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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

The relativistic Lorentz-covariant quantum space-times obtained by Snyder can be described by the coset generators of (anti) de-Sitter algebras. Similarly, the Lorentz-covariant quantum phase spaces introduced by Yang, which contain additionally quantum curved fourmomenta and quantum-deformed relativistic Heisenberg algebra, can be defined by suitably chosen coset generators of conformal algebras. We extend such algebraic construction to the respective superalgebras, which provide quantum Lorentz-covariant superspaces (SUSY Snyder model) and indicate also how to obtain the quantum relativistic phase superspaces (SUSY Yang model). In last Section we recall briefly other ways of deriving quantum phase (super)spaces and we compare the spinorial Snyder type models defining bosonic or fermionic quantum-deformed spinors., 15 pages, LaTeX

We consider the generalized (10+10)-dimensional D=4 quantum phase spaces containing translational and Lorentz spin sectors associated with the dual pair of twist-quantized Poincare Hopf algebra $\mathbb{H}$ and quantum Poincare Hopf group $\widehat{\mathbb{G}}$. Two Hopf algebroid structures of generalized phase spaces with spin sector will be investigated: first one $% \mathcal{H}^{(10,10)}$ describing dynamics on quantum group algebra $% \widehat{\mathbb{G}}$ provided by the Heisenberg double algebra $\mathcal{HD=% }\mathbb{H}\rtimes \widehat{\mathbb{G}}$, and second, denoted by $\mathcal{% \tilde{H}}^{(10,10)}$, describing twisted Hopf algebroid with base space containing twisted noncommutative Minkowski space $\hat{x}_{\mu }$. We obtain the first explicit example of Hopf algebroid structure of relativistic quantum phase space which contains quantum-deformed Lorentz spin sector., Comment: 15 pages

In the current version of QCD the quarks are described by ordinary Dirac fields, organized in the following internal symmetry multiplets: the $SU(3)$ colour, the $SU(2)$ flavour, and broken $SU(3)$ providing the family triplets. \noindent In this paper we argue that internal and external (i.e. space-time) symmetries are entangled at least in the colour sector in order to introduce the spinorial quark fields in a way providing all the internal quark's degrees of freedom which do appear in the Standard Model. Because the $SU(3)$ colour algebra is endowed with natural $Z_3$-graded discrete automorphisms, in order to introduce entanglement the $Z_3$-graded version of Lorentz and Poincar\'e algebras with their realizations are considered. The colour multiplets of quarks are described by $12$-component colour Dirac equations, with a $Z_3$-graded triplet of masses (one real and a Lee-Wick complex conjugate pair). We argue that all quarks in the Standard Model can be described by the $72$-component master quark sextet of $12$-component coloured Dirac fields., Comment: 39 pages, 3 figures. This version is conformal with the revised version published in the journal (Nucl. Phys. B)

We consider two quantum phase spaces which can be described by two Hopf algebroids linked with the well-known ${\ensuremath{\theta}}_{\ensuremath{\mu}\ensuremath{\nu}}$-deformed $D=4$ Poincar\'e-Hopf algebra $\mathbb{H}$. The first algebroid describes ${\ensuremath{\theta}}_{\ensuremath{\mu}\ensuremath{\nu}}$-deformed relativistic phase space with canonical NC space-time (constant ${\ensuremath{\theta}}_{\ensuremath{\mu}\ensuremath{\nu}}$ parameters) and the second one incorporates dual to $\mathbb{H}$ quantum ${\ensuremath{\theta}}_{\ensuremath{\mu}\ensuremath{\nu}}$-deformed Poincar\'e-Hopf group algebra $\mathbb{G}$, which contains noncommutative space-time translations given by $\mathrm{\ensuremath{\Lambda}}$-dependent ${\mathrm{\ensuremath{\Theta}}}_{\ensuremath{\mu}\ensuremath{\nu}}$ parameters ($\mathrm{\ensuremath{\Lambda}}\ensuremath{\equiv}{\mathrm{\ensuremath{\Lambda}}}_{\ensuremath{\mu}\ensuremath{\nu}}$ parametrize classical Lorentz group). The canonical ${\ensuremath{\theta}}_{\ensuremath{\mu}\ensuremath{\nu}}$-deformed space-time algebra and its quantum phase space extension is covariant under the quantum Poincar\'e transformations described by $\mathbb{G}$. We will also comment on the use of Hopf algebroids for the description of multiparticle structures in quantum phase spaces.

In our previous paper [1] we obtained a full classification of nonequivalent quasitriangular quantum deformations for the complex D = 4 Euclidean Lie symmetry o ( 4 ; C ) . The result was presented in the form of a list consisting of three three-parameter, one two-parameter and one one-parameter nonisomorphic classical r-matrices which provide ‘directions’ of the nonequivalent quantizations of o ( 4 ; C ) . Applying reality conditions to the complex o ( 4 ; C ) r-matrices we obtained the nonisomorphic classical r-matrices for all possible real forms of o ( 4 ; C ) : Euclidean o ( 4 ) , Lorentz o ( 3 , 1 ) , Kleinian o ( 2 , 2 ) and quaternionic o ⋆ ( 4 ) Lie algebras. In the case of o ( 4 ) and o ( 3 , 1 ) real symmetries these r-matrices give the full classifications of the inequivalent quasitriangular quantum deformations, however for o ( 2 , 2 ) and o ⋆ ( 4 ) the classifications are not full. In this paper we complete these classifications by adding three new three-parameter o ( 2 , 2 ) -real r-matrices and one new three-parameter o ⋆ ( 4 ) -real r-matrix. All nonisomorphic classical r-matrices for all real forms of o ( 4 ; C ) are presented in the explicit form what is convenient for providing the quantizations. We will mention also some applications of our results to the deformations of space–time symmetries and string σ-models.

Following the recently obtained complete classification of quantum-deformed $\mathfrak{o}(4)$, $\mathfrak{o}(3,1)$ and $\mathfrak{o}(2,2)$ algebras, characterized by classical $r$-matrices, we study their inhomogeneous $D = 3$ quantum IW contractions (i.e. the limit of vanishing cosmological constant), with Euclidean or Lorentzian signature. Subsequently, we compare our results with the complete list of $D = 3$ inhomogeneous Euclidean and $D = 3$ Poincar\'{e} quantum deformations obtained by P.~Stachura. It turns out that the IW contractions allow us to recover all Stachura deformations. We further discuss the applicability of our results in the models of 3D quantum gravity in the Chern-Simons formulation (both with and without the cosmological constant), where it is known that the relevant quantum deformations should satisfy the Fock-Rosly conditions. The latter deformations in part of the cases are associated with the Drinfeld double structures, which also have been recently investigated in detail., Comment: 28 pages, v2 analysis of applicability of r-matrices in 3D gravity with $\Lambda \neq 0$ in Subsec. VI.C, generalization of procedure of quantum IW contractions in Subsec. IV.E, some conventions changed, redundant formulae removed, references added

We study Yang-Baxter sigma models with deformed 4D Minkowski spacetimes arising from classical $r$-matrices associated with $\kappa$-deformations of the Poincar\'e algebra. These classical $\kappa$-Poincar\'e $r$-matrices describe three kinds of deformations: 1) the standard deformation, 2) the tachyonic deformation, and 3) the light-cone deformation. For each deformation, the metric and two-form $B$-field are computed from the associated $r$-matrix. The first two deformations, related to the modified classical Yang-Baxter equation, lead to T-duals of dS$_4$ and AdS$_4$\,, respectively. The third deformation, associated with the homogeneous classical Yang-Baxter equation, leads to a time-dependent pp-wave background. Finally, we construct a Lax pair for the generalized $\kappa$-Poincar\'e $r$-matrix that unifies the three kinds of deformations mentioned above as special cases., Comment: 31 pages, v2: some clarifications and references added, published version

We employ new calculational technique and present complete list of classical r-matrices for D = 4 complex homogeneous orthogonal Lie algebra o ( 4 ; C ) , the rotational symmetry of four-dimensional complex space–time. Further applying reality conditions we obtain the classical r-matrices for all possible real forms of o ( 4 ; C ) : Euclidean o ( 4 ) , Lorentz o ( 3 , 1 ) , Kleinian o ( 2 , 2 ) and quaternionic o ⋆ ( 4 ) Lie algebras. For o ( 3 , 1 ) we get known four classical D = 4 Lorentz r-matrices, but for other real Lie algebras (Euclidean, Kleinian, quaternionic) we provide new results and mention some applications.

We propose a modification of standard QCD description of the colour triplet of quarks describing quark fields endowed with colour degree of freedom by introducing a 12-component colour generalization of Dirac spinor, with built-in Z_3 grading playing an important algebraic role in quark confinement. In "colour Dirac equations" the SU(3) colour symmetry is entangled with the Z_3-graded generalization of Lorentz symmetry, containing three 6-parameter sectors related by Z_3 maps. The generalized Lorentz covariance requires simultaneous presence of 24 colour Dirac multiplets, which lead to the description of all internal symmetries of quarks: besides SU(3) \times SU(2) \times U(1), the flavour symmetries and three quark families., LaTeX, 14 pages, no figures; v2 improved LaTeX format and small corrections in Sect 3, v3 changes in Introduction: mainly formula (2) deleted and formula (3) shifted to Sect.2;v4 small change in formula (27) and exposed role of Lorentz-covariant doublets of colour multiplets v5 extended version by three pages, new content of Sect 4

We consider the general D=4 (10+10)-dimensional kappa-deformed quantum phase space as given by Heisenberg double \mathcal{H} of D=4 kappa-deformed Poincare-Hopf algebra H. The standard (4+4) -dimensional kappa - deformed covariant quantum phase space spanned by kappa - deformed Minkowski coordinates and commuting momenta generators ({x}_{\mu },{p}_{\mu }) is obtained as the subalgebra of \mathcal{H}. We study further the property that Heisenberg double defines particular quantum spaces with Hopf algebroid structure. We calculate by using purely algebraic methods the explicite Hopf algebroid structure of standard kappa - deformed quantum covariant phase space in Majid-Ruegg bicrossproduct basis. The coproducts for Hopf algebroids are not unique, determined modulo the coproduct gauge freedom. Finally we consider the interpretation of the algebraic description of quantum phase spaces as Hopf algebroids., Comment: 14 pages,v4; this version appeared in Physics Letters B

We consider new Abelian twists of Poincare algebra describing non-symmetric generalization of the ones given in [1], which lead to the class of Lie-deformed quantum Minkowski spaces. We apply corresponding twist quantization in two ways: as generating quantum Poincare-Hopf algebra providing quantum Poincare symmetries, and by considering the quantization which provides Hopf algebroid describing the class of quantum relativistic phase spaces with built-in quantum Poincare covariance. If we assume that Lorentz generators are orbital i.e.do not describe spin degrees of freedom, one can embed the considered generalized phase spaces into the ones describing the quantum-deformed Heisenberg algebras., 15 pages,no figures; v2 with completed references, which appeared in PLB

