲ in (6) provided by the formulae (1), (3). In such a way one can express the noncommutative x µ in terms of classical phase space coordinates (x µ , p µ ) and generators M µν . In this paper we plan to consider the generalization of twist considered in [1], with arbitrary symmetry of log F as tensor product. In such a way we introduce additional real parameter u which for u = 1 2 leads to antisymmetric log F tensor and antisymmetric classical r-matrix (this was the case considered in [1] and [19]); other cases u = 0 and u = 1 correspond to maximally nonsymmetric twists, which for u = 0 was discussed as well in the literature [20].
The aim of our paper is to describe the twist quantizations in two frameworks: first is based entirely on Hopf-algebraic techniques, which provides quantum Poincare-Hopf algebra H F and second, which leads for any value of parameter u to the embedding of H F into the Hopf algebroids describing suitably deformed smash product T #M. New results in the present paper are provided by the second method by providing the Hopf algebroid structure: construction of source, target and antipode maps (for their definition see [15], [16]) and by 4 The semidirect product of two Lie algebras is again a Lie algebra, what is generalized by the notion of smash product, which describes the algebraic structure on the vector space H ⊕ V , where H = (M H , ∆ H , ǫ H , 1 H ) is (unital and counital) bialgebra (in particular Lie bialgebra) and V is a unital H-module algebra, which may be noncommutative. Cross-product algebra can be endowed with Hopf algebroid structure [11], [12].
determining the coproduct freedom for twisted bialgebroids H F = (P#M) F (so-called coproduct gauges, see [21]). We add that the Hopf algebroid techniques were extensively studied in mathematics (see e.g. [22], [11]- [16], [23]) and applied to the description of quantum-deformed relativistic phase spaces by some of the present authors [24]- [28], [21]. The novelty of our discussion of Hopf algebroid structure in comparison with our earlier efforts [24]- [28] is to consider Lorentz generators M µν as independent -we shall not assume the standard orbital phase space realization of M µν If relation (7) is valid, the Hopf-algebraic Poincare algebra twist F as well as noncommutative Minkowski space coordinates x µ can be expressed in terms of phase space variables i.e. the Hopf-algebraic formulae are realized in terms of canonical Heisenberg algebra, which provides a classical example of Hopf algebroid. The noncommutative Minkowski coordinates x µ can be expressed as the following functions of classical phase space variables (x µ , p µ ) We mention that the formula (8) was considered [20], [29]- [30] as well for other Lie-algebraic quantum deformations of Poincare algebra, not necessarily described by twist quantization 5 (see [28,14]).
2 Twist-deformed Poincaré Hopf algebra and quantum Minkowski spaces Poincaré algebra P, generated by Lorentz generators M µν and momentum generators p µ is defined by where η µν = diag(−1, 1, ..., 1). Classical Poincaré Hopf algebra is defined by the universal enveloping algebra U(P) of the Poincaré algebra P, together with the coproduct ∆ 0 , antipode S 0 and counit ǫ 0 , given by Twist F is an invertible element of U(P) ⊗ U(P), satisfying the cocycle condition and the normalization condition Twists deform coproducts (12) and antipodes (13) of h ∈ U(P) as follows: where χ F = m[(S 0 ⊗ 1)F ] and the deformed coproduct ∆ F is coassociative due to the cocycle condition (15).
Here we consider the following families of Abelian twists, for all dimensions n ≥ 3: where a · p = a µ η µν p ν . The parameter u ∈ [0, 1], κ is the deformation parameter with dimension of mass, a 2 ∈ {−1, 0, 1}, and the following conditions hold: These twists are a generalization of the twist proposed in [1] -being Abelian, they automatically satisfy the cocycle condition. Because under flip transfor- Using equation (17) for the family of twists (19) we obtain the following deformed coproducts (F ≡ F u ) where K µν is given in equation (20).
Corresponding antipodes (18) are: The counit is trivial: It is interesting to note that coproduct and antipode of K µν remain classical, i.e.
Coproduct and antipode of (e K ) µ ν are If noncommutativity is introduced through Hopf-algebraic twist quantization one can introduce the star product realization of the algebraÂ of functions on quantum Minkowski space in terms of ⋆-algebra of classical functions f (x), g(x) where the action ⊲ is defined by eq. (1), (3) and (Â, ·) algebra is represented as (A, ⋆) algebra. Alternatively, one can write equation (32) in the following form wheref is a noncommutative counterpart of f (x), described as the functions of classical Following (6), if we put in (34) f = x µ and F = F u the non-commutative coordinates are given bŷ The non-commutative coordinates (35) close to a Lie algebra Note that the structure constants C µν α = 1 κ (a µ θ ν α − a ν θ µ α ) do not depend on the parameter u and satisfy Jacobi identities.
