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Recently it was shown that by using two different realizations of $\hat{o}(1,4)$ Lie algebra one can describe one-parameter standard Snyder model and two-parameter $\kappa$-deformed Snyder model. In this paper, by using the generalized Born duality and Jacobi identities we obtain from the $\kappa$-deformed Snyder model the doubly $\kappa$-deformed Yang model which provides the new class of quantum relativistic phase spaces. These phase spaces contain as subalgebras the $\kappa$-deformed Minkowski space-time as well as quantum $\tilde{\kappa}$-deformed fourmomenta and are depending on five independent parameters. Such a large class of quantum phase spaces can be described in $D=4$ by particular realizations of $\hat{o}(1,5)$ algebra, what illustrates the property that in noncommutative geometry different $D=4$ physical models may be described by various realizations of the same algebraic structure. Finally, in the last Section we propose two new ways of generalizing Yang models: by introducing $\hat o(1,3+2N)$ algebras ($N=1,2\ldots$) we provide internal symmetries $O(N)$ symmetries in Kaluza-Klein extended Yang model, and by replacing the classical $\hat{o}(1,5)$ algebras which describe the algebraic structure of Yang models by $\hat o(1,5)$ quantum groups with suitably chosen nonprimitive coproducts., Comment: 15 pages, Contribution to PoS Corfu 2023

We propose the doubly $\kappa$-dependent Yang quantum phase space which describes the generalization of $D = 4$ Yang model. We postulate that such model is covariant under the generalized Born map, what permits to derive this new model from the earlier proposed $\kappa$-Snyder model. Our model of $D=4$ relativistic Yang quantum phase space depends on five deformation parameters which form two Born map-related dimensionful pairs: $(M,R)$ specifying the standard Yang model and $(\kappa,\tilde{\kappa})$ characterizing the Born-dual $\kappa$-dependence of quantum space-time and quantum fourmomenta sectors; fifth parameter $\rho$ is dimensionless and Born-selfdual. In the last section, we propose the Kaluza-Klein generalization of $D=4$ Yang model and the new quantum Yang models described algebraically by quantum-deformed $\hat{o}(1,5)$ algebras., Comment: 10 pages; published version

This study investigates possibility of placing bounds on the parameters, arising from the non-commutative Snyder space-time model and Generalized Uncertainty Principle (GUP) approach, by utilizing seismic data. We investigate the dependence of constraints on the type of realization used for the quantum phase space. Results indicate improved bounds compared to prior studies, with the model parameter $\beta_0$ constrained to be less than $5.2\times 10^{44}$ for certain choice of realizations. This approach demonstrates the potential for using Earth's empirical data to refine constraints on GUP parameters., Comment: 8 pages, 3 figures, typos corrected

We introduce the perturbative $\hbar$-power series ($\hbar$ = Planck constant) providing the algebraic solutions of $D=4$ quantum Snyder and Yang models which describe relativistic quantum space-times and Lorentz-covariant quantum phase spaces. We argue that if in these series the zero order ($\hbar $-independent) terms are non-vanishing they describe the spontaneous symmetry breaking (SSB) parameters of Lie-algebraic symmetries which characterize the considered models ($D=4$ dS symmetry in Snyder and $D=5$ dS symmetry in Yang cases). The consecutive terms in $\hbar$-power series can be calculated explicitly if we supplement the SSB order parameters (Nambu-Goldstone or NG modes) by dual set of commutative momenta, which together define the canonical tensorial Heisenberg algebra., Comment: Contribution to PoS Corfu 2022

We investigate the application of an equation of state that incorporates corrections derived from the Snyder model (and the Generalized Uncertainty Principle) to describe the behavior of matter in a low-mass star. Remarkably, the resulting equations exhibit striking similarities to those arising from modified Einstein gravity theories. By modeling matter with realistic considerations, we are able to more effectively constrain the theory parameters, surpassing the limitations of existing astrophysical bounds. The bound we obtain is $\beta_0 \leq 1.36 \times 10^{48}$. We underline the significance of realistic matter modeling in order to enhance our understanding of effects arising in quantum gravity phenomenology and implications of quantum gravitational corrections in astrophysical systems., Comment: 11 pages, 2 figures, 1 table, version compatible with Eur.Phys.J.C