Using the isomorphism $${\mathfrak {o}}(3;{\mathbb {C}})\simeq {\mathfrak {sl}}(2;{\mathbb {C}})$$ o ( 3 ; C ) ≃ sl ( 2 ; C ) we develop a new simple algebraic technique for complete classification of quantum deformations (the classical r-matrices) for real forms $${\mathfrak {o}}(3)$$ o ( 3 ) and $${\mathfrak {o}}(2,1)$$ o ( 2 , 1 ) of the complex Lie algebra $${\mathfrak {o}}(3;{\mathbb {C}})$$ o ( 3 ; C ) in terms of real forms of $${\mathfrak {sl}}(2;{\mathbb {C}})$$ sl ( 2 ; C ) : $${\mathfrak {su}}(2)$$ su ( 2 ) , $${\mathfrak {su}}(1,1)$$ su ( 1 , 1 ) and $${\mathfrak {sl}}(2;{\mathbb {R}})$$ sl ( 2 ; R ) . We prove that the $$D=3$$ D = 3 Lorentz symmetry $${\mathfrak {o}}(2,1)\simeq {\mathfrak {su}}(1,1)\simeq {\mathfrak {sl}}(2;{\mathbb {R}})$$ o ( 2 , 1 ) ≃ su ( 1 , 1 ) ≃ sl ( 2 ; R ) has three different Hopf-algebraic quantum deformations, which are expressed in the simplest way by two standard $${\mathfrak {su}}(1,1)$$ su ( 1 , 1 ) and $${\mathfrak {sl}}(2;{\mathbb {R}})$$ sl ( 2 ; R ) q-analogs and by simple Jordanian $${\mathfrak {sl}}(2;{\mathbb {R}})$$ sl ( 2 ; R ) twist deformation. These quantizations are presented in terms of the quantum Cartan–Weyl generators for the quantized algebras $${\mathfrak {su}}(1,1)$$ su ( 1 , 1 ) and $${\mathfrak {sl}}(2;{\mathbb {R}})$$ sl ( 2 ; R ) as well as in terms of quantum Cartesian generators for the quantized algebra $${\mathfrak {o}}(2,1)$$ o ( 2 , 1 ) . Finally, some applications of the deformed $$D=3$$ D = 3 Lorentz symmetry are mentioned.

We present new model of D=4 relativistic massive particle with spin and we describe its quantization. The model is obtained by an extension of standard relativistic phase space description of massive spinless particle by adding new topological Souriau-Wess-Zumino term which depends on spin fourvector variable. We describe equivalently our model as given by the free two-twistor action with suitable constraints. An important tool in our derivation is the spin-dependent twistor shift, which modifies standard Penrose incidence relations. The quantization of the model provides the wave function with correct mass and spin eigenvalues., 1+15 pages

Colour $SU(3)$ group is an exact symmetry of Quantum Chromodynamics, which describes strong interactions between quarks and gluons. Supplemented by two internal symmetries, $SU(2)$ and $U(1)$, it serves as the internal symmetry of the Standard Model, describing as well the electroweak interactions of quarks and leptons. The colour$SU(3)$ symmetry is exact, while two other symmetries are broken by means of the Higgs-Kibble mechanism. The three colours and fractional quarks charges with values $1/3$ and $2/3$ suggest that the cyclic group $Z_3$ may play a crucial role in quark field dynamics. In this paper we consequently apply the $Z_3$ symmetry to field multiplets describing colour quark fields. Generalized Dirac equation for coloured $12$-component spinors is introduced and its properties are discussed. Imposing $Z_3$-graded Lorentz and Poincar\'e covariance leads to enlargement of quark fields multiplets and incorporates additional $Z_2 \times Z_3$ symmetry which leads to the appearance of three generations (families) of distinct quark doublets., Comment: 20 pages, 1 figure; submitted to Proceedings of ISOS26 Conference held in Prague, Czech Republik (08-012.07.2019); to appear in Journal of Physics (Conference Series). arXiv admin note: text overlap with arXiv:1801.01403; v2 corrected formula (31), version to appear in Journal of Physics, Conference Series

