κ-Deformed Phase Space, Hopf Algebroid and Twisting

Hopf algebroid structures on the Weyl algebra (phase space) are presented. We define the coproduct for the Weyl generators from Leibniz rule. The codomain of the coproduct is modified in order to obtain an algebra structure. We use the dual base to construct the target map and antipode. The notion o...

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Опубліковано в: :	Symmetry, Integrability and Geometry: Methods and Applications
Дата:	2014
Автори:	Jurić, T., Kovačević, D., Meljanac, S.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут математики НАН України 2014
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Цитувати:	κ-Deformed Phase Space, Hopf Algebroid and Twisting / T. Jurić, D. Kovačević, S. Meljanac // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 65 назв. — англ.

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spelling	Jurić, T. Kovačević, D. Meljanac, S. 2019-02-09T20:58:50Z 2019-02-09T20:58:50Z 2014 κ-Deformed Phase Space, Hopf Algebroid and Twisting / T. Jurić, D. Kovačević, S. Meljanac // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 65 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 81R60; 17B37; 81R50 DOI:10.3842/SIGMA.2014.106 https://nasplib.isofts.kiev.ua/handle/123456789/146538 Hopf algebroid structures on the Weyl algebra (phase space) are presented. We define the coproduct for the Weyl generators from Leibniz rule. The codomain of the coproduct is modified in order to obtain an algebra structure. We use the dual base to construct the target map and antipode. The notion of twist is analyzed for κ-deformed phase space in Hopf algebroid setting. It is outlined how the twist in the Hopf algebroid setting reproduces the full Hopf algebra structure of κ-Poincaré algebra. Several examples of realizations are worked out in details. This paper is a contribution to the Special Issue on Deformations of Space-Time and its Symmetries. The full collection is available at http://www.emis.de/journals/SIGMA/space-time.html. The authors would like to thank A. Borowiec, J. Lukierski, A. Pachol, R. Strajn and Z. ˇ Skoda for ˇ useful discussions and comments. The authors would also like to thank the anonymous referee for useful comments and suggestions. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications κ-Deformed Phase Space, Hopf Algebroid and Twisting Article published earlier
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
title	κ-Deformed Phase Space, Hopf Algebroid and Twisting
spellingShingle	κ-Deformed Phase Space, Hopf Algebroid and Twisting Jurić, T. Kovačević, D. Meljanac, S.
title_short	κ-Deformed Phase Space, Hopf Algebroid and Twisting
title_full	κ-Deformed Phase Space, Hopf Algebroid and Twisting
title_fullStr	κ-Deformed Phase Space, Hopf Algebroid and Twisting
title_full_unstemmed	κ-Deformed Phase Space, Hopf Algebroid and Twisting
title_sort	κ-deformed phase space, hopf algebroid and twisting
author	Jurić, T. Kovačević, D. Meljanac, S.
author_facet	Jurić, T. Kovačević, D. Meljanac, S.
publishDate	2014
language	English
container_title	Symmetry, Integrability and Geometry: Methods and Applications
publisher	Інститут математики НАН України
format	Article
description	Hopf algebroid structures on the Weyl algebra (phase space) are presented. We define the coproduct for the Weyl generators from Leibniz rule. The codomain of the coproduct is modified in order to obtain an algebra structure. We use the dual base to construct the target map and antipode. The notion of twist is analyzed for κ-deformed phase space in Hopf algebroid setting. It is outlined how the twist in the Hopf algebroid setting reproduces the full Hopf algebra structure of κ-Poincaré algebra. Several examples of realizations are worked out in details.
issn	1815-0659
url	https://nasplib.isofts.kiev.ua/handle/123456789/146538
citation_txt	κ-Deformed Phase Space, Hopf Algebroid and Twisting / T. Jurić, D. Kovačević, S. Meljanac // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 65 назв. — англ.
work_keys_str_mv	AT jurict κdeformedphasespacehopfalgebroidandtwisting AT kovacevicd κdeformedphasespacehopfalgebroidandtwisting AT meljanacs κdeformedphasespacehopfalgebroidandtwisting
first_indexed	2025-11-25T23:46:46Z
last_indexed	2025-11-25T23:46:46Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 10 (2014), 106, 18 pages κ-Deformed Phase Space, Hopf Algebroid and Twisting? Tajron JURIĆ †, Domagoj KOVAČEVIĆ ‡ and Stjepan MELJANAC † † Rudjer Bošković Institute, Bijenička cesta 54, HR-10000 Zagreb, Croatia E-mail: tajron.juric@irb.hr, meljanac@irb.hr ‡ University of Zagreb, Faculty of Electrical Engineering and Computing, Unska 3, HR-10000 Zagreb, Croatia E-mail: domagoj.kovacevic@fer.hr Received February 21, 2014, in final form November 11, 2014; Published online November 18, 2014 http://dx.doi.org/10.3842/SIGMA.2014.106 Abstract. Hopf algebroid structures on the Weyl algebra (phase space) are presented. We define the coproduct for the Weyl generators from Leibniz rule. The codomain of the coproduct is modified in order to obtain an algebra structure. We use the dual base to construct the target map and antipode. The notion of twist is analyzed for κ-deformed phase space in Hopf algebroid setting. It is outlined how the twist in the Hopf algebroid setting reproduces the full Hopf algebra structure of κ-Poincaré