If u = 1 the twist F 1 is special, because only such value of u gives the particular choice described by formula (8) for any choice of the Lorentz generators M αβ . If u = 0, eq. (35) reduces tô This case was considered in [20] and it was related to twisted statistics. For u = 1/2, twist F τ = F −1 and because F † = F τ (where † denotes Hermitian conjugation), for such a choice of u the twist F 1 2 is unitary. This case was considered in [1], [19] . In [19] it was related to non-Pauli effects in noncommutative spacetimes.
In order to obtain the coproduct sector and consistent bialgebroid relations forx µ defined by (35) we should look for the Hopf algebroid structure of twistdeformed cross product (P#M) F .

Deformed Heisenberg algebras and twisted cross products in algebroid approach
Quantum-mechanical phase-space coordinates x µ and momenta p µ describe canonical undeformed Heisenberg algebra, given by: If we deal with Hopf-algebraic scheme of Poincare symmetries the relations (39) can be derived by the identification of x µ ∈ M with Abelian space-time translations of classical Poincare group and p µ ∈ T with the generators of dual Abelian fourmomenta subalgebra acting on M. The standard quantummechanical phase-space with basic algebra (39) is provided by smash product H 0 = T #M, defining Heisenberg double with undeformed (canonical) Heisenberg Hopf algebroid structure 6 of two Abelian dual Hopf algebras which describe respectively the functions of coordinates x µ and momenta p µ . The cross multiplication rules in H 0 are given by the Heisenberg double formula [10], [18] where ·, · describes the canonical duality pairing, with ∆ 0 (p µ ) = p given by (12) and The relation (41) follows as well from the coproduct of classical Poincare group describing space-time translations, after contraction of the Lorentz group parameters Λ µ ν −→ δ µ ν . The action p µ ⊲ x ν , given by formula (3), in Hopf-algebraic scheme can be identified with the binary duality map T ⊗ M −→ C : p ⊗ x → p, x , which provides the differential realization of fourmomenta p µ Finally, one can easily deduce from (40-41) and (3) the set of canonical commutation relations given by eq. (39).
In H 0 one can choose different bases, in particular it is possible to incorporate the change x µ −→ x µ = x ρ ϕ ρ µ (p) (see (8)) and employ the fourmomenta p µ , satisfying the relations (9) with twisted coproduct (23) of p µ denoted as follows As follows from (35) the deformed Heisenberg algebra basis ( x µ , p µ ) satisfies the standard duality relations p µ , x ν = −iη µν . Further one can show that the algebraic relation (36) are dual to the coproducts (23) in accordance with Hopf-algebraic duality, e.g.
Subsequently, introducing the coproduct which is dual to commuting fourmomenta p µ , one can show that we deal with Heisenberg double H with the basis ( x µ , p µ ) describing deformed Heisenberg algebra. Therefore, the basic relation (40) remains valid, i.e.
We get from decomposition (43) that the term ∆ 0 (p ν ) gives the contribution η µν + x ν p µ , and relation (46) takes the form of deformed canonical commutation relations where due to the duality of coordinates x ν and momenta p µ , we use the formula The formula (47) can be also written in the form which was derived in alternative way also in [14]. If we use the formulae (35), (2) and (39) we can directly calculate the commutator (49). Such a method leads to the same deformed Heisenberg algebrâ H, given by relations (36), i.e. the cross commutator [p µ , x ν ] which for any u does not depend on Lorentz generators M µν . We obtain [p µ , p ν ] = 0.
Additionally, commutation relations between (e K ) µ ν andx λ and M αβ after using (1)-(3) are given by and the commutation relations between Lorentz generators M µν and non-commutative coordinatesx ρ are the following form The relation (53) in generalized quantum-deformed phase space (x µ , p µ , S µν ) where describe the noncommutativity of translational and spin degrees of freedom. Let us consider now the Hopf algebroid H F with algebraic structure described by classical Poincare algebra P supplemented by the noncommutative space-time coordinatesx µ ∈M satisfying the relations (36), (50) and (53) The total algebra A with the basis (x µ , p µ , M µν ) is given by the smash product U(P)#U(M) and base algebra B F (x µ ∈ B F ) is provided by the algebra of functions onM with the multiplication in B F represented by star product formula (32). The source map s F : B F → A (algebra homomorphism) and target map t F : B F → A (algebra antihomomorphism) introduce in A the (B F , B F ) bimodule structure, namely for any a ∈ A and b, b ′ ∈ B F one gets the formula bab ′ = s F (b)t F (b ′ )a, i.e. we consider H F as left bialgebroid [13], [11].The comultiplication map∆ F : A → A⊗ BF A with nonstandard tensor product introduced firstly in [22] is a coassociative bimodule map with the elements a⊗ BF a ′ ∈ A⊗ BF A defined in the description using standard tensor product A ⊗ A by the equivalence class generated by the following condition [11] m(I F (b⊗b ′ )) = 0, where A⊗ BF A = (A ⊗ A) I F and s F and t F are respectively the twisted source and target maps defined below (see (61)-(62)). If we describe coproducts 7 ∆ F using standard tensor products one can treat the elements (a⊗a ′ ) satisfying (56) as defining coproduct gauges, with gauge-invariant elements described by a⊗ BF a ′ . In particular forx µ ∈ B F we shall choose the special coproduct gauge defined by the formula (see e.g. [11], [21]) The canonical choice of the coproduct given by the formula (57) can be obtained if we insert the twisted coproducts (23)- (24) and into the relation (35), in accordance with the equality One can check further that the coproducts (57) and (23) The twisted source and target maps are introduced as follows [11], [15], [16] Due to the model-independent relation (see e.g. [11], proof of preposition (2.4)) it follows that the formulae (61) and (57) are consistent as expected. Further, it can be shown that the source and target maps satisfy the relations (C µν The coproduct∆ F : A → A ⊗ A is only coassociative when A ⊗ A is projected into equivalence classes A⊗ BF A generated by the ideal I F . The choice of representatives in the equivalence class defines the coproduct gauge. The simplest choice of the coproduct gauge transformation is obtained by adding to (57) the ideal I F multiplied by a constant α One can check that the coproducts∆ (α) ( x µ ) together with the Hopf-algebraic coproducts (23)-(24) satisfy the algebraic relations which are homomorphic to the relations (36), (50) and (53). The coproduct gauge can be generalized by introducing powers of ideal I F as well as powers of coproducts (60). In such a case the homomorphism between the algebraic and coalgebraic relations of Hopf algebroid H F will be only valid in standard tensor notation modulo the choices of coproduct gauge transformation [28], [21].
