# ℓ-oscillators from second-order invariant PDEs of the centrally extended conformal Galilei algebras

@article{Aizawa2015oscillatorsFS, title={ℓ-oscillators from second-order invariant PDEs of the centrally extended conformal Galilei algebras}, author={Naruhiko Aizawa and Zhanna Kuznetsova and Francesco Toppan}, journal={Journal of Mathematical Physics}, year={2015}, volume={56}, pages={031701} }

We construct, for any given l=12+N0, the second-order, linear partial differential equations (PDEs) which are invariant under the centrally extended conformal Galilei algebra. At the given l, two invariant equations in one time and l+12 space coordinates are obtained. The first equation possesses a continