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New classes of Lie-Hamilton systems are obtained from the six-dimensional fundamental representation of the symplectic Lie algebra $\mathfrak{sp}(6,\mathbb{R})$. The ansatz is based on a recently proposed procedure for constructing higher-dimensional Lie-Hamilton systems through the representation theory of Lie algebras. As applications of the procedure, we study a time-dependent electromagnetic field and several types of coupled oscillators. The irreducible embedding of the special unitary Lie algebra $\mathfrak{su}(3)$ into $\mathfrak{sp}(6, \mathbb{R})$ is also considered, yielding Lie-Hamilton systems arising from the sum of the quark and antiquark three-dimensional representations of $\mathfrak{su}(3)$, which are applied in the construction of t-dependent coupled systems. In addition, t-independent constants of the motion are obtained explicitly for all these Lie-Hamilton systems, which allows the derivation of a nonlinear superposition rule, Comment: 18 pages. Based on the contribution presented at the "XXIVth International Conference on Geometry, Integrability and Quantization" held in Varna, Bulgaria, June 6-13, 2024

The exploration of the universe has recently entered a new era thanks to the multi-messenger paradigm, characterized by a continuous increase in the quantity and quality of experimental data that is obtained by the detection of the various cosmic messengers (photons, neutrinos, cosmic rays and gravitational waves) from numerous origins. They give us information about their sources in the universe and the properties of the intergalactic medium. Moreover, multi-messenger astronomy opens up the possibility to search for phenomenological signatures of quantum gravity. On the one hand, the most energetic events allow us to test our physical theories at energy regimes which are not directly accessible in accelerators; on the other hand, tiny effects in the propagation of very high energy particles could be amplified by cosmological distances. After decades of merely theoretical investigations, the possibility of obtaining phenomenological indications of Planck-scale effects is a revolutionary step in the quest for a quantum theory of gravity, but it requires cooperation between different communities of physicists (both theoretical and experimental). This review is aimed at promoting this cooperation by giving a state-of-the art account of the interdisciplinary expertise that is needed in the effective search of quantum gravity footprints in the production, propagation and detection of cosmic messengers., Comment: This review was written by participants of the COST Action CA18108. Further information on the review can be found at https://qg-mm.unizar.es/review/, updated to published version

We bring the concept that quantum symmetries describe theories with nontrivial momentum space properties one step further, looking at quantum symmetries of spacetime in presence of a nonvanishing cosmological constant $\Lambda$. In particular, the momentum space associated to the $\kappa$-deformation of the de Sitter algebra in (1+1) and (2+1) dimensions is explicitly constructed as a dual Poisson-Lie group manifold parametrized by $\Lambda$. Such momentum space includes both the momenta associated to spacetime translations and the `hyperbolic' momenta associated to boost transformations, and has the geometry of (half of) a de Sitter manifold. Known results for the momentum space of the $\kappa$-Poincar\'e algebra are smoothly recovered in the limit $\Lambda\to 0$, where hyperbolic momenta decouple from translational momenta. The approach here presented is general and can be applied to other quantum deformations of kinematical symmetries, including (3+1)-dimensional ones., Comment: 13 pages

Noncommutative spacetimes are widely believed to model some properties of the quantum structure of spacetime at the Planck regime. In this contribution the construction of (anti-)de Sitter noncommutative spacetimes obtained through quantum groups is reviewed. In this approach the quantum deformation parameter $z$ is related to a Planck scale, and the cosmological constant $\Lambda$ plays the role of a second deformation parameter of geometric nature, whose limit $\Lambda\to 0$ provides the corresponding noncommutative Minkowski spacetimes., Comment: 10 pages. Based on the contribution presented at "Eighth International Workshop DICE2016", Castello Pasquini in Castiglioncello, Italy, September 12-16, 2016