We construct firstly the complete list of five quantum deformations of $D=4$ complex homogeneous orthogonal Lie algebra $\mathfrak{o}(4;\mathbb{C})\cong \mathfrak{o}(3;\mathbb{C})\oplus \mathfrak{o}(3;\mathbb{C})$, describing quantum rotational symmetry of four-dimensional complex space-time, in particular we provide the corresponding universal quantum $R$-matrices. Further applying four possible reality conditions we obtain all sixteen Hopf-algebraic quantum deformations for the real forms of $\mathfrak{o}(4;\mathbb{C})$: Euclidean $\mathfrak{o}(4)$, Lorentz $\mathfrak{o}(3,1)$, Kleinian $\mathfrak{o}(2,2)$ and quaternionic $\mathfrak{o}^{\star}(4)$. For $\mathfrak{o}(3,1)$ we only recall well-known results obtained previously by the authors, but for other real Lie algebras (Euclidean, Kleinian, quaternionic) as well as for the complex Lie algebra $\mathfrak{o}(4;\mathbb{C})$ we present new results., Comment: 32 pages, v2 minor improvements, added references, new formulas on p.23

The $(4+4)$-dimensional $\kappa$-deformed quantum phase space as well as its $(10+10)$-dimensional covariant extension by the Lorentz sector can be described as Heisenberg doubles: the $(10+10)$-dimensional quantum phase space is the double of $D=4$ $\kappa$-deformed Poincar\'e Hopf algebra $\mathbb{H}$ and the standard $(4+4)$-dimensional space is its subalgebra generated by $\kappa$-Minkowski coordinates $\hat{x}_\mu$ and corresponding commuting momenta $\hat{p}_\mu$. Every Heisenberg double appears as the total algebra of a Hopf algebroid over a base algebra which is in our case the coordinate sector. We exhibit the details of this structure, namely the corresponding right bialgebroid and the antipode map. We rely on algebraic methods of calculation in Majid-Ruegg bicrossproduct basis. The target map is derived from a formula by J-H. Lu. The coproduct takes values in the bimodule tensor product over a base, what is expressed as the presence of coproduct gauge freedom., Comment: 11 pages, RevTeX4, to appear in Proceedings of IX-th International Symposium "Quantum Theory and Symmetries" (QTS-9), held July 13-18, 2015, Yerevan; to be published in "Physics of Atomic Nuclei" (English Version of "Jadernaja Fizika"), ed. G. Pogosyan

I shall recall in historical perspective some results from nineties and show further how $\kappa$-deformed symmetries and $\kappa$-Minkowski space inspired DSR (Doubly of Deformed Special Relativity) approach proposed after 2000. As very recent development I shall show how to describe quantum-covariant $\kappa$-deformed phase spaces by passing from Hopf algebras to Hopf algebroids (arXiv:1507.02612) and I will briefly describe the $\kappa$-deformations of $AdS_5 \times S^5$ superstring target spaces (arXiv:1510.030.83)., Comment: 19 pages; presented at ISQS24 Conference on Integrable Systems and Quantum Symmetries, Prague (Czech Republik), 14.06-18.06.2016, 3-rd POTOR (Polish Society on Relativity) Conference, Cracov (Poland), 25-29.09.2016 and 5-th Conference "New Trends in Field Theories", Varanasi (India), 06.11-10.11.2016. To be published in JPhysA, Conf.Series

We introduce the Maxwell-invariant extension of D=4 relativistic free particle model into ten-dimensional Maxwell tensorial space. The new model after first quantization describes in particular Lorentz frame the planar dynamics providing Landau orbits in the presence of constant magnetic field., LaTeX, 16 p. Slightly extended version which will appear in Physics Letters B

The Maxwell algebra, an enlargement of Poincare algebra by Abelian tensorial generators, can be obtained in arbitrary dimension D by the suitable contraction of O(D-1,1) \oplus O(D-1,2) (Lorentz algebra \oplus AdS algebra). We recall that in D=4 the Lorentz algebra O(3,1) is described by the realification Sp_R(2|C) of complex algebra Sp(2|C)\simeq Sl(2|C) and O(3,2)\simeq Sp(4). We study various D=4 N-extended Maxwell superalgebras obtained by the contractions of real superalgebras OSp_R(2N-k; 2|C)\oplus OSp(k;4), (k=1,2,...,2N) (extended Lorentz superalgebra \oplus extended AdS superalgebra). If N=1 (k=1,2) one arrives at two different versions of simple Maxwell superalgebra. For any fixed N we get 2N different superextensions of Maxwell algebra with n-extended Poincare superalgebras (1\leq n \leq N) and the internal symmetry sectors obtained by suitable contractions of the real algebra O_R(2N-k|C)\oplus O(k). Finally the comments on possible applications of Maxwell superalgebras are presented., 14 pages, LaTeX

