with Stjepan Meljanac

A Snyder model generated by the noncommutative coordinates and Lorentz generators which close a Lie algebra can be equipped in a Hopf algebra structure. The Heisenberg double is described for the dual pair of generators: noncommutative Snyder coordinates and momenta (with non-coassociative coproducts). The phase space of the Snyder model is obtained as a result. 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 further Heisenberg doubles are considered, one with the conjugate tensorial momenta and another with the Lorentz matrices. Explicit formulae for all Heisenberg doubles are given.


with Paolo Aschieri and Andrzej Borowiec

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 κ-Minkowski spacetime in flat space in the presence of a Friedman-Lemaître-Robertson-Walker (FLRW) cosmological background. We obtain a new expression for the variation of the speed of light, depending linearly on the ratio Eph/ELV (photon energy / Lorentz violation scale), but also linearly on the cosmological time, the Hubble parameter and inversely proportional to the scale factor.

with Shahn Majid

We find and classify all bialgebras and Hopf algebras or `quantum groups’ of dimension 4 over the field F_2={0,1}. We summarise our results as a quiver, where the vertices are the inequivalent algebras and there is an arrow for each inequivalent bialgebra or Hopf algebra built from the algebra at the source of the arrow and the dual of the algebra at the target of the arrow. There are 314 distinct bialgebras, and among them 25 Hopf algebras with at most one of these from one vertex to another. We find a unique smallest noncommutative and noncocommutative one, which is moreover self-dual and resembles a digital version of u_q(sl_2). We also find a unique self-dual Hopf algebra in one anyonic variable x^4=0. For all our Hopf algebras we determine the integral and associated Fourier transform operator, viewed as a representation of the quiver. We also find all quasitriangular or `universal R-matrix’ structures on our Hopf algebras. These induce solutions of the Yang-Baxter or braid relations in any representation.

Journal of Mathematical Physics 61, 103510 (2020)


with Andrzej Borowiec, Stjepan Meljanac and Daniel Meljanac

We propose a new generalization of the Jordanian twist (building on the previous idea from [J.Phys.A50,26(2017),arXiv:1612.07984]). 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. Exponential formulae, used to obtain coproducts and star products, are presented with details.

SIGMA 15 (2019), 054, 22 pages

with Shahn Majid

We study parallelisable bimodule noncommutative Riemannian geometries in small dimensions, working over the field F2 of 2 elements and with coordinate algebras up to dimension n3. We find a rich moduli of examples for n=3 and top form degree 2, many of them not flat. Their coordinate algebras are commutative but their differentials are not. We also study the quantum Laplacian Δ=( , )d on our models and characterise when it has a massive mode.

Journal of Physics A: Mathematical and Theoretical


with Paolo Aschieri and Andrzej Borowiec

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 d’Alembert operator coincides with the quadratic Casimir of quantum translations and it is deformed as in Deformed Special Relativity theories. Also momenta, i.e. 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 phenomenomenological literature on the subject.

JHEP 1710 (2017) 152

with Shahn Majid

It is known that connected translation invariant n-dimensional noncommutative differentials dxi on the algebra k[x1,,xn] 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 [xμ,xν]=λΘμν where Θ is an antisymmetric matrix as well as to Lie algebras with pre-Lie algebra structures. We specialise the general theory to the field k= F2 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 n2 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 F2 we construct some of our algebras and associated structures by digital gates, opening up the possibility of ‘digital geometry’.

Journal of Mathematical Physics 59, 033505 (2018)

by Patrizia Vitale

We derive an expression for the k-Minkowski star product in d space-time dimensions as a symplectic reduction of a normal ordered star product of the Wick type on the tangent bundle T*d.

Proceedings of the Fourteenth Marcel Grossmann Meeting


with Stjepan Meljanac, Daniel Meljanac and Danijel Pikutic

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 κ-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.

J. Phys .A: Math. Theor., 50, 26 (2017)

with Andrzej Borowiec

Bialgebroids (resp. Hopf algebroids) are bialgebras (Hopf algebras) over noncommutative rings. Drinfeld twist techniques are particularly useful in (deformation) quantization of Lie algebras as well as underlying module algebras (=quantum spaces). Cross (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 bialgebroid from twisted bialgebra give rise to the same result in the case of normalized cocycle twist. This can be useful for better description of 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.