A Lie system is a system of differential equations describing the integral curves of a $t$-dependent vector field taking values in a finite-dimensional real Lie algebra of vector fields, a Vessiot-Guldberg Lie algebra. We define and analyze Lie systems possessing a Vessiot-Guldberg Lie algebra of Hamiltonian vector fields relative to a Jacobi manifold, the hereafter called Jacobi-Lie systems. We classify Jacobi-Lie systems on $\mathbb{R}$ and $\mathbb{R}^2$. Our results shall be illustrated through examples of physical and mathematical interest., Comment: 15 pages. Examples, references and comments added. Based on the contribution presented at "The 10th AIMS Conference on Dynamical Systems, Differential Equations and Applications", July 07-11, 2014, Madrid, Spain. To appear in the Proceedings of the 10th AIMS Conference

A Lie-Hamilton system is a nonautonomous system of first-order ordinary differential equations describing the integral curves of a $t$-dependent vector field taking values in a finite-dimensional real Lie algebra of Hamiltonian vector fields with respect to a Poisson structure. We provide new algebraic/geometric techniques to easily determine the properties of such Lie algebras on the plane, e.g., their associated Poisson bivectors. We study new and known Lie-Hamilton systems on $\mathbb{R}^2$ with physical, biological and mathematical applications. New results cover Cayley-Klein Riccati equations, the here defined planar diffusion Riccati systems, complex Bernoulli differential equations and projective Schr\"odinger equations. Constants of motion for planar Lie-Hamilton systems are explicitly obtained which, in turn, allow us to derive superposition rules through a coalgebra approach., Comment: 33 pages. New contents added covering formalism, invariants and superposition rules

A {\it Lie system} is a nonautonomous system of first-order differential equations admitting a {\it superposition rule}, i.e., a map expressing its general solution in terms of a generic family of particular solutions and some constants. Using that a Lie system can be considered as a curve in a finite-dimensional Lie algebra of vector fields, a so-called {\it Vessiot--Guldberg Lie algebra}, we associate every Lie system with a Lie algebra of Lie point symmetries induced by the Vessiot--Guldberg Lie algebra. This enables us to derive Lie symmetries of relevant physical systems described by first- and higher-order systems of differential equations by means of Lie systems in an easier way than by standard methods. A generalization of our results to partial differential equations is introduced. Among other applications, Lie symmetries for several new and known generalizations of the real Riccati equation are studied.

We study Lie-Hamilton systems on the plane, i.e. systems of first-order differential equations describing the integral curves of a $t$-dependent vector field taking values in a finite-dimensional real Lie algebra of planar Hamiltonian vector fields with respect to a Poisson structure. We start with the local classification of finite-dimensional real Lie algebras of vector fields on the plane obtained in [A. Gonz\'alez-L\'opez, N. Kamran and P.J. Olver, Proc. London Math. Soc. 64, 339 (1992)] and we interpret their results as a local classification of Lie systems. Moreover, by determining which of these real Lie algebras consist of Hamiltonian vector fields with respect to a Poisson structure, we provide the complete local classification of Lie-Hamilton systems on the plane. We present and study through our results new Lie-Hamilton systems of interest which are used to investigate relevant non-autonomous differential equations, e.g. we get explicit local diffeomorphisms between such systems. In particular, the Milne-Pinney, second-order Kummer-Schwarz, complex Riccati and Buchdahl equations as well as some Lotka-Volterra and nonlinear biomathematical models are analysed from this Lie-Hamilton approach., Comment: 37 pages

A Lie system is a nonautonomous system of first-order differential equations possessing a superposition rule, i.e. a map expressing its general solution in terms of a generic finite family of particular solutions and some constants. Lie-Hamilton systems form a subclass of Lie systems whose dynamics is governed by a curve in a finite-dimensional real Lie algebra of functions on a Poisson manifold. It is shown that Lie-Hamilton systems are naturally endowed with a Poisson coalgebra structure. This allows us to devise methods to derive in an algebraic way their constants of motion and superposition rules. We illustrate our methods by studying Kummer-Schwarz equations, Riccati equations, Ermakov systems and Smorodinsky-Winternitz systems with time-dependent frequency., Comment: 30 pages