We perform a nonrelativistic contraction of N-extended Poincare superalgebra with internal symmetry U(N) and general set of N(N-1) real central charges. We show that for even N=2k and particular choice of the dependence of Z_{ij} on light velocity c one gets the N-extended Galilei superalgebra with unchanged number of central charges and compact internal symmetry algebra U(k;H)=USp(2k). The Hamiltonian positivity condition is modified only by one central charge. If we put all the central charges equal to zero one gets the 2k-extended Galilei superalgebra as the subalgebra of recently introduced extended Galilei conformal superalgebra [1,2]., 13 pages

This paper together with the previous one (Borowiec, Eur. Phys. J. C 48:633, 2006) presents a detailed description of all quantum deformations of the D=4 Lorentz algebra as a Hopf algebra in terms of complex and real generators. We describe here in detail two quantum deformations of the D=4 Lorentz algebra $\mathfrak{o}(3,1)$ , obtained by twisting of the standard q-deformation $U_{q}(\mathfrak{o}(3,1))$ . For the first twisted q-deformation an Abelian twist depending on the Cartan generators of $\mathfrak{o}(3,1)$ is used. The second example of twisting provides a quantum deformation of Cremmer–Gervais type for the Lorentz algebra. For completeness we describe also twisting of the Lorentz algebra by a standard Jordanian twist. By twist quantization techniques we obtain for these deformations new explicit formulae for the deformed coproducts and antipodes of the $\mathfrak{o}(3,1)$ -generators.

By considering the nonrelativistic limit of de-Sitter geometry one obtains the nonrelativistic space-time with a cosmological constant and Newton-Hooke (NH) symmetries. We show that the NH symmetry algebra can be enlarged by the addition of the constant acceleration generators and endowed with central extensions (one in any dimension (D) and three in D=(2+1)). We present a classical Lagrangian and Hamiltonian framework for constructing models quasi-invariant under enlarged NH symmetries which depend on three parameters described by three nonvanishing central charges. The Hamiltonian dynamics then splits into external and internal sectors with new non-commutative structures of external and internal phase spaces. We show that in the limit of vanishing cosmological constant the system reduces to the one presented in [1] which possesses accelaration-enlarged Galilean symmetries., Comment: 13 pages; small changes like a couple of footnotes etc

We present D=3 and D=4 models for massive particles moving in a new type of enlarged spacetime, with D-1 additional vector coordinates, which after quantization lead to the towers of massive higher spin (HS) free fields. Two classically equivalent formulations are presented: one with a hybrid spacetime/bispinor geometry and a second described by a free two-twistor dynamics with constraints. After quantization in the D=3 and D=4 cases, the wave functions are given as functions on the SL(2,R) and SL(2,C) group manifolds respectively, and describe arbitrary on-shell momenta and spin degrees of freedom. Finally, the D=6 case and possible supersymmetric extensions are mentioned., Comment: 37 pages, plain latex, v2. Text in Secs. 1 nd 4 enlarged, references added. Version to appear in JHEP

