Within the so-called group geometric approach to (super)gravity and (super)string theories, any compact Lie manifold $G_{c}$ can be smoothly deformed into a $G_{c}^{\mu }$ (locally diffeomorphic itself), which is `soft', namely, based on non-left-invariant, intrinsic one-form Vielbein $\mu $, violates Maurer-Cartan equations consequently has non-vanishing associated curvature two-form. framework above deformation (`softening'), we show how construct an infinite-dimensional (infinite-rank),...

We consider the quantum analog of generalized Zernike systems given by Hamiltonian: $$ \hat{\mathcal{H}} _N =\hat{p}_1^2+\hat{p}_2^2+\sum_{k=1}^N \gamma_k (\hat{q}_1 \hat{p}_1+\hat{q}_2 \hat{p}_2)^k , with canonical operators $\hat{q}_i,\, \hat{p}_i$ and arbitrary coefficients $\gamma_k$. This two-dimensional model, besides conservation angular momentum, exhibits higher-order integrals motion within enveloping algebra Heisenberg $\mathfrak h_2$. By constructing suitable combinations these...

We propose an adaptation of the notion scaling symmetries for case Lie-Hamilton systems, allowing their subsequent reduction to contact Lie systems. As illustration procedure, time-dependent frequency oscillators and thermodynamic systems are analyzed from this point view. The formalism provides a novel method constructing on three-dimensional sphere, derived recently established arising fundamental four-dimensional representation symplectic algebra $\mathfrak{sp}(4,\mathbb{R})$. It is shown...

The indecomposable solvable Lie algebras with graded nilradical of maximal nilindex and a Heisenberg subalgebra codimension one are analyzed, their generalized Casimir invariants calculated. It is shown that rank have contact form, which implies the existence an associated dynamical system. Moreover, due to structure quadratic operator nilradical, these contain non-abelian quasi-classical algebra dimension $2n-1$, indicating gauge theories (with ghosts) possible on subalgebras.

The class of solvable Lie algebras with an -graded nilradical maximal nilpotency index is classified. It shown that such extensions are unique up to isomorphism. generalized Casimir invariants for the nilradicals and their associated computed by method moving frames.

Starting from a purely algebraic procedure based on the commutant of subalgebra in universal enveloping algebra given Lie algebra, notion Hamiltonians and constants motion generating polynomial symmetry is proposed. The case special linear $\mathfrak{sl}(n)$ discussed detail, where an explicit basis for with respect to Cartan obtained, order computed. It further shown that, appropriate realization $\mathfrak{sl}(n)$, this provides connection generic superintegrable model $(n-1)$-dimensional...

Using the theory of Lie-Hamilton systems, formal generalized time-dependent Hamiltonian systems that enlarge a recently proposed SIS epidemic model with variable infection rate are considered. It is shown that, independently on particular interpretation coefficients, these generally admit an exact solution, up to case maximal extension within classification for which superposition rule constructed. The method provides algebraic frame any preserves above mentioned properties subjected. In...

The notion of color algebras is generalized to the class F -ary algebras, and corresponding decoloration theorems are established.This used give a construction colored structures by means tensor products with Clifford-like algebras.It moreover shown that admit realisations as q = 0 quon algebras.

Abstract The Universe expansion rate has two different but very precise values ( <?CDATA $67.4~\pm~0.5$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mn>67.4</mml:mn><mml:mtext> </mml:mtext><mml:mo>±</mml:mo><mml:mtext> </mml:mtext><mml:mn>0.5</mml:mn></mml:math> and $73.30~\pm~1.04$?> overflow="scroll"><mml:mn>73.30</mml:mn><mml:mtext> </mml:mtext><mml:mn>1.04</mml:mn></mml:math> km s −1 Mpc ) that are not compatible. This problem, known as a Hubble...

Let g = s n r be an indecomposable Lie algebra with nontrivial semisimple Levi subalgebra and solvable radical r. In this note it is proved that cannot isomorphic to a filiform nilpotent algebra. The proof uses the fact any snr would degenerate (even contract) snfn, where fn standard graded filiform Lie of dimension dim This leads contradiction, since no such exists

We study and classify kinematical algebras which appear in the framework of Lie superalgebras or order three. All these are related through generalised Inonü-Wigner contractions from either orthosymplectic superalgebra de Sitter algebra

Hopf algebra deformations are merged with a class of Lie systems Hamiltonian type, the so-called Lie–Hamilton systems, to devise novel formalism: Poisson–Hopf systems. This approach applies any deformation system. Remarkably, transforms system, whose dynamic is governed by finite-dimensional functions, into non-Lie–Hamilton system associated functions that allows for explicit description its t-independent constants motion from deformed Casimir functions. We illustrate our considering...

The invariants of all complex solvable rigid Lie algebras up to dimension 8 are computed. Moreover we show, for rank 1 algebras, some criteria deduce the non-existence nontrivial or existence fundamental sets formed by rational functions Casimir associated nilradical.

Combining the decomposition of Casimir operators induced by embedding a subalgebra into semisimple Lie algebra with properties commutators subgroup scalars, an analytical algorithm for computation missing label commutativity requirement is proposed. Two new criteria subgroups scalars to commute are given. The completed recursive method construct orthonormal bases states. As examples illustrate procedure, four labelling problems explicitly studied.

In a recent paper, Post and Winternitz studied the properties of two-dimensional Euclidean potentials that are linear in one two Cartesian variables. particular, they proved existence potential endowed with an integral third-order fourth-order. this paper we show these results can be obtained more simple direct way by noting is directly related Holt potential. It higher order superintegrability consequence integrability family type potentials.

A direct approach to non-linear second-order ordinary differential equations admitting a superposition principle is developed by means of Vessiot-Guldberg-Lie algebras dimension not exceeding three. This procedure allows us describe generic types subjected some constraints and given Lie algebra as algebra. In particular, well-known types, such the Milne-Pinney or Kummer-Schwarz equations, are recovered special cases this classification. The analogous problem for systems in real plane...