Higgs algebras are used to construct rotational Hamiltonians. The correspondence between the spectrum of a triaxial rotor and the spectrum of a cubic Higgs algebra is demonstrated. It is shown that a suitable choice of the parameters of the polynomial algebra allows for a precise identification of rotational properties. The harmonic limit is obtained by a contraction of the algebra, leading to a linear symmetry., Comment: 3 figures, 6 pages, 15 references. Phys. Rev. C (in press, ms CZ10038)

We present a systematic procedure to obtain singular solutions of the constant quantum Yang-Baxter equation in arbitrary dimension. This approach, inspired in the Lie (super)algebra structure, is explicitly applied to the particular case of (graded) contractions of the orthogonal real algebra ${\mathfrak{so}}(N+1)$. In this way we show that "classical" contraction parameters which appear in the commutation relations of the contracted Lie algebras, become quantum deformation parameters, arising as entries of the resulting quantum $R$-matrices., Comment: 8 pages

Quantum Algebras (q-algebras) are used to describe interactions between fermions and bosons. Particularly, the concept of a su_q(2) dynamical symmetry is invoked in order to reproduce the ground state properties of systems of fermions and bosons interacting via schematic forces. The structure of the proposed su_q(2) Hamiltonians, and the meaning of the corresponding deformation parameters, are discussed., Comment: 20 pages, 10 figures. Physical Review C (in press)

A unified algebraic construction of the classical Smorodinsky-Winternitz systems on the ND sphere, Euclidean and hyperbolic spaces through the Lie groups SO(N+1), ISO(N), and SO(N,1) is presented. Firstly, general expressions for the Hamiltonian and its integrals of motion are given in a linear ambient space $R^{N+1}$, and secondly they are expressed in terms of two geodesic coordinate systems on the ND spaces themselves, with an explicit dependence on the curvature as a parameter. On the sphere, the potential is interpreted as a superposition of N+1 oscillators. Furthermore each Lie algebra generator provides an integral of motion and a set of 2N-1 functionally independent ones are explicitly given. In this way the maximal superintegrability of the ND Euclidean Smorodinsky-Winternitz system is shown for any value of the curvature., Comment: 8 pages, LaTeX

A new twisted deformation, U_z(so(4,2)), of the conformal algebra of the (3+1)-dimensional Minkowskian spacetime is presented. This construction is provided by a classical r-matrix spanned by ten Weyl-Poincare generators, which generalizes non-standard quantum deformations previously obtained for so(2,2) and so(3,2). However, by introducing a conformal null-plane basis it is found that the twist can indeed be supported by an eight-dimensional carrier subalgebra. By construction the Weyl-Poincare subalgebra remains as a Hopf subalgebra after deformation. Non-relativistic limits of U_z(so(4,2)) are shown to be well defined and they give rise to new twisted conformal algebras of Galilean and Carroll spacetimes. Furthermore a difference-differential massless Klein-Gordon (or wave) equation with twisted conformal symmetry is constructed through deformed momenta and position operators. The deformation parameter is interpreted as the lattice step on a uniform Minkowskian spacetime lattice discretized along two basic null-plane directions., Comment: 20 pages, LaTeX

The minimal twist map introduced by B. Abdesselam, A. Chakrabarti, R. Chakrabarti and J. Segar (Mod. Phys. Lett. A 14 (1999) 765) for the non-standard (Jordanian) quantum sl(2,R) algebra is used to construct the twist maps for two different non-standard quantum deformations of the (1+1) Schrodinger algebra. Such deformations are, respectively, the symmetry algebras of a space and a time uniform lattice discretization of the (1+1) free Schrodinger equation. It is shown that the corresponding twist maps connect the usual Lie symmetry approach to these discrete equations with non-standard quantum deformations. This relationship leads to a clear interpretation of the deformation parameter as the step of the uniform (space or time) lattice., Comment: 16 pages, LaTeX