We investigate the impact of the deformed phase space associated with the quantum Snyder space on microphysical systems. The general Fermi-Dirac equation of state and specific corrections to it are derived. We put emphasis on non-relativistic degenerate Fermi gas as well as on the temperature-finite corrections to it. Considering the most general one-parameter family of deformed phase spaces associated with the Snyder model allows us to study whether the modifications arising in physical effects depend on the choice of realization. It turns out that we can distinguish three different cases with radically different physical consequences., Comment: 15 pages; version accepted for publication in CQG

We propose $\hbar$-expansions as perturbative solutions of quantum extended Snyder and Yang models, with $\hbar$-independent classical zero-th order terms responsible for the spontaneous breaking of $D=4$ and $D=5$ de Sitter symmetries. In such models, with algebraic basis spanned by $\hat o(D,1)$ Lie algebra generators, we relate the vacuum expectation values (VEV) of the spontaneously broken generators with the Abelian set of ten (Snyder, $D=4$) or fifteen (Yang, $D=5$) antisymmetric tensorial generalized coordinates, which are also used as zero order input for obtaining the perturbative solutions of quantum extended Snyder and Yang models. In such a way we will attribute to these Abelian generalized coordinates the physical meaning of the order parameters describing spontaneous symmetry breaking (SSB). It appears that the consecutive terms in $\hbar$-power series can be calculated explicitly if we supplement the SSB order parameters by the dual set of tensorial commutative momenta., Comment: 11 pages; published version

We describe, in an algebraic way, the $\kappa$-deformed extended Snyder models, that depend on three parameters $\beta, \kappa$ and $\lambda$, which in a suitable algebra basis are described by the de Sitter algebras ${o}(1,N)$. The commutation relations of the algebra contain a parameter $\lambda$, which is used for the calculations of perturbative expansions. For such $\kappa$-deformed extended Snyder models we consider the Heisenberg double with dual generalized momenta sector, and provide the respective generalized quantum phase space depending on three parameters mentioned above. Further, we study for these models an alternative Heisenberg double, with the algebra of functions on de Sitter group. In both cases we calculate the formulae for the cross commutation relations between generalized coordinate and momenta sectors, at linear order in $\lambda$. We demonstrate that in the commutators of quantum space-time coordinates and momenta of the quantum-deformed Heisenberg algebra the terms generated by $\kappa$-deformation are dominating over $\beta$-dependent ones for small values of $\lambda$., Comment: 11 pages, no figures, accepted version

A Snyder model generated by the noncommutative coordinates and Lorentz generators close a Lie algebra. The application of the Heisenberg double construction is investigated for the Snyder coordinates and momenta generators. It leads to the phase space of the Snyder model. Further, the extended Snyder algebra is constructed by using the Lorentz algebra, in one dimension higher. The dual pair of extended Snyder algebra and extended Snyder group is then formulated. Two Heisenberg doubles are considered, one with the conjugate tensorial momenta and another with the Lorentz matrices. Explicit formulae for all Heisenberg doubles are given., Comment: 19 pages, no figures, 1 Appendix; version accepted for publication

We study noncommutative deformations of the wave equation in curved backgrounds and discuss the modification of the dispersion relations due to noncommutativity combined with curvature of spacetime. Our noncommutative differential geometry approach is based on Drinfeld twist deformation, and can be implemented for any twist and any curved background. We discuss in detail the Jordanian twist $-$giving $\kappa$-Minkowski spacetime in flat space$-$ in the presence of a Friedman-Lema\^{i}tre-Robertson-Walker (FLRW) cosmological background. We obtain a new expression for the variation of the speed of light, depending linearly on the ratio $E_{ph}/E_{LV}$ (photon energy / Lorentz violation scale), but also linearly on the cosmological time, the Hubble parameter and inversely proportional to the scale factor., Comment: 20 pages. New version: 23 pages, added 4-dim. dispersion relations and numerical estimates

