κ-Poincaré–Hopf algebra and Hopf algebroid structure of phase space from twist

Jurić, Tajron; Meljanac, Stjepan; Štrajn, Rina

doi:10.1016/j.physleta.2013.07.021

Cited by 32 publications

(45 citation statements)

References 86 publications

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“…In [40], a phenomenological analysis related to vector-like deformations of the relativistic quantum phase space and relativistic kinematics was elaborated up to first order in the deformation, particularly on particle propagation in spacetime. Note that if NC coordinatesx μ close a Lie algebra inx μ , then the corresponding deformed quantum phase space has a Hopf algebroid structure [46][47][48][49]. Otherwise, the coproduct is non-coassociative and the corresponding structure should be quasi-bialgebroid.…”

Section: Outlook and Discussionmentioning

confidence: 99%

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Families of vector-like deformations of relativistic quantum phase spaces, twists and symmetries

Meljanac

Pikutić

2017

Eur. Phys. J. C

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Families of vector-like deformed relativistic quantum phase spaces and corresponding realizations are analyzed. A method for a general construction of the star product is presented. The corresponding twist, expressed in terms of phase space coordinates, in the Hopf algebroid sense is presented. General linear realizations are considered and corresponding twists, in terms of momenta and Poincaré-Weyl generators or gl(n) generators are constructed and R-matrix is discussed. A classification of linear realizations leading to vector-like deformed phase spaces is given. There are three types of spaces: (i) commutative spaces, (ii) κ-Minkowski spaces and (iii) κ-Snyder spaces. The corresponding star products are (i) associative and commutative (but non-local), (ii) associative and noncommutative and (iii) non-associative and non-commutative, respectively. Twisted symmetry algebras are considered. Transposed twists and left-right dual algebras are presented. Finally, some physical applications are discussed.

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Section: Outlook and Discussionmentioning

confidence: 99%

“…If the star product is associative, the twist operator Eq. (18) satisfies the cocycle condition in the Hopf algebroid sense and vice versa [47][48][49][50][51][52].…”

Section: Star Product and Twist Operatormentioning

confidence: 99%

Families of vector-like deformations of relativistic quantum phase spaces, twists and symmetries

Meljanac

Pikutić

2017

Eur. Phys. J. C

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“…The corresponding twist does not satisfy the cocycle condition in the Hopf algebra sense, but it does in the Hopf algebroid sense [31,35,36,37].Generally, for noncommutative coordinates (7), the corresponding twist in the Hopf algebroid approach is given by [23,38]…”

Section: κ-Deformed Relativistic Quantum Phase Spacementioning

confidence: 99%

“…As in the previous subsection, it is easily shown that using the relations between the two realizations and the homomorphism property of the coproduct, one obtains the same expressions. However, a crucial point in showing this for the coproduct of M N µν are the so-called tensor identities [31,35,36,37]. They can be calculated from the undeformed ones, R 0 = x µ ⊗ 1 − 1 ⊗ x µ = 0, using the twist operator.…”

Section: Relation Between Natural and Left Covariant Realizationsmentioning

confidence: 99%

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κ -deformed phase spaces, Jordanian twists, Lorentz-Weyl algebra, and dispersion relations

Meljanac

Mignemi

et al. 2019

Phys. Rev. D

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We consider κ-deformed relativistic quantum phase space and possible implementations of the Lorentz algebra. There are two ways of performing such implementations. One is a simple extension where the Poincaré algebra is unaltered, while the other is a general extension where the Poincaré algebra is deformed. As an example we fix the Jordanian twist and the corresponding realization of noncommutative coordinates, coproduct of momenta and addition of momenta. An extension with a one-parameter family of realizations of the Lorentz generators, dilatation and momenta closing the Poincaré-Weyl algebra is considered. The corresponding physical interpretation depends on the way the Lorentz algebra is implemented in phase space. We show how the spectrum of the relativistic hydrogen atom depends on the realization of the generators of the Poincaré-Weyl algebra. * Daniel.Meljanac@irb.hr † meljanac@irb.hr ‡ smignemi@unica.it § rina.strajn@unidu.hr

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Lie algebra type noncommutative phase spaces are Hopf algebroids

2016

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For a noncommutative configuration space whose coordinate algebra is the universal enveloping algebra of a finite dimensional Lie algebra, it is known how to introduce an extension playing the role of the corresponding noncommutative phase space, namely by adding the commuting deformed derivatives in a consistent and nontrivial way, therefore obtaining certain deformed Heisenberg algebra. This algebra has been studied in physical contexts, mainly in the case of the kappa-Minkowski space-time. Here we equip the entire phase space algebra with a coproduct, so that it becomes an instance of a completed variant of a Hopf algebroid over a noncommutative base, where the base is the enveloping algebra.Comment: uses kluwer.cls; v. 2: 25 pages, significant corrections, 3 authors; version 3: significant revision, 32 pages, corrections and added geometrical viewpoint and preliminaries on formal differential operators; version 4: final corrections and slightly improved readability; accepted in Letters in Mathematical Physic

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κ-Poincaré–Hopf algebra and Hopf algebroid structure of phase space from twist

Cited by 32 publications

References 86 publications

Families of vector-like deformations of relativistic quantum phase spaces, twists and symmetries

Families of vector-like deformations of relativistic quantum phase spaces, twists and symmetries

κ -deformed phase spaces, Jordanian twists, Lorentz-Weyl algebra, and dispersion relations

Lie algebra type noncommutative phase spaces are Hopf algebroids

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