The most general possible central extensions of two whole families of Lie algebras, which can be obtained by contracting the special pseudo-unitary algebras su(p,q) of the Cartan series A_l and the pseudo-unitary algebras u(p,q), are completely determined and classified for arbitrary p,q. In addition to the su(p,q) and u({p,q}) algebras, whose second cohomology group is well known to be trivial, each family includes many non-semisimple algebras; their central extensions, which are explicitly given, can be classified into three types as far as their properties under contraction are involved. A closed expression for the dimension of the second cohomology group of any member of these families of algebras is given., Comment: 23 pages. Latex2e file

Two quantum Hopf structures for the Schr\"odinger algebra as well as their corresponding differential-difference realizations are presented. For each case a (space or time) discretization of the Schr\"odinger equation is deduced and the quantum Schr\"odinger generators are shown to be symmetry operators., Comment: Communication presented in the 5th Wigner Symposium, Vienna, 1997; 4 pages, LaTeX

A non-standard quantum deformation of the Poincar\'e algebra is presented in a null-plane framework for 1+1, 2+1 and 3+1 dimensions. Their corresponding universal $R$-matrices are obtained in a factorized form by choosing suitable bases related to the $T$-matrix formalism., Comment: 8 pages, LaTeX; communication presented in the XXI ICGTMP, Goslar (Germany) 1996

We determine the central extensions of a whole family of Lie algebras, obtained by the method of graded contractions from so(N+1), N arbitrary. All the inhomogeneous orthogonal and pseudo-orthogonal algebras are members of this family, as well as a large number of other non-semisimple algebras, all of which have at least a semidirect structure (in some cases two or more). The dimensions of their second cohomology groups H^2(G,R) and the explicit expression of their central extensions are given., Comment: Latex file. 32 pages. Minor changes. To appear in J. Phys. A

A non-linear map is applied onto the (non-standard) null-plane deformation of (3+1) Poincar\'e algebra giving rise to a simpler form of this triangular quantization. A universal $R$-matrix for the null plane quantum algebra is then obtained from a universal $T$-matrix corresponding to a Hopf subalgebra. Finally, the associated Poincar\'e Poisson--Lie group is quantized by using the FRT approach., Comment: 8 pages, LaTeX

The general solution of the graded contraction equations for a $\zz_2^{\otimes N}$ grading of the real compact simple Lie algebra $so(N+1)$ is presented in an explicit way. It turns out to depend on $2^N-1$ independent real parameters. The structure of the general graded contractions is displayed for the low dimensional cases, and kinematical algebras are shown to appear straightforwardly. The geometrical (or physical) meaning of the contraction parameters as curvatures is also analysed; in particular, for kinematical algebras these curvatures are directly linked to geometrical properties of possible homogeneous space-times., Comment: 12 pages, LaTeX; explicit proofs and applications to kinematical algebras are added

A new quantum deformation, which we call null-plane, of the (3+1) Poincar\'e algebra is obtained. The algebraic properties of the classical null-plane description are generalized to this quantum deformation. In particular, the classical isotopy subalgebra of the null-plane is deformed into a Hopf subalgebra, and deformed spin operators having classical commutation rules can be defined. Quantum Hamiltonian, mass and position operators are studied, and the null-plane evolution is expressed in terms of a deformed Schr\"odinger equation., Comment: 13 pages, LaTeX

The Hopf algebra dual form for the non--standard uniparametric deformation of the (1+1) Poincar\'e algebra $iso(1,1)$ is deduced. In this framework, the quantum coordinates that generate $Fun_w(ISO(1,1))$ define an infinite dimensional Lie algebra. A change in the basis of the dual form is obtained in order to compare this deformation to the standard one. Finally, a non--standard quantum Heisenberg group acting on a quantum Galilean plane is obtained., Comment: 17 pages, LaTeX.

A universal quasitriangular $R$--matrix for the non-standard quantum (1+1) Poincar\'e algebra $U_ziso(1,1)$ is deduced by imposing analyticity in the deformation parameter $z$. A family $g_\mu$ of ``quantum graded contractions" of the algebra $U_ziso(1,1)\oplus U_{-z}iso(1,1)$ is obtained; this set of quantum algebras contains as Hopf subalgebras with two primitive translations quantum analogues of the two dimensional Euclidean, Poincar\'e and Galilei algebras enlarged with dilations. Universal $R$--matrices for these quantum Weyl algebras and their associated quantum groups are constructed., Comment: 12 pages, LaTeX.