algebra. Several examples of realizations are worked out in details. Key words: noncommutative space; κ-Minkowski spacetime; Hopf algebroid; κ-Poincaré algebra; realizations; twist 2010 Mathematics Subject Classification: 81R60; 17B37; 81R50 1 Introduction Motivation for studying noncommutative (NC) spaces is related to the fact that general theory of relativity together with Heisenberg uncertainty principle leads to the uncertainty of position coordinates itself 4xµ 4 xν > l2Planck [22, 23]. This uncertainty in the position can be reali- zed via NC coordinates. There are also arguments based on quantum gravity [22, 23, 36], and string theory models [20, 61], which suggest that the spacetime at the Planck length is quantum, i.e. noncommutative. We will consider a particular example of NC space, the so called κ-Minkowski spacetime [13, 17, 37, 40, 41, 49, 50, 51, 53, 54, 56, 57, 58], which is a Lie algebraic deformation of the usual Minkowski spacetime. Here, κ is the deformation parameter usually interpreted as Planck mass or the quantum gravity scale. Investigations of physical theories on κ-Minkowski spacetime leads to many new properties, such as: modification of particle statistics [5, 18, 24, 27, 64, 65], deformed electrodynamics [28, 29], NC quantum mechanics [3, 4, 31, 46, 47], and quantum gravity effects [11, 21, 26, 30, 60]. κ-Minkowski spacetime is also related to doubly-special and deformed relativity theories [1, 2, 10, 43, 44]. The symmetries of κ-Minkowski spacetime are described via Hopf algebra setting and they are encoded in the κ-Poincaré–Hopf algebra (in the same sense as are the symmetries of Minkowski spacetime encoded in the Poincaré–Hopf algebra). A Hopf algebra is a bialgebra equipped with an antipode map satisfying the Hopf axiom. The bialgebra is an (unital, associative) algebra which is also a (conunital, coassociative) coalgebra such that certain compatibility conditions are satisfied. The antipode is an antihomomorphism of the algebra structure (an antialgebra ?This paper is a contribution to the Special Issue on Deformations of Space-Time and its Symmetries. The full collection is available at http://www.emis.de/journals/SIGMA/space-time.html mailto:tajron.juric@irb.hr mailto:meljanac@irb.hr mailto:domagoj.kovacevic@fer.hr http://dx.doi.org/10.3842/SIGMA.2014.106 http://www.emis.de/journals/SIGMA/space-time.html 2 T. Jurić, D. Kovačević and S. Meljanac homomorphism). Hopf algebras are used in various areas of mathematics and physics for fifty years. See [8, 52] for some examples. It turns out that the notion of the Hopf algebra is too restrictive and it has to be generalized. For example, it is shown that the Weyl algebra (quantum phase space) can not have a structure of a Hopf algebra. Namely, the whole phase space (Weyl algebra) generated by pµ and xµ (or x̂µ) can not be equipped with the Hopf algebra structure, since one can not include 4xµ in a satisfactory way, i.e. the notion of Hopf algebra is too restrictive for the whole phase space (Weyl algebra). Several types of generalizations are possible: quasi-Hopf algebras, multiplier Hopf algebras and weak Hopf algebras. Our construction is very similar to the structure of the Hopf algebroid defined by Lu in [48]. Lu was inspired by the notion of the Poisson algebroid from the Poisson geometry. Namely, some Hopf algebras are quantization of the Poisson groups. Now, Hopf algebroids can be con- sidered as the quantization of the Poisson groupoids. Lu introduces two algebras: the base algebra A and the total algebra H. One can consider the total algebra H as the algebra over the base algebra A. The left and right multiplications are given by the source and the target maps. Hence, the coproduct4 is defined on the total algebraH and the image lies inH⊗AH which is an (A,A)-bimodule but not an algebra. Namely, H⊗AH is the quotient of H⊗H by the right ideal. G. Böhm and K. Szlachányi in [9] considered the same structure as Lu did, but they changed the definition of the antipode. For more comprehensive approach, see [8]. Let us mention that some ideas existed before the definition of Lu in which the base algebra or both the base algebra and the total algebra had to be commutative (see [8, 48] and references therein). Bialgebroid is equivalent to the notion of ×A-bialgebra introduced much earlier by Takeuchi in [62]. One can analyze the structure of the Hopf algebra by twists. See [6, 7] for more details. P. Xu in [63] applies the twist to the bialgebroid (which he calls Hopf algebroid although he does not have the antipode). It is important to mention that Xu uses the definition of the bialgebroid which is equivalent to the definition from [48]. In [40] κ-Minkowski spacetime and Lorentz algebra are unified in a unique Lie algebra. Realizations and star products are defined and analyzed in general and specially, their relation to coproduct of the momenta is pointed out. The deformation of Heisenberg algebra and the corresponding coalgebra by twist is performed in [57]. Here, the so called tensor exchange identities are introduced and coalgebras for the generalized Poincaré algebras are constructed. The exact universal R-matrix for the deformed Heisenberg (co)algebra is found. The quantum phase space (Weyl algebra) and its Hopf algebroid structure is analyzed in [33]. Unification of κ-Poincaré algebra and κ-Minkowski spacetime is done via embedding into quan- tum phase space. The construction of κ-Poincaré–Hopf algebra and κ-Minkowski spacetime using Abelian twist in the Hopf algebroid approach has been elaborated. Twists, realizations and Hopf algebroid structure of κ-deformed phase space are discussed in [34]. It is shown that starting from a given deformed coalgebra of commuting coordinates and momenta one can construct the corresponding twist operator. In the present paper, the total algebra is the Weyl algebra Ĥ and the base algebra is the subalgebra Â generated by noncommutative coordinates x̂µ. The construction of the target map is obtained via dual realizations. The codomain of the coproduct is changed. We take a quotient of the image of the coproduct instead of quotient of Ĥ ⊗ Ĥ. As a consequence, the right ideal by which Lu [48] has taken the quotient is now two-sided and the codomain of the coproduct has the algebra structure. The notion of the counit is related to realizations. Furthermore, we manage to incorporate the twist in our construction, obtaining the Hopf algebroid structure from the twist. This paper is structured as follows. In Section 2 we introduce the κ-Minkowski spacetime and κ-deformed phase space, and we establish the connection between Leibniz rule and coproduct κ-Deformed Phase Space, Hopf Algebroid and Twisting 3 for the Weyl generators. Also, the dual basis is introduced and elaborated. The Hopf algebroid structure of κ-deformed phase space Ĥ and undeformed phase space H is presented in Section 3. In Section 4 we first discuss the realizations and then we provide the twist operator in the Hopf algebroid approach. It is shown that the twisted Hopf algebroid structure of phase space H is isomorphic to the Hopf algebroid structure of Ĥ. Finally, in Section 5 we consider the κ- Poincaré–Hopf algebra in the natural realization (classical basis). It is outlined how the twist in Hopf algebroid setting reproduces the full Hopf algebra structure of κ-Poincaré algebra. Also, we discuss the existence and properties of twist in all types of deformations (space-, time- and light-like). 2 κ-deformed phase space 2.1 κ-Minkowski spacetime Let us denote coordinates of the κ-Minkowski spacetime by x̂µ. Latin indices will be used for the set {1, . . . , n − 1} and Greek indices will be used for the set {0, . . . , n − 1}. The Lorentz signature of the κ-Minkowski spacetime is defined by [ηµν ] = diag(−1, 1, . . . , 1). Let gκ be the Lie algebra generated by x̂µ such that [x̂µ, x̂ν ] = i (aµx̂ν − aν x̂µ) , (2.1) where a ∈ Mn. The relation to κ mass parameter is aµ = 1 κuµ, uµ ∈ Mn (u2 = −1 time-like, u2 = 1 space-like and u2 = 0 light-like). The enveloping algebra U(gκ) of gκ will be denoted by Â. 2.2 Phase space The momentum space T = C[[pµ]] is the commutative space generated by pµ such that [pµ, x̂ν ] = −iϕµν(p) (2.2) is satisfied for some set of real functions ϕµν (see [33, 34, 40] for details). Let us recall that lim a→0 ϕµν = ηµν1 and detϕ 6= 0. We also require that generators x̂µ and pµ satisfy Jacobi identities. This gives the set of restrictions on functions ϕµν (see equation (11) in [40] or equation (4) in [34]). The existence of such space T is analyzed in several papers [40, 54]. One particularly interesting solution is the set {pLµ} which is related to the so called left covariant realization [40, 54] where ϕµν = ηµνZ −1, i.e. (2.2) leads to[ pLµ, x̂ν ] = −iηµνZ−1. (2.3) Here Z denotes the shift operator defined by [Z, x̂µ] = iaµZ, [Z, pµ] = 0, and for the left covariant realization is given by Z−1 = 1 + ( apL ) , where we used ( apL ) ≡ aαpLα. The phase space Ĥ is generated as an algebra by Â and T such that (2.1) and (2.2) are satisfied. Let I be the unique action of Ĥ on Â, such that Â acts on itself by left multiplication and t I f̂ = [t, f̂ ] I 1 for all t ∈ T and f̂ ∈ Â. Â can be considered as an Ĥ-module. 4 T. Jurić, D. Kovačević and S. Meljanac 2.3 Leibniz rule We have already mentioned that Ĥ does not have the structure of the Hopf algebra, but it is possible to construct the structure of the Hopf algebroid. In this subsection we do the preparation for the coproduct which will be completely defined in Section 3. The formula for the coproduct can be built from the action I and the Leibniz rule (see [40, Section 2.3] and [34]). In κ- Poincaré–Hopf algebra Uκ(P) (where P is generated by momenta pµ and Lorentz generators Mµν) the coproducts of momenta and Lorentz generators are unique and 4\|Uκ(P) : Uκ(P) → Uκ(P)⊗Uκ(P). However in the Hopf algebroid structure the coproduct of generators pµ and x̂µ are not unique, modulo the right ideal K̂ in (2.10). Let 4(ĥ) = ĥ(1) ⊗ ĥ(2) for ĥ(1), ĥ(2) ∈ Ĥ (using Sweedler notation). Then ĥI ( f̂ ĝ ) = m ( 4 ( ĥ ) I ( f̂ ⊗ ĝ )) = ( ĥ(1)If̂ )( ĥ(2)Iĝ ) (2.4) for f̂ , ĝ ∈ Â. Now we recall the formula for the coproduct of pµ defined by 4\|T : T → T ⊗ T . Then pµI ( f̂ ĝ ) = [ pµ, f̂ ĝ ] I1 = ([ pµ, f̂ ] ĝ + f̂ [pµ, ĝ] ) I1 = [ pµ, f̂ ] Iĝ + f̂pµIĝ. (2.5) For example let us write the coproduct of pLµ. One finds by induction, starting with (2.3) that[ pLµ, f̂ ] = ( pLµ I f̂ ) Z−1, ∀ f̂ ∈ Â. Inserting this result in the r.h.s. of (2.5) and comparison with r.h.s. of (2.4) for ĥ = pLµ gives 4 ( pLµ ) = pLµ ⊗ Z−1 + 1⊗ pLµ. Now, let us consider elements x̂µ. It is clear that 4(x̂µ) = x̂µ ⊗ 1 (2.6) since x̂µI(f̂ ĝ) = (x̂µf̂)ĝ. Formula (33) from [40] shows that x̂µI ( f̂ ĝ ) = ( Z−1If̂ ) (x̂µIĝ)− aµ ( pLαIf̂ ) (x̂αIĝ) and1 4′(x̂µ) = Z−1 ⊗ x̂µ − aµpLα ⊗ x̂α. (2.7) It is convenient to write (2.7) in the form 4′(x̂µ) = Oµα ⊗ x̂α, where Oµα = Z−1ηµα − aµpLα. (2.8) Hence, elements R̂µ = x̂µ ⊗ 1−Oµα ⊗ x̂α (2.9) satisfy m(R̂µI(f̂ ⊗ ĝ)) = 0 for all f̂ , ĝ ∈ Â where m denotes the multiplication (m(f̂ ⊗ ĝ) = f̂ ĝ) and (a⊗ b)I(f̂ ⊗ ĝ) = (aIf̂)⊗ (bIĝ). Then K̂ = U+(R̂µ)Ĥ ⊗ Ĥ (2.10) 1In Section 3.1, the coproduct will be defined and (2.6) and (2.7) will be equal, since both choices of coproducts of x̂µ belong to the same congruence class. κ-Deformed Phase Space, Hopf Algebroid and Twisting 5 is the right ideal in Ĥ ⊗ Ĥ. Here we used that U+(R̂µ) is the universal enveloping algebra generated by R̂µ but without the unit element. It is important to emphasize that such derived coproduct is an algebra homomorphism 4 ( ĥ1ĥ2 ) = 4 ( ĥ1 ) 4 ( ĥ2 ) for any ĥ1, ĥ2 ∈ Ĥ which enables us to define the formula for the coproduct for all elements of Ĥ. 2.4 Dual basis In [40] we have introduced the notion of the dual basis. Let us recall some basic facts since it will be used for the definition of the target map. We define elements ŷµ = x̂αO−1µα , (2.11) where O−1µα = ( ηµα + aµp L α ) Z (2.12) (it would be more precise to write (O−1)µα). They have some interesting properties. Since x̂µ = ŷαOµα, (2.13) ŷµ and pµ form a basis of Ĥ (it would be more correct to say that power series in ŷµ and pµ form a basis of Ĥ). Elements ŷµ satisfy commutation relations similar to (2.1): [ŷµ, ŷν ] = −i(aµŷν − aν ŷµ). We call this basis the dual basis. It is easy to check that x̂µ and ŷν commute, i.e. [x̂µ, ŷν ] = 0. (2.14) Also, the straightforward calculation shows that Oµν and Oλρ commute. It remains to consider commutation relations among Oµν , x̂µ and ŷµ. The definition of Oµν yields [Oµν , x̂λ] = i(aµηλν− aληµν)Z−1 = i(aµOλν − aλOµν) and it shows that [Oµν , x̂λ] = iC α µλ Oαν , (2.15) where Cµλα = aµηλα − aληµα stands for structure constants. One can easily obtain[ O−1µν , x̂λ ] = i ( −aµηλν + aλO −1 µν ) , [Oµν , ŷλ] = i(aµηλν − aλOµν) and [ O−1µν , ŷλ ] = i ( −aµO−1λν + aλO −1 µν ) = −iCµλα ( O−1 )αν . The commutation relation [O−1µν , x̂λ] can be also obtained from (2.15) multiplying by O−1µα and O−1βν and using aαO−1µα = aµ. Let us mention that elements Oµν satisfy Oµν = ηµν + Cαµνp L α. (2.16) One can easily check that ŷµI1 = x̂µ. Using (2.14) and (2.16), it is easy to obtain that ŷµIx̂ν = x̂ν x̂µ and f̂(ŷ)Iĝ(x̂) = ĝ(x̂)f̂op(x̂). Here f̂op stands for the opposite polynomial ((x̂µx̂ν)op = x̂ν x̂µ). Hence, the action I of f̂(ŷ) can be understood as a multiplication from the right with f̂op(x̂). One can show that4(ŷµ) = 1⊗ŷµ. Note that the same construction as for κ-Minkowski space (2.1) could be generalized to arbitrary Lie algebra defined by structure constants Cµνλ. 6 T. Jurić, D. Kovačević and S. Meljanac 3 Hopf algebroid 3.1 Hopf algebroid structure of Ĥ We define the source map, target map, coproduct, counit and antipode such that Ĥ has the structure of the Hopf algebroid. In Hopf algebroid, the unit map is replaced by the source and target maps. In our case Ĥ is the total algebra and Â is the base algebra. The source map α̂ : Â → Ĥ is defined by α̂ ( f̂(x̂) ) = f̂(x̂). The target map β̂ : Â → Ĥ is defined by β̂ ( f̂(x̂) ) = f̂op(ŷ). Let us recall that the source map is the homomorphism while the target map is the antihomo- morphism. Relation (2.14) shows that α̂ ( f̂(x̂) ) β̂(ĝ(x̂)) = β̂(ĝ(x̂))α̂ ( f̂(x̂) ) . In order to define the coproduct on Ĥ, we consider the subspace B̂ of Ĥ ⊗ Ĥ: B̂ = U ( R̂µ )( Â ⊗ C ) 4 T , where U(R̂µ) denotes the universal enveloping algebra generated by R̂µ (see (3.1)). Here, 4T denotes the subalgebra of Ĥ ⊗ Ĥ generated by 1⊗ 1 and elements 4(pµ). For example, we can consider pLµ and then 4T is generated by 1⊗ 1 and pLµ ⊗ Z−1 + 1⊗ pLµ. Since[ R̂µ, R̂ν ] = i ( aµR̂ν − aνR̂µ ) = iCµναR̂α, (3.1)[ x̂µ ⊗ 1, R̂ν ] = i ( aµR̂ν − aνR̂µ ) , (3.2)[ Oµα ⊗ x̂α, R̂ν ] = 0, (3.3)[ 4 pLµ, R̂ν ] = 0 and [ x̂µ ⊗ 1, pLν ⊗ Z−1 + 1⊗ pLν ] = iηµνZ −1 ⊗ Z−1 ∈ 4T , (3.4) B̂ is a subalgebra of Ĥ ⊗ Ĥ. It is obvious that (3.3) is a consequence of (3.1) and (3.2) but we write it for completeness. Now, let us consider the subspace Î of B̂ defined by Î = U+ ( R̂µ )( Â ⊗ C ) 4 T , where U+(R̂µ) is the universal enveloping algebra generated by R̂µ but without the unit element. Using (3.1)–(3.4) one can check that Î = K̂ ∩ B̂ and Î is the twosided ideal in B̂. Remark. We could also define the subalgebra B̂3 in Ĥ ⊗ Ĥ ⊗ Ĥ by B̂3 = U [( R̂µ ) 1,2 , ( R̂µ ) 2,3 ]( Â ⊗ C⊗ C ) (4⊗ 1)(4T ), where U [(R̂µ)1,2, (R̂µ)2,3] denotes the universal enveloping algebra generated by 1 ⊗ 1 ⊗ 1, (R̂µ)1,2 = R̂µ ⊗ 1 and (R̂µ)2,3 = 1 ⊗ R̂µ and we have that (4 ⊗ 1)(4T ) = (1 ⊗ 4)(4T ) since T is a Hopf algebra. Similarly, we can define B̂n and then B̂ would correspond to B̂2. Also, K̂n and În = K̂n ∩ B̂n can be defined. See [48] for the similar discussion. κ-Deformed Phase Space, Hopf Algebroid and Twisting 7 Now, we define the coproduct 4 : Ĥ → B̂/Î = 4Ĥ by 4(x̂µ) = x̂µ ⊗ 1 + Î = Z−1 ⊗ x̂µ − aµpLα ⊗ x̂α + Î = Oµα ⊗ x̂α + Î, (3.5) 4 ( pLµ ) = pLµ ⊗ Z−1 + 1⊗ pLµ + Î. Notice that B̂/Î is the “restriction” of Lu’s Ĥ ⊗ Ĥ/K̂, or in other words an (Â, Â)-submodule of Ĥ⊗ Ĥ/K̂ that turns out to be an algebra, which, in turn, allows us to define 4 as an algebra homomorphism 4 ( f̂ ĝ ) = 4 ( f̂ ) 4 (ĝ). The coproduct of ŷµ is given by 4(ŷµ) = 1⊗ ŷµ + Î = ŷα ⊗O−1µα + Î. One can check that such defined coproduct is coassociative. The counit ε̂ : Ĥ → Â is defined by ε̂ ( ĥ ) = ĥI1. This map is not a homomorphism. It is easy to check that m(α̂ε̂⊗1)4 = 1 and m(1⊗ β̂ε̂)4 = 1. In order to check the first identity, we write elements of Ĥ in the form f̂(x̂)g(p) and for the second identity in the form f̂(ŷ)g(p). The antipode S : Ĥ → Ĥ is defined by S(ŷµ) = x̂µ and S ( pLµ ) = −pLµZ. The antipode S(x̂µ) can be calculated from (2.13). One obtains that S(x̂µ) = ŷµ + iaµ(1− n). (3.6) It follows that S2(ŷµ) = ŷµ + iaµ(1 − n) (and S2(x̂µ) = x̂µ + iaµ(1 − n)) and S2(pµ) = pµ. Previous two formulas can be written also as S2(ĥ) = Z1−nĥZn−1. It is enough to check it for the elements x̂µ and pµ since S2 is a homomorphism. The expression of S2(x̂µ) can be written in terms of structure constants: S2(x̂µ) = x̂µ + iC α αµ . (3.7) A nice way to check the consistency of the antipode is to start with (2.13) and apply the antipode S (note that S(Oµα) = O−1µα): S(x̂µ) = O−1µα x̂ α = O−1µα ŷβO αβ. It produces S2(x̂µ) = ( O−1 )αβ x̂βOµα. It remains to apply expressions for (O−1)αβ and Oµα (see (2.12) and (2.8)), use the abbreviation AL = −aαpLα = −(apL) and recall the identity Z = (1−AL)−1 (see [40]). Let P ⊂ Ĥ be the enveloping algebra of the Poincaré algebra p. It is possible to define the Hopf algebra structure on the subalgebra P [40]. It is interesting to note that the coproduct and the antipode map defined above on Ĥ and restricted to P coincides with the coproduct and the antipode map on the Hopf algebra P [33]. For more details see Section 5. It is easy to check that Sβ̂ = α̂, m(1⊗ S)4 = α̂ε̂, m(S ⊗ 1)4 = β̂ε̂S. (3.8) 8 T. Jurić, D. Kovačević and S. Meljanac The first identity is obvious, the second one can be easily checked for the base elements and the third identity can be easily checked using the dual basis. In [48], Lu analyzes the right ideal K̂ generated by Q̂µ = ŷµ ⊗ 1 − 1 ⊗ x̂µ (right ideal K̂ is denoted by I2 in [48]). These elements are equal to R̂α((O−1)µα⊗1). It is important to mention that the identity m(1 ⊗ S)4 = α̂ε̂ is not satisfied in [48], because m(1 ⊗ S)K̂ 6= 0 and this is why the section γ is needed. In our approach, since we have 4 : Ĥ → B̂/Î = 4Ĥ and m(1⊗ S)Î = 0, it is easy to see that (3.8) holds ∀h ∈ Ĥ. Let us point out that [R̂µ, Q̂ν ] = 0 and [Q̂µ, Q̂ν ] = i(−aµQ̂ν +aνQ̂µ). Also, it is easy to check that [Q̂µ,4pLν ] = 0 and [Q̂µ, x̂ν ⊗ 1] = 0. 3.2 Hopf algebroid structure of H Now, let us consider the case when the deformation vector aµ is equal to 0. Then (2.1) trans- forms to [x̂µ, x̂ν ] = 0, the algebra Ĥ becomes the Weyl algebra which we denote by H and write xµ instead of x̂µ. We have already mentioned that it is not possible to construct the Hopf algebra structure on H. Let us repeat the Hopf algebroid structure on H and set the terminology. Now, ϕµν = Oµν = ηµν , Z = 1 and ŷµ = xµ. Let A (the base algebra) be the subalgebra of H generated by 1 and xµ. We define the action � of H on A in the same way as we did it in Section 2.2: f(x) � g(x) = f(x)g(x), pµ � 1 = 0 and pµ � g(x) = [pµ, g(x)] � 1 = pµg(x) � 1. Then A can be considered as an H-module. It is clear that the action I transforms to the action � when the vector a is equal to 0. The source and the target map are now equal α0 = β0 and α0;β0 : A → H reduces to the natural inclusion. The counit ε0 : H → A is defined by ε0(h) = h� 1. In order to define the coproduct, let us define relations (R0)µ by (R0)µ = xµ ⊗ 1− 1⊗ xµ. Let U [(R0)µ] be the universal enveloping algebra generated by 1 ⊗ 1 and (R0)µ, U+[(R0)µ] be the universal enveloping algebra generated by (R0)µ but without the unit element, and 40T be the algebra generated by 1 ⊗ 1 and pµ ⊗ 1 + 1 ⊗ pµ. Note that T is isomorphic to 40T . Now, we define B0, the subalgebra of H⊗H of the form B0 = U [(R0)µ](A⊗ C)40 T and twosided ideal I0 of B0 by I0 = U+[(R0)µ](A⊗ C)40 T . The coproduct 40 : H → B0/I0 = 40H is a homomorphism defined by 40(xµ) = xµ ⊗ 1 + I0, 40(pµ) = pµ ⊗ 1 + 1⊗ pµ + I0. κ-Deformed Phase Space, Hopf Algebroid and Twisting 9 One checks that the coproduct 40 and the counit ε0 satisfy m(α0ε0 ⊗ 1)40 = 1 and m(1 ⊗ β0ε0)40 = 1. The antipode S0 : H → H transforms to S0(xµ) = xµ, S0(pµ) = −pµ. (3.9) It is easy to check that m(1⊗ S0)40 = α0ε0, m(S0 ⊗ 1)40 = β0ε0S0. (3.10) Similarly as in the deformed case, the expression m(1⊗S0)40 is not well defined in [48], because m(1⊗ S0)K0 6= 0 and this is why the section γ is needed. In our approach, since m(1⊗ S0)I0 = 0 holds, one can check (3.10) ∀h ∈ H. 4 Twisting Hopf algebroid structure 4.1 Realizations The phase space satisfying (2.1) and (2.2) can be analyzed by realizations (see [40, 42, 56]). In Section 3.2, we have analyzed the Weyl algebra H generated by pµ and commutative coordinates xµ satisfying [pµ, xν ] = −iηµν1. Then, the noncommutative coordinates x̂µ are expressed in the form x̂µ = xαϕαµ(p) (4.1) such that (2.1) and (2.2) are satisfied. It is important to observe that the space H is isomorphic to Ĥ as an algebra. Hence, we set Ĥ = H and treat sets {xµ, pν} and {x̂µ, pν} as different bases of the same algebra. However, we will use both symbols, Ĥ and H in order to emphasize the basis. The action �, defined in Section 3.2 corresponds to H. However, H and Ĥ, considered as Hopf algebroids are different. The restriction of the counit ε0\|Â, introduced in Section 3.2, defines the bijection of vector spaces Â and A. By the abuse of notation, we denote it by ε0 or �. Let us mention that the inverse map is simply ε̂\|A. Then, the star product ? on A is defined by (f?g)(x) = f̂(x̂)ĝ(x̂)�1 = f̂(x̂)�g(x) where f = f̂�1 and g = ĝ�1. The algebra A equipped with the star product instead of pointwise multiplication will be denoted by A? and the map ε0 : Â → A? is an isomorphism of algebras. It is possible to construct the dual realization ϕ̃µν and the dual star product ?ϕ̃ such that (f ?ϕ g)(x) = (g ?ϕ̃ f)(x) is satisfied (see [40, Section 5]). Now, elements ŷµ are given by ŷµ = xαϕ̃αµ(p). It is easy to check the following properties: x̂µ � f(x) = xµ ?ϕ f(x) = f(x) ?ϕ̃ xµ and ŷµ � f(x) = xµ ?ϕ̃ f(x) = f(x) ?ϕ xµ. 10 T. Jurić, D. Kovačević and S. Meljanac 4.1.1 Similarity transformations The relation between realizations is given by the similarity transformations [34]. Let us consider two realizations. The first one is denoted by xµ and pµ and given by the set of functions {ϕµν} (and (2.2) or (4.1)). The second realization is denoted by Xµ, Pµ and Φµν (x̂µ = XαΦαµ(P )). The similarity transformation E is given by E = exp{xαΣα(p)} such that lim a→0 Σα = 0. Now, the relation between realizations is given by Pµ = EpµE−1, Xµ = ExµE−1. It is easy to see that Pµ = Pµ(p). Since [Pµ, x̂ν ] = −iΦµν(P ), ∂Pµ ∂pα ϕαν = Φµν(P (p)) and ϕαν = [ ∂P ∂p ]−1 αµ Φµν(P (p)). It follows that the set of functions ϕµν can be obtained from the set of functions Φµν and the expressions of P in terms of p. Since Oµν = Oµν(P (p)), it is easy to express Oµν in the realization determined by xµ and pµ. 4.1.2 Examples Let us consider three examples of realizations. The noncovariant λ-family of realizations is given by x̂0 = x (λ) 0 − a0 (1− λ)x (λ) k p (λ) k , x̂k = x (λ) k Z−λ, (4.2) and ŷ0 = x̂0Z − ia0 + a0 ( x̂pL ) Z, ŷj = x̂jZ, (4.3) where Z = eA (λ) and λ ∈ R. For this family we assume that a = (a0, 0, . . . , 0). Here, (λ) denotes the label. Generic realizations are denoted without the label. It is easy to obtain pL0 = 1 a0 (1 − Z−1) and pLk = p (λ) k Zλ−1. Now, one calculates Oµν (see (2.8)) in terms of p (λ) µ : Okν = Z−1ηkν , O00 = −1 and O0k = Z−1η0k − a0pLk = ( η0k − a0p (λ) k Zλ ) Z−1. The left covariant realization is defined by x̂µ = xLµ ( 1−AL ) , where Z = (1 − AL)−1. The element pL that we have mentioned in Section 2.2 corresponds to the left covariant realization. It is easy to obtain that ŷµ = xLµ + aµ ( xLpL ) (see (2.11) for the definition of ŷµ). The right covariant realization is defined by x̂µ = xRµ − aµ ( xRpR ) , where Z = 1 +AR. The relation between pLµ and pRµ is given by pRµ = pLµZ. Now, ŷµ = xRµ ( 1 +AR ) . Also, it easy to calculate Oµν in terms of pRµ : Oµν = Z−1ηµν − aµpLν = ( ηµν − aµpRν ) Z−1. One should notice the duality between the left covariant and the right covariant realizations. κ-Deformed Phase Space, Hopf Algebroid and Twisting 11 4.2 Twist and Hopf algebroid For each realization, there is the corresponding twist and vice versa [34]. The relation between the star product and twist is given by f ? g = m ( F−1 � (f ⊗ g) ) for f, g ∈ A. It follows that F−1 ∈ H ⊗ H/K0. Now, we will use twists to reconstruct the Hopf algebroid structure described in Section 3.1, from the Hopf algebroid structure analyzed in Section 3.2. That is we will show that by twisting the Hopf algebroid structure of H one can obtain the Hopf algebroid structure of Ĥ. Hence, we will consider twists F such that F : 40H → 4H. Here I ∼= Î and 4H ∼= 4Ĥ. More precisely, I is the twosided ideal generated by elements Rµ which are defined by Rµ = F(R0)µF−1. Let us mention that the relation between R̂µ and Rµ is given by R̂µ = Rα 4 (ϕαµ). Also, it is easy to rebuild the realization from the twist. For the given twist F , the corresponding realization is obtained by x̂µ = m ( F−1(�⊗ 1)(xµ ⊗ 1) ) . Similarly, ŷµ = m ( F̃−1(�⊗ 1)(xµ ⊗ 1) ) , where F̃−1 is given by F̃−1 = τ0F−1τ0 (τ0 stands for the flip operator with the property τ0(h1 ⊗ h2) = h2 ⊗ h1, ∀h1, h2 ∈ H). The noncovariant λ-family of realizations have twists of the form F (λ) = exp ( i ( λx (λ) k p (λ) k ⊗A (λ) − (1− λ)A(λ) ⊗ x(λ)k p (λ) k )) . (4.4) These twists belong to the family of Abelian twists (see [24]). The left covariant and the right covariant realizations, respectively, have twists of the form FL = exp ( i ( xLpL ) ⊗ lnZ ) and FR = exp ( − lnZ ⊗ i ( xRpR )) . These two twists belong to the family of Jordanian twists (see [13]). Let us reconstruct the source and the target maps from the twist. First, we define α and β, α : A? → U(x̂µ) ⊂ H, β : A? → U(ŷµ) ⊂ H by α(f(x)) = m ( F−1(�⊗ 1)(α0(f(x))⊗ 1) ) , α0(f(x)) = f(x), and β(f(x)) = m ( F̃−1(�⊗ 1)(β0(f(x))⊗ 1) ) , β0(f(x)) = f(x). Now, the source and the target maps are given by α̂ = αε0\|Â and β̂ = βε0\|Â. 12 T. Jurić, D. Kovačević and S. Meljanac The counit ε̂ : H → Â is given by ε̂(h) = m ( F−1(�⊗ 1)(ε0(h)⊗ 1) ) . The coproduct can be calculated by the formula: 4(h) = F(40(h))F−1. For the noncovariant λ-family of realizations 4 ( x (λ) j ) = x (λ) j ⊗ Z λ = Zλ−1 ⊗ x(λ)j , (4.5) 4 ( x (λ) 0 ) = x (λ) 0 ⊗ 1 + a0(1− λ)⊗ x(λ)k p (λ) k = 1⊗ x(λ)0 − a0λx (λ) k p (λ) k ⊗ 1, (4.6) 4 ( p (λ) j ) = p (λ) j ⊗ Z −λ + Z1−λ ⊗ p(λ)j , (4.7) and 4 ( p (λ) 0 ) = p (λ) 0 ⊗ 1 + 1⊗ p(λ)0 . (4.8) It is a nice exercise to express x̂µ in terms of x (λ) α