We describe in detail two-parameter nonstandard quantum deformation of D=4 Lorentz algebra $\mathfrak{o}(3,1)$, linked with Jordanian deformation of $\mathfrak{sl} (2;\mathbb{C})$. Using twist quantization technique we obtain the explicit formulae for the deformed coproducts and antipodes. Further extending the considered deformation to the D=4 Poincar\'{e} algebra we obtain a new Hopf-algebraic deformation of four-dimensional relativistic symmetries with dimensionless deformation parameter. Finally, we interpret $\mathfrak{o}(3,1)$ as the D=3 de-Sitter algebra and calculate the contraction limit $R\to\infty$ ($R$ -- de-Sitter radius) providing explicit Hopf algebra structure for the quantum deformation of the D=3 Poincar\'{e} algebra (with masslike deformation parameters), which is the two-parameter light-cone $\kappa$-deformation of the D=3 Poincar\'{e} symmetry., Comment: 13 pages, no figures

We extend the Shirafuji model for massless particles with primary spacetime coordinates and composite four-momenta to a model for massive particles with spin and electric charge. The primary variables in the model are the spacetime four-vector, four scalars describing spin and charge degrees of freedom as well as a pair of Weyl spinors. The geometric description proposed in this paper provides an intermediate step between the free purely twistorial model in two-twistor space in which both spacetime and four-momenta vectors are composite, and the standard particle model, where both spacetime and four-momenta vectors are elementary. We quantize the model and find explicitly the first-quantized wavefunctions describing relativistic particles with mass, spin and electric charge. The spacetime coordinates in the model are not commutative; this leads to a wavefunction that depends only on one covariant projection of the spacetime four-vector (covariantized time coordinate) defining plane wave solutions., Latex, 27 pages, appendix.sty, newlfont.sty (attached)

We consider two new classes of twisted D=4 quantum Poincar\'{e} symmetries described as the dual pairs of noncocommutative Hopf algebras. Firstly we investigate a two-parameter class of twisted Poincar\'{e} algebras which provide the examples of Lie-algebraic noncommutativity of the translations. The corresponding associative star-products and new deformed Lie-algebraic Minkowski spaces are introduced. We discuss further the twist deformations of Poincar\'{e} symmetries generated by the twist with its carrier in Lorentz algebra. We describe corresponding deformed Poincar\'{e} group which provides the quadratic deformations of translation sector and define the quadratically deformed Minkowski space-time algebra., Comment: LaTeX 16 pages. The version which appear in PLB

We exhibit surprising relations between higher spin theory and non-linear realizations of the supergroup OSp ( 1 | 8 ) , a minimal superconformal extension of N = 1 , 4D supersymmetry with tensorial charges. We construct a realization of OSp ( 1 | 8 ) on the coset supermanifold OSp ( 1 | 8 ) / SL ( 4 , R ) which involves the tensorial superspace R ( 10 | 4 ) and Goldstone superfields given on it. The covariant superfield equation encompassing the component ones for all integer and half-integer massless higher spins amounts to the vanishing of covariant spinor derivatives of the suitable Goldstone superfields, and, via Maurer–Cartan equations, to the vanishing of SL ( 4 , R ) supercurvature in odd directions of R ( 10 | 4 ) . Aiming at higher spin extension of the Ogievetsky–Sokatchev formulation of N = 1 supergravity, we generalize the notion of N = 1 chirality and construct first examples of invariant superfield actions involving a non-trivial interaction. Some other potential implications of OSp ( 1 | 8 ) in the proposed setting are briefly outlined.

We consider a superextension of the extended Jordanian twist, describing nonstandard quantization of anti-de-Sitter ($AdS$) superalgebra $osp(1|4)$ in the form of Hopf superalgebra. The super-Jordanian twisting function and corresponding basic coproduct formulae for the generators of $osp(1|4)$ are given in explicit form. The nonlinear transformation of the classical superalgebra basis not modifying the defining algebraic relations but simplifying coproducts and antipodes is proposed. Our physical application is to interpret the new super-Jordanian deformation of $osp(1|4)$ superalgebra as deformed D=4 $AdS$ supersymmetries. Subsequently we perform suitable contraction of quantum Jordanian $AdS$ superalgebra and obtain new $\kappa$-deformation of D=4 Poincare superalgebra, with the bosonic sector describing the light cone $\kappa$-deformation of Poincare symmetries., Comment: LaTeX,13 pages, comments and references added, to be published in Eur.Phys.J.C