J. Phys. A: Math. Theor., 50, 5 (2016)

with A. Borowiec, T. Juric and S. Meljanac

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 κ-Minkowski type in four dimensions. The differential calculus corresponding to a class of Jordanian κ-deformations provide metrics which lead either to cosmological constant or spatial-curvature type solutions of non-vacuum Einstein equations. Among vacuum solutions one finds pp-waves.

Int. J. Geom. Methods Mod. Phys. 13, 1640005 (2016)


with Stijn J. van Tongeren

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.

Phys. Rev. D 93, 026008

with S. Meljanac and D. Pikutic

Twisted deformations of the conformal symmetry in the Hopf algebraic framework are constructed. The first one is obtained by Jordanian twist built up from dilatation and momenta generators. The second is the light-like κ-deformation of the Poincare algebra extended to the conformal algebra, obtained by twist corresponding to the extended Jordanian r-matrix. The κ-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, respectively. The differential realizations for κ-Minkowski coordinates, as well as their left-right dual counterparts, are also included.

Phys. Rev. D 92, 105015 (2015) [arXiv:1510.02389]

with P. Vitale

We derive an explicit expression for the star product reproducing the κ-Minkowski Lie algebra in any dimension n. The result is obtained by suitably reducing the Wick-Voros star product defined on C^d_θ with n=d+1. It is thus shown that the new star product can be obtained from a Jordanian twist.

J. Phys. A: Math. Theor. 48 (2015) 445202 [arXiv:1507.03523]


with A. Borowiec

We extend our previous study of Hopf-algebraic kappa-deformations of all inhomogeneous orthogonal Lie algebras iso(g) as written in a tensorial and unified form. Such deformations are determined by a vector 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.

SIGMA 10, 107 (2014) [arXiv:1404.2916]

with M. Dimitrijevic and L. Jonke

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.

SIGMA 10 , 063 (2014) [arXiv:1403.1857]


with A. Borowiec and J. Lukierski

We demonstrate that the coproduct of D = 2 and D = 4 quantum kappa-Poincare algebras in a classical algebra basis cannot be obtained by a cochain twist depending only on Poincare algebra generators. We also argue that the nonexistence of such a twist does not imply the nonexistence of a universal R-matrix.

J. Phys. A: Math. Theor. 47, 405203 (2014) [arXiv:1312.7807]

with A. Borowiec

In this paper we provide universal formulae 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 formulae depend on the choice of an additional vector field which parameterizes classical r-matrices. Nonequivalent deformations are then labeled by the corresponding type of stability subgroups. For the Lorentzian signature it covers three (nonequivalent) Hopf-algebraic deformations: time-like, space-like (aka tachyonic) and light-like (aka light cone) quantizations of the Poincare algebra. In the first case this vector can be identified with relativistic observer. Finally the existence of the so-called Majid-Ruegg (non-classical) basis is reconsidered.

Eur. Phys. J. C 74, 2812 (2014) [arXiv:1311.4499]

This review presents noncommutative spacetimes as one of the approaches to Planck scale physics, with the main assumption that at this energy scale spacetime becomes quantized. Spacetime coordinates become noncommutative as observables in Quantum Mechanics. The basic elements of Drinfeld twist deformation theory are reminded. The Hopf algebra language provides natural framework for deformed relativistic symmetries which constitute Quantum Group of symmetry and noncommutative spacetime is in fact Hopf module algebra. The notion of realization for noncommutative coordinates in terms of di erential operators is also presented.

J. Phys.: Conf. Ser. 442 012039 (2013)


We show that the different realizations for momentum sector of kappa-Poincare Hopf algebra can be interpreted in the framework of a curved momentum space leading to the relativity of locality. The mass of a particle seems to be realization independent (up to linear order in deformation parameter l), therefore it indicates the existence of a universal Casimir element for a wide class of realizations. On the other hand, the time delay formula clearly shows a dependence on the choice of realization.