Finally we complete the description of Hopf bialgebroid H F structure if we define (ǫ 0 (x µ ) = x µ for undeformed case) (see e.g. [28]) Hopf algebroid is a bialgebroid with antipode (coinverse). In order to obtain the antipode S F one can use the formula (a ∈ A) (see (18)) with Note that S 2 F = 1. The antipodes S F (p µ ), S F (M µν ) in Hopf algebroid (55) remain the same as for the twisted Poincare -Hopf algebra (see (25)- (26)) and are also involutive (see (18)). Further it can be shown that Note that in second formula the introduction of anchor projection γ (∆ F → γ∆ F , where γ is a section of the projection A ⊗ A → A⊗ BF A, see [11]- [12]) is not needed 8 . Finally we recall that for spinless systems one can introduce the orbital realization of Lorentz generators M µν , described by formula (7). Inserting (7) in formula (19) one gets the u-dependent twist of canonical Heisenberg-Hopf with the coproducts∆F of x µ and p ν obtained from (23) and (57) after inserting the substitution (75). Further, it follows that the two-cocycle condition of F (see [10]) is becoming a two-cocycle condition for bialgebroid twist F (see 9 ), given by (75). It is easy to see that after inserting (7) in relation (35) the formula (8) becomes valid for all values of u. Concluding, from the Hopf algebroid (55) with independent Lorentz generators by using (7) one obtains twisted Heisenberg-Hopf algebroid with the formulae for source and target maps, antipodes and the ideal describing coproduct gauges expressed only in terms of phase space variables (x µ , p µ ) or (x µ , p µ ) 10 .

Final Remarks
The cross product algebra P#M, with the algebra basis described by generators (p µ , M µν ,x µ ), can be named Poincare-Heisenberg algebra [34] or DSR algebra [35], [36] 11 . In this paper we provide a particular example of quantum twist-deformed DSR algebra P#M and present explicitly its algebraic and coalgebraic Hopf algebroid structure. It should be observed that DSR algebra can be obtained by the contraction of full generalized relativistic quantum phase space described as the Heisenberg double (see e.g. [21]), i.e. the cross product H#H of deformed Poincare-Hopf algebra H (with basis (p µ , M µν )) and quantum Poincare-Hopf quantum groupH (with basis (x µ , Λ µν ), where Λ µν (Λ µα Λ α ν = η µν ) describe the Lorentz 4 × 4 matrix group elements). Because D = 4 Heisenberg algebra is described as well by the cross-product T 4 #M 4 , one can represent D = 4 DSR algebraic structure as the following composition of cross products DSR algebra = SO(3, 1)#(T 4 #M 4 ).
It follows from (76) that the Lorentz generators SO(3, 1) act covariantly on the standard (without spin degrees of freedom) quantum phase space T 4 #M 4 . We add that the cross-product structures presented in (76) are preserved for twist quantum-deformed phase space (T 4 #M 4 ) F endowed with quantum-relativistic covariance under the action of twisted Poincare -Hopf algebra There remain some questions which should be further studied. In particular one should elaborate more on the role of Heisenberg algebra twists (see e.g. (75)), in the construction of Hopf algebroids which provide the quantum deformed relativistic phase space frameworks. In this paper the advantage of our approach to quantum phase space formulation is the appearance of spin degrees of freedom S µν as independent phase space coordinates. The extension of quantum phase spaces with spin degrees of freedom still remains quite open subject, and we plan to study the relation of such extended phase spaces (see e.g. [37], [38] in undeformed case) with the Hopf algebroid constructions.