Lie bialgebra contractions are introduced and classified. A non-degenerate coboundary bialgebra structure is implemented into all pseudo-orthogonal algebras $so(p,q)$ starting from the one corresponding to $so(N+1)$. It allows to introduce a set of Lie bialgebra contractions which leads to Lie bialgebras of quasi-orthogonal algebras. This construction is explicitly given for the cases $N=2,3,4$. All Lie bialgebra contractions studied in this paper define Hopf algebra contractions for the Drinfel'd-Jimbo deformations $U_z so(p,q)$. They are explicitly used to generate new non-semisimple quantum algebras as it is the case for the Euclidean, Poincar\'e and Galilean algebras., Comment: 26 pages LATEX

A new "non-standard" quantization of the universal enveloping algebra of the split (natural) real form $so(2,2)$ of $D_2$ is presented. Some (classical) graded contractions of $so(2,2)$ associated to a $Z_2 \times Z_2$ grading are studied, and the automorphisms defining this grading are generalized to the quantum case, thus providing quantum contractions of this algebra. This produces a new family of "non-standard" quantum algebras. Some of these algebras can be realized as (2+1) kinematical algebras; we explicitly introduce a new deformation of Poincar\'e algebra, which is naturally linked to the null plane basis. Another realization of these quantum algebras as deformations of the conformal algebras for the two-dimensional Euclidean, Galilei and Minkowski spaces is given, and its new properties are emphasized., Comment: 19 pages, LaTeX; final version

An embedding method to get $q$-deformations for the non--semisimple algebras generating the motion groups of $N$--dimensional flat spaces is presented. This method gives a global and simultaneous scheme of $q$-deformation for all $iso(p,q)$ algebras and for those ones obtained from them by some In\"on\"u--Wigner contractions, such as the $N$--dimensional Euclidean, Poincar\'e and Galilei algebras., Comment: 10 pages, AMS-TEX file

A Poisson--Hopf algebra of smooth functions on the (1+1) Cayley--Klein groups is constructed by using a classical $r$--matrix which is invariant under contraction. The quantization of this algebra for the Euclidean, Galilei and Poincar\'e cases is developed, and their duals are also computed. Contractions on these quantum groups are studied., Comment: 7 pages

A universal R--matrix for the quantum Heisenberg algebra h(1)q is presented. Despite of the non--quasitriangularity of this Hopf algebra, the quantum group induced from it coincides with the quasitriangular deformation already known., Comment: 5 pages

The N-dimensional Cayley-Klein scheme allows the simultaneous description of $3^N$ geometries (symmetric orthogonal homogeneous spaces) by means of a set of Lie algebras depending on $N$ real parameters. We present here a quantum deformation of the Lie algebras generating the groups of motion of the two and three dimensional Cayley-Klein geometries. This deformation (Hopf algebra structure) is presented in a compact form by using a formalism developed for the case of (quasi) free Lie algebras. Their quasitriangularity (i.e., the most usual way to study the associativity of their dual objects, the quantum groups) is also discussed., Comment: 7 pages, AMS-TEX file, UVA/93-102

We study $\Bbb Z_2^{\otimes N}$ graded contractions of the real compact simple Lie algebra $so(N+1)$, and we identify within them the Cayley-Klein algebras as a naturally distinguished subset., Comment: 13 pages, Preprint UVA, AMS-TeX

A global model of $q$-deformation for the quasi--orthogonal Lie algebras generating the groups of motions of the four--dimensional affine Cayley--Klein geometries is obtained starting from the three dimensional deformations. It is shown how the main algebraic classical properties of the CK systems can be implemented in the quantum case. Quantum deformed versions of either the space--time or space symmetry algebras (Poincar\'e (3+1), Galilei (3+1), 4D Euclidean as well as others) appear in this context as particular cases and several $q$-deformations for them are directly obtained., Comment: 13 pages, AMS-TEX file