and p (λ) α (see (4.2)), use (4.5)–(4.8) and obtain (3.5). Similarly, 4 ( xLµ ) = xLµ ⊗ Z = 1⊗ ( xLµ + iaµZ ) and 4 ( pLµ ) = pLµ ⊗ Z−1 + 1⊗ pLµ for the left covariant realization and 4 ( xRµ ) = ( xRµ − iaµZ ) ⊗ 1 = Z−1 ⊗ xRµ and 4 ( pRµ ) = pRµ ⊗ 1 + Z ⊗ pRµ for the right covariant realization. It remains to consider the antipode. Let χ−1 = m(S0 ⊗ 1)F−1, then S(h) = χ(S0(h))χ−1 (4.9) where S0 denotes the undeformed antipode map defined by (3.9) (S0(xµ) = xµ and S0(pµ) = −pµ). For the similar approach regarding Hopf algebras, see [7, 6]. For the noncovariant λ-family of realizations, χ has the form χ(λ) = exp ( i(1− 2λ)A(λ)x (λ) k p (λ) k + λ(1− n)A(λ) ) . Then S ( p (λ) j ) = −p(λ)j Z2λ−1, (4.10) S ( p (λ) 0 ) = −p(λ)0 , (4.11) S ( x (λ) j ) = x (λ) j Z1−2λ, (4.12) S ( x (λ) 0 ) = x (λ) 0 − (1− 2λ)a0x (λ) k p (λ) k + λia0(1− n). (4.13) Again, it is an exercise to express x̂µ in terms of x (λ) α and p (λ) α (see (4.2)), use (4.10)–(4.13) and obtain (3.6). The antipode is given by S(x̂j) = x̂jZ (4.14) κ-Deformed Phase Space, Hopf Algebroid and Twisting 13 and S(x̂0) = x̂0 + a0x (λ) k p (λ) k + ia0(1− n). (4.15) Let us recall that for the noncovariant λ-family of realizations we set aµ = (a0, 0, ..., 0). Now, one can compare (4.14) and (4.15) with (3.6). The formula for the antipode of x̂µ can be also obtained from the formula S(ŷµ) = x̂µ, formulas for the realization of x̂µ and ŷµ, (4.2) and (4.3) and formulas for S(pµ). For all examples, it is easy to check that S(ŷµ) = χ(S0(ŷµ))χ−1 = x̂µ. For the left covariant realization( χL )−1 = exp ( i ( pLxL ) AL ) . For the right covariant realization( χR )−1 = exp ( −iAR ( xRpR )) . There is a natural question if the antipode map on the Hopf algebroid Ĥ defined by (4.9) and the antipode map defined on the Hopf algebra U(igl(n)) coincide (see [38] for the formulas of the antipode). They coincide for h ∈ Ĥ for which αε(h) = βεS0(h). For elements h for which αε(h) 6= βεS0(h), the antipode maps do not coincide. For example, S0(xjpj) = −xjpj + i in the Hopf algebroid, while S0(xjpj) = −xjpj in the Hopf algebra (here no summation is assumed). See also [33]. Using (4.9), it is easy to obtain the expression for S−1: S−1(h) = S0(χ)S0(h)S0 ( χ−1 ) . One can show that S0(χ) = Zn−1χ. Then S−1(h) = Zn−1S(h)Z1−n and S2(h) = Z1−nhZn−1. For example, S2(pµ) = pµ, S2(x̂µ) = x̂µ+iaµ(1−n) and S2(ŷµ) = ŷµ+iaµ(1−n). This coincides with results in Section 3 (see (3.6) and (3.7)). 5 κ-Poincaré Hopf algebra from κ-deformed phase space and twists Let us consider the κ-Poincaré Hopf algebra in natural realization [54, 55, 56] (or classical basis [15, 44]). We start with the undeformed Poincaré algebra generated by Lorentz genera- tors Mµν and translation generators (momentum) Pµ [Mµν ,Mλρ] = ηνλMµρ − ηµλMνρ − ηνρMµλ + ηµρMνλ, [Pµ, Pν ] = 0, [Mµν , Pλ] = ηνλPµ − ηµλPν . The corresponding κ-deformed Poincaré–Hopf algebra can be written in a unified covariant way [25, 35, 40, 54, 56]. The coproduct 4 is given by 4Pµ = Pµ ⊗ Z−1 + 1⊗ Pµ − aµpLαZ ⊗ Pα, 4Mµν = Mµν ⊗ 1 + 1⊗Mµν − aµ ( pL )α Z ⊗Mαν + aν ( pL )α Z ⊗Mαµ, (5.1) as well as the antipode S and counit ε S(Pµ) = ( −Pµ − aµpLαPα ) Z, S(Mµν) = −Mµν − aµ ( pL )α Mαν + aν ( pL )α Mαµ, ε(Pµ) = ε(Mµν) = 0, (5.2) 14 T. Jurić, D. Kovačević and S. Meljanac where the momentum Pµ is related to pLµ via Pµ = pLµ − aµ 2 (pL)2Z. The above Hopf algebra structure unifies all three types of deformations aµ, i.e. time-like (a2 < 0), space-like (a2 > 0) and light-like (a2 = 0). Using the action I and coproduct 4 we can get the whole algebra {x̂µ,Mµν , Pµ} (for details see [32, 33]) [Mµν , x̂λ] = ηνλx̂µ − ηµλx̂ν − iaµMνλ + iaνMµλ, [Pµ, x̂ν ] = −i ( ηµνZ −1 − aµPν ) , (5.3) where Z−1 = (aP ) + √ 1 + a2P 2 and from (5.3) it follows that the NC coordinates x̂µ can be written in terms of canonical Xα and Pα ([Xα, Xβ] = 0, [Pµ, Xν ] = −iηµν1) via x̂µ = XµZ −1 − (aX)Pµ and satisfies (2.1). Now we will discuss the realization of κ-Poincaré–Hopf algebra via phase space Ĥ and discuss the issue of the twist in the Hopf algebroid approach. Realization ofMµν in terms of canonicalXα and Pα is given by Mµν = i(XµPν −XνPµ) which for κ-deformed phase space variables x̂µ, Pµ reads Mµν = i(x̂µPν − x̂νPµ)Z ∈ Ĥ. This is a unique realization in Ĥ (see [54]). Using 4Pµ (5.1), 4x̂µ (3.5), 4Z = Z ⊗ Z and relations R̂µ (2.9) we obtain coproduct 4Mµν as in Hopf algebra (5.1). Note that the result for 4Mµν is unique in the κ-Poincaré–Hopf algebra Uκ(P) since Uκ(P)⊗Uκ(P)∩ K̂ = 0 (which is obvious). Similarly we find S(Mµν) within Hopf algebroid which coincides with S(Mµν) in Hopf algebra (5.2) (for details see [33, 57]). There is a question whether 4Pµ and 4Mµν could be obtained from twist F expressed in terms of Poincaré generators only. 1. For aµ light-like, a2 = 0, such cocycle twist within Hopf algebra approach exists [35] F = exp { aαP β ln[1 + (aP )] (aP ) ⊗Mαβ } . (5.4) The cocycle condition for twist F (5.4) can be checked using the results by Kulish et al. [45] in the Hopf algebra setting (see also [16]). 