We propose a new generalisation of the Jordanian twist (building on the previous idea from [Meljanac S., Meljanac D., Pachol A., Pikutic D., J. Phys. A: Math. Theor. 50 (2017), 265201, 11 pages]). Obtained this way, the family of the Jordanian twists allows for interpolation between two simple Jordanian twists. This new version of the twist provides an example of a new type of star product and the realization for noncommutative coordinates. Real forms of new Jordanian deformations are also discussed. Exponential formulae, used to obtain coproducts and star products, are presented with details., Comment: the final (journal; extended and revised) version

We study bimodule quantum Riemannian geometries over the field $\Bbb F_2$ of two elements as the extreme case of a finite-field adaptation of noncommutative-geometric methods for physics. We classify all parallelisable such geometries for coordinate algebras up to vector space dimension $n\le 3$, finding a rich moduli of examples for $n=3$ and top form degree 2, including many that are not flat. Their coordinate algebras are commutative but their differentials are not. We also study the quantum Laplacian $\Delta=(\ ,\ )\nabla{\rm d}$ on our models and characterise when it has a massive eigenvector., Comment: 37 pages latex, one figure. Corrected some of the curvature formulae in line with final version in press

We revisit the notion of quantum Lie algebra of symmetries of a noncommutative spacetime, its elements are shown to be the generators of infinitesimal transformations and are naturally identified with physical observables. Wave equations on noncommutative spaces are derived from a quantum Hodge star operator. This general noncommutative geometry construction is then exemplified in the case of k-Minkowski spacetime. The corresponding quantum Poincare'-Weyl Lie algebra of infinitesimal translations, rotations and dilatations is obtained. The d'Alembert wave operator coincides with the quadratic Casimir of quantum translations and it is deformed as in Deformed Special Relativity theories. Also momenta (infinitesimal quantum translations) are deformed, and correspondingly the Einstein-Planck relation and the de Broglie one. The energy-momentum relations (dispersion relations) are consequently deduced. These results complement those of the phenomenological literature on the subject., Comment: 30 pages. Revised version further motivating Poincare'-Weyl (conformal) symmetry. Added explicit proof of covariance of massive scalar fields wave equation under quantum translations (eq. (76)). To appear in JHEP

It is known that connected translation invariant $n$-dimensional noncommutative differentials $d x^i$ on the algebra $k[x^1,\cdots,x^n]$ of polynomials in $n$-variables over a field $k$ are classified by commutative algebras $V$ on the vector space spanned by the coordinates. This data also applies to construct differentials on the Heisenberg algebra `spacetime' with relations $[x^\mu,x^\nu]=\lambda\Theta^{\mu\nu}$ where $ \Theta$ is an antisymmetric matrix as well as to Lie algebras with pre-Lie algebra structures. We specialise the general theory to the field $k={\ \mathbb{F}}_2$ of two elements, in which case translation invariant metrics (i.e. with constant coefficients) are equivalent to making $V$ a Frobenius algebras. We classify all of these and their quantum Levi-Civita bimodule connections for $n=2,3$, with partial results for $n=4$. For $n=2$ we find 3 inequivalent differential structures admitting 1,2 and 3 invariant metrics respectively. For $n=3$ we find 6 differential structures admitting $0,1,2,3,4,7$ invariant metrics respectively. We give some examples for $n=4$ and general $n$. Surprisingly, not all our geometries for $n\ge 2$ have zero quantum Riemann curvature. Quantum gravity is normally seen as a weighted `sum' over all possible metrics but our results are a step towards a deeper approach in which we must also `sum' over differential structures. Over ${\mathbb{F}}_2$ we construct some of our algebras and associated structures by digital gates, opening up the possibility of `digital geometry'., Comment: 30 pages LATEX 2 pdf figure files

In this paper, we propose a simple generalization of the locally r-symmetric Jordanian twist, resulting in the one-parameter family of Jordanian twists. All the proposed twists differ by the coboundary twists and produce the same Jordanian deformation of the corresponding Lie algebra. They all provide the $\kappa$-Minkowski spacetime commutation relations. Constructions from noncommutative coordinates to the star product and coproduct, and from the star product to the coproduct and the twist are presented. The corresponding twist in the Hopf algebroid approach is given. Our results are presented symbolically by a diagram relating all of the possible constructions., Comment: 12 pages