The sixteen real coordinates of two-twistor space are transformed by a nonlinear mapping into an enlarged space–time framework. The standard relativistic phase space of coordinates (Xμ,Pμ) is supplemented by a six-parameter spin phase manifold (two pairs (ηα,σα) and ( η α , σ α ) of canonically conjugated Weyl spinors constrained by two second class constraints) and the electric charge phase space (e,φ). The free two-twistor classical mechanics is rewritten in this enlarged relativistic phase space as a model for a relativistic particle. Definite values for the mass, spin and the electric charge of the particle are introduced by means of three first class constraints.

The twisting function describing a nonstandard (super-Jordanian) quantum deformation of $osp(1|2)$ is given in explicite closed form. The quantum coproducts and universal R-matrix are presented. The non-uniqueness of the twisting function as well as two real forms of the deformed $osp(1|2)$ superalgebras are considered. One real quantum $osp(1|2)$ superalgebra is interpreted as describing the $\kappa$-deformation of D=1, N=1 superconformal algebra, which can be applied as a symmetry algebra of N=1 superconformal mechanics., Comment: 13 pages,LaTeX, v2. References added, typos corrected

We describe the classical $o(3,2)$ $r$-matrices as generating the quantum deformations of either D=3 conformal algebra with mass-like deformation parameters or D=4 $AdS$ algebra with dimensionless deformation parameters. We describe the quantization of classical $o(3,2)$ $r$-matrices via Drinfeld twist method which locates the deformation in the coalgebra sector. Further we obtain the quantum $o(3,2)$ algebra in a convenient Hopf algebra form by considering suitable deformation maps from classical to deformed $o(3,2)$ algebra basis. It appears that if we pass from $\kappa$-deformed D =3 conformal algebra basis to the deformed D=4 $AdS$ generators basis the role of dimensionfull parameter is taken over by the $AdS$ radius $R$. We provide also the bilinear $o(3,2)$ Casimir which we express using the deformed D=3 conformal basis., Comment: LaTeX, 15 pages. The address of the author (V.D. L.) corrected and references are completed

We provide the classification of real forms of complex D=4 Euclidean algebra $\mathcal{\epsilon}(4; \mathbb{C}) = \mathfrak{o}(4;\mathbb{C})) \ltimes \mathbf{T}_{\mathbb{C}}^4$ as well as (pseudo)real forms of complex D=4 Euclidean superalgebras $\mathcal{\epsilon}(4|N; \mathbb{C})$ for N=1,2. Further we present our results: N=1 and N=2 supersymmetric D=4 Poincare and Euclidean r-matrices obtained by using D= 4 Poincare r-matrices provided by Zakrzewski [1]. For N=2 we shall consider the general superalgebras with two central charges., Comment: 20 pages, presented at XXIII-th International Conference on Integrable Systems and Quantum Symmetries, Prague, June 2015; to be published in JPhysA Conference Series, ed. C. Burdik, S.Posta and O.Navratil

We provide first explicite examples of quantum deformations of D=4 conformal algebra with mass-like deformation parameters, in applications to quantum gravity effects related with Planck mass. It is shown that one of the classical $r$-matrices defined on the Borel subalgebra of $sl(4)$ with $o(4,2)$ reality conditions describes the light-cone $\kappa$-deformation of D=4 Poincar\'{e} algebra. We embed this deformation into the three-parameter family of generalized $\kappa$-deformations, with $r$-matrices depending additionally on the dilatation generator. Using the extended Jordanian twists framework we describe these deformations in the form of noncocommutative Hopf algebra. We describe also another four-parameter class of generalized $\kappa$-deformations, which is obtained by continuous deformation of distinguished $\kappa$-deformation of D=4 Weyl algebra, called here the standard $\kappa$-deformation of Weyl algebra., Comment: LaTeX, 14 pages, corrected some typos