Phys. Rev. D 87, 125009 (2013) [arXiv:1210.6814]

We propose a generalized description for the kappa-Poincare Hopf algebra as symmetry quantum group of underlying kappa-Minkowski spacetime. We investigate all the possible implementations of (deformed) Lorentz algebras which are compatible with the given choice of kappa-Minkowski algebra realization. For the given realization of kappa-Minkowski spacetime there is a unique kappa-Poincare Hopf algebra with undeformed Lorentz algebra. We have constructed a two-parameter family of deformed Lorentz generators with kappa-Poincare algebras which are related to kappa-Poincare Hopf algebra with undeformed Lorentz algebra. Known bases of kappa-Poincare Hopf algebra are obtained as special cases. We have also presented an infinite class of basis of kappa-deformed igl(1,3) Hopf algebra which are compatible with the kappa-Minkowski spacetime. Some physical applications are briefly discussed.

Physics Letters B 711 (2012) 122-127; [arXiv:1202.3305]

with A. Borowiec

We are focused on detailed analysis of the Weyl-Heisenberg algebra in the framework of bicrossproduct construction. We argue that however it is not possible to introduce full bialgebra structure in this case, it is possible to introduce non-counital bialgebra counterpart of this construction. Some remarks concerning bicrossproduct basis for kappa-Poincare Hopf algebra are also presented.

J. Phys.: Conf. Ser. 343 012090 (2012)


[PhD Thesis]

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.

with A. Borowiec

We briefly analyze some general questions concerning the twist deformation of the Heisenberg double. We reconsider Heisenberg doubles based on quantized Poincare (Hopf) algebras as illustrative examples.

Theoretical and Mathematical Physics 169 (Issue: 2), 1620 (2011)


with A. Borowiec

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-Poincare 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.

SIGMA 6, 086 (2010) [arXiv:1005.4429]

with A. Borowiec, Kumar S. Gupta and S. Meljanac

We compare two versions of deformed dispersion relations (energy vs momenta and momenta vs energy) and the corresponding time delay up to the second order accuracy in the quantum gravity scale (deformation parameter). A general framework describing modified dispersion relations and time delay with respect to different noncommutative kappa -Minkowski spacetime realizations is firstly proposed here and it covers all the cases introduced in the literature. It is shown that some of the realizations provide certain bounds on quadratic corrections, i.e. on quantum gravity scale, but it is not excluded in our framework that quantum gravity scale is the Planck scale. We also show how the coefficients in the dispersion relations can be obtained through a multiparameter fit of the gamma ray burst (GRB) data.

EPL 92, 20006 (2010) [arXiv:0912.3299]


with A. Borowiec

Several issues concerning quantum kappa-Poincare algebra are discussed and reconsidered here. We propose two different formulations of kappa-Poincare quantum algebra. Firstly we present a complete Hopf algebra formulae of kappa-Poincare in classical Poincare basis. Further by adding one extra generator, which modifies the classical structure of Poincare algebra, we eliminate non polynomial functions in the kappa-parameter. Hilbert space representations of such algebras make Doubly Special Relativity (DSR) similar to the Stueckelberg’s version of (proper-time) relativistic Quantum Mechanics.

J. Phys. A: Math. Theor. 43, 045203 (2010) [arXiv:0903.5251]

with A. Borowiec

    Two one-parameter families of twists providing kappa-Minkowski * -product deformed spacetime are considered: Abelian and Jordanian. We compare the derivation of quantum Minkowski space from two perspectives. The first one is the Hopf module algebra point of view, which is strictly related with Drinfeld’s twisting tensor technique. The other one relies on an appropriate extension of “deformed realizations” of nondeformed Lorentz algebra by the quantum Minkowski algebra. This extension turns out to be de Sitter Lie algebra. We show the way both approaches are related. The second path allows us to calculate deformed dispersion relations for toy models ensuing from different twist parameters. In the Abelian case one recovers kappa-Poincar’e dispersion relations having numerous applications in doubly special relativity. Jordanian twists provide a new type of dispersion relations which in the minimal case (related to Weyl-Poincare algebra) takes an energy-dependent linear mass deformation form.

Phys. Rev. D 79, 045012 (2009) [arXiv:0812.0576]