Bialgebroids (resp. Hopf algebroids) are bialgebras (Hopf algebras) over noncommutative rings. Drinfeld twist techniques are particularly useful in the (deformation) quantization of Lie algebras as well as underlying module algebras (=quantum spaces). Smash product construction combines these two into the new algebra which, in fact, does not depend on the twist. However, we can turn it into bialgebroid in the twist dependent way. Alternatively, one can use Drinfeld twist techniques in a category of bialgebroids. We show that both techniques indicated in the title: twisting of a bialgebroid or constructing a bialgebroid from the twisted bialgebra give rise to the same result in the case of normalized cocycle twist. This can be useful for better description of a quantum deformed phase space. We argue that within this bialgebroid framework one can justify the use of deformed coordinates (i.e. spacetime noncommutativity) which are frequently postulated in order to explain quantum gravity effects., Comment: 13 pages, version accepted for publication

In this paper, we perform a parallel analysis to the model proposed in [25]. By considering the central co-tetrad (instead of the central metric), we investigate the modifications in the gravitational metrics coming from the noncommutative spacetime of the $\kappa$-Minkowski type in four dimensions. The differential calculus corresponding to a class of Jordanian $\kappa$-deformations provides metrics, which lead either to cosmological constant or spatial curvature type solutions of non-vacuum Einstein equations. Among vacuum solutions, we find pp-wave type., Comment: 12 pages, published version, title modified

We discuss a quantum deformation of the Green-Schwarz superstring on flat space, arising as a contraction limit of the corresponding deformation of AdS_5 x S^5. This contraction limit turns out to be equivalent to a previously studied limit that yields the so-called mirror model - the model obtained from the light cone gauge fixed AdS_5 x S^5 string by a double Wick rotation. Reversing this logic, the AdS_5 x S^5 superstring is the double Wick rotation of a quantum deformation of the flat space superstring. This quantum deformed flat space string realizes symmetries of timelike kappa-Poincare type, and is T dual to dS_5 x H^5, indicating interesting relations between symmetry algebras under T duality. Our results directly extend to AdS_2 x S^2 x T^6 and AdS_3 x S^3 x T^4, and beyond string theory to many (semi)symmetric space coset sigma models, such as for example a deformation of the four dimensional Minkowski sigma model with timelike kappa-Poincare symmetry. We also discuss possible null and spacelike deformations., Comment: v3, published version, 12 pages

Twisted deformations of the conformal symmetry in the Hopf algebraic framework are constructed. The first one is obtained by a Jordanian twist built up from dilatation and momenta generators. The second is the light-like $\kappa$-deformation of the Poincare algebra extended to the conformal algebra, obtained by a twist corresponding to the extended Jordanian r-matrix. The $\kappa$-Minkowski spacetime is covariant quantum space under both of these deformations. The extension of the conformal algebra by the noncommutative coordinates is presented in two cases. The differential realizations for $\kappa$-Minkowski coordinates, as well as their left-right dual counterparts, are also included., Comment: 8 pages; version accepted for publication in Phys. Rev. D

We derive an explicit expression for the star product reproducing the $\kappa$-Minkowski Lie algebra in any dimension $n$. The result is obtained by suitably reducing the Wick-Voros star product defined on $\mathbb{C}^{d}_\theta$ with $n=d+1$. It is thus shown that the new star product can be obtained from a Jordanian twist., Comment: published version

We extend our previous study of Hopf-algebraic $\kappa$-deformations of all inhomogeneous orthogonal Lie algebras ${\rm iso}(g)$ as written in a tensorial and unified form. Such deformations are determined by a vector $\tau$ which for Lorentzian signature can be taken time-, light- or space-like. We focus on some mathematical aspects related to this subject. Firstly, we describe real forms with connection to the metric's signatures and their compatibility with the reality condition for the corresponding $\kappa$-Minkowski (Hopf) module algebras. Secondly, $h$-adic vs $q$-analog (polynomial) versions of deformed algebras including specialization of the formal deformation parameter $\kappa$ to some numerical value are considered. In the latter the general covariance is lost and one deals with an orthogonal decomposition. The last topic treated in this paper concerns twisted extensions of $\kappa$-deformations as well as the description of resulting noncommutative spacetime algebras in terms of solvable Lie algebras. We found that if the type of the algebra does not depend on deformation parameters then specialization is possible., Comment: new extended version with new material added and with title changed

We review the application of twist deformation formalism and the construction of noncommutative gauge theory on $\kappa$-Minkowski space-time. We compare two different types of twists: the Abelian and the Jordanian one. In each case we provide the twisted differential calculus and consider ${U}(1)$ gauge theory. Different methods of obtaining a gauge invariant action and related problems are thoroughly discussed.