2. For aµ time- and space-like such twist does not exist within Hopf algebra. Namely, starting from 4Pµ = F 40 PµF−1 and 4Mµν = F 40 MµνF−1 one can construct an operator F = ef , where f = f1 + f2 + · · · is expanded in aµ and expressed in terms of Poincaré generators and dilatation only. In the first order we found that the result is not unique, namely we have a one parameter family of solutions f1 = aαP β ⊗Mαβ + u ( Mαβ ⊗ aαP β − aαP β ⊗Mαβ −D ⊗ (aP ) + (aP )⊗D ) , where u ∈ R is a free parameter. However there is one solution (u = 0) that can be ex- pressed in terms of Poincaré generators only. Also up to first order in aµ cocycle condition is satisfied and one obtains the correct classical r-matrix (see equation (65) in [25]). In the second order for f2 we found a two parameter family of solutions. Here there is no solution without including dilatation, that is the operator F can not be expressed in terms of Poincaré generators only. We have checked that the corresponding quantum R-matrix obtained using f1 and f2 is correct up to the second order. The cocycle condition is no longer satisfied in the Hopf algebra approach, that is F is not a twist in the Drinfeld sense. However, after using tensor exchange identities [33, 34, 57] the cocycle condition is satis- fied and F is a twist in Hopf algebroid approach. It also reproduces the κ-Poincaré–Hopf κ-Deformed Phase Space, Hopf Algebroid and Twisting 15 algebra (when applied to Poincaré generators) (see [34]). In [34], we have developed a ge- neral method for calculating operator F for a given coproducts of xµ and pµ. In Section 3 of [34] the operator F is constructed up to the third order for natural realization (classi- cal basis) and it is shown that this operator F gives the correct coproduct for Mµν (see equation (59) in [34]) and R-matrix (see equation (61) in [34]). We also stated that this operator F can not be expressed in terms of κ-Poincaré generators only (see [34, p. 16]). From the results for f1, f2 and f3 (see equations (42), (46), (49) in [34]) one can show that they could be rewritten in terms of Poincaré generators and dilatation only (after using tensor exchange identities). For alternative arguments on nonexistence of cocycle twist for κ-Poincaré–Hopf algebra see [12, 14]. The main point that we want to emphasize is that the twist operator exists within Hopf algebroid approach, that the cocycle condition is satisfied [33, 34, 57] and that this twist gives the full κ-Poincaré–Hopf algebra (when applied to the generators of Poincaré algebra). General statements on associativity of star product, twist and cocycle condition in Hopf algebroid approach are: 1. Lorentz generators Mµν can be written in terms of x(λ) and p(λ) (4.5)–(4.8). This defines the family of basis labeled by λ. The momenta p(λ) do not transform as vectors under Mµν . The star product is associative for all λ ∈ R. The corresponding twist F (λ) given in (4.4) is Abelian and satisfies the cocycle condition for all λ ∈ R. Applying F (λ) to primitive coproduct 40Mµν leads to κ-deformed igl(n) Hopf algebra (see [17, 25, 32, 38, 41]). How- ever, if we apply conjugation by F (λ) to 4(λ) 0 M (λ) µν (which is not primitive coproduct) we obtain, in the Hopf algebroid approach [57], the correct coproduct4(λ)M (λ) µν corresponding to λ basis (for λ = 0 see [33]). Similarly for the antipode S. Hence, the κ-Poincaré–Hopf algebra can be obtained by twist F in the more generalized sense, i.e. in the Hopf algebroid approach. 2. If star product is associative in one base, then it is associative in any other base obtained by similarity transformations [34]. 3. If star product is associative, then the corresponding twist F satisfies cocycle condition in the Hopf algebroid approach, and vice versa. Note that, there exist star products which are associative but the corresponding twist operator F does not satisfy the cocycle condition in the Hopf algebra approach. 6 Final remarks It is important to note that the work presented in this paper is not genuinely different from Lu’s construction of Hopf algebroid [48] and that we use a particular choice of the algebra which makes it easier to construct the coproduct as an algebra homomorphism to the subalgebra B̂/Î. By this particular choice of algebra we are able to satisfy m(1⊗ S)4 = α̂ε̂, m(S ⊗ 1)4 = β̂ε̂S, while in [48] m(1 ⊗ S)4 is not well defined (for the version of coproduct in [48]) because m(1⊗S)K̂ 6= 0, while in our case m(1⊗S)Î = 0. Therefore we do not need the section γ in the first identity for the antipode. In our approach, since we have 4 : Ĥ → B̂/Î = 4Ĥ and m(1⊗ S)Î = 0, it is easy to see that (3.8) holds ∀h ∈ Ĥ. We are doing this in order to explain the structure of quantum phase space, i.e. Weyl algebra Ĥ. 16 T. Jurić, D. Kovačević and S. Meljanac An axiomatic treatment of the Hopf algebroid structure on general Lie algebra type non- commutative phase spaces, involving completed tensor products, has recently been proposed in [59]. The construction of QFT suitable for κ-Minkowski spacetime is still under active research [19, 39, 55]. We plan to apply κ-deformed phase space, Hopf algebroid approach and twisting to NCQFT and NC (quantum) gravity. Acknowledgements The authors would like to thank A. Borowiec, J. Lukierski, A. Pachol, R. Štrajn and Z. Škoda for useful discussions and comments. The authors would also like to thank the anonymous referee for useful comments and suggestions. References [1] Amelino-Camelia G., Testable scenario for relativity with minimum-length, Phys. Lett. B 510 (2001), 255– 263, hep-th/0012238. 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κ-Deformed Phase Space, Hopf Algebroid and Twisting

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