We demonstrate that the coproduct of D=2 and D=4 quantum kappa-Poincare algebra in classical algebra basis can not be obtained by the cochain twist depending only on Poincare algebra generators. We also argue that nonexistence of such a twist does not imply the nonexistence of universal R-matrix., Comment: p.13, v3: extended version, with enlarged final Sect.4 and added Appendix, to appear in Journ. Phys.A

In this paper we provide universal formulas describing Drinfeld-type quantization of inhomogeneous orthogonal groups determined by a metric tensor of an arbitrary signature living in a spacetime of arbitrary dimension. The metric tensor does not need to be in diagonal form and kappa-deformed coproducts are presented in terms of classical generators. It opens the possibility for future applications in deformed general relativity. The formulas depend on the choice of an additional vector field which parameterizes classical r-matrices. Non-equivalent deformations are then labeled by the corresponding type of stability subgroups. For the Lorentzian signature it covers three (non-equivalent) Hopf-algebraic deformations: time-like, space-like (a.k.a. tachyonic) and light-like (a.k.a. light-cone) quantizations of the Poincare algebra. Finally the existence of the so-called Majid-Ruegg (non-classical) basis is reconsidered., Comment: 9 pages, typos corrected, references added, Conclusions substantially changed, replaced to match published version

We consider different realizations for the momentum sector of kappa-Poincare Hopf algebra, which is associated with a curved momentum space. We show that the notion of the particle mass as introduced recently by Amelino-Camelia et al. in the context of relative-locality is realization independent for a wide class of realizations, up to linear order in deformation parameter l. On the other hand, the time delay formula clearly shows a dependence on the choice of realization., Comment: 12 pages; extended version of the paper published in Phys. Rev. D 87, 125009 (2013)

The dissertation presents possibilities of applying noncommutative spacetimes description, particularly kappa-deformed Minkowski spacetime and Drinfeld's deformation theory, as a mathematical formalism for Doubly Special Relativity theories (DSR), which are thought as phenomenological limit of quantum gravity theory. Deformed relativistic symmetries are described within Hopf algebra language. In the case of (quantum) kappa-Minkowski spacetime the symmetry group is described by the (quantum) kappa-Poincare Hopf algebra. Deformed relativistic symmetries were used to construct the DSR algebra, which unifies noncommutative coordinates with generators of the symmetry algebra. It contains the deformed Heisenberg-Weyl subalgebra. It was proved that DSR algebra can be obtained by nonlinear change of generators from undeformed algebra. We show that the possibility of applications in Planck scale physics is connected with certain realizations of quantum spacetime, which in turn leads to deformed dispersion relations., Comment: PhD thesis, 110 pages

Some classes of Deformed Special Relativity (DSR) theories are reconsidered within the Hopf algebraic formulation. For this purpose we shall explore a minimal framework of deformed Weyl-Heisenberg algebras provided by a smash product construction of DSR algebra. It is proved that this DSR algebra, which uniquely unifies $\kappa$-Minkowski spacetime coordinates with Poincar\'e generators, can be obtained by nonlinear change of generators from undeformed one. Its various realizations in terms of the standard (undeformed) Weyl-Heisenberg algebra opens the way for quantum mechanical interpretation of DSR theories in terms of relativistic (St\"uckelberg version) Quantum Mechanics. On this basis we review some recent results concerning twist realization of $\kappa$-Minkowski spacetime described as a quantum covariant algebra determining a deformation quantization of the corresponding linear Poisson structure. Formal and conceptual issues concerning quantum $\kappa$-Poincar\'e and $\kappa$-Minkowski algebras as well as DSR theories are discussed. Particularly, the so-called "$q$-analog" version of DSR algebra is introduced. Is deformed special relativity quantization of doubly special relativity remains an open question. Finally, possible physical applications of DSR algebra to description of some aspects of Planck scale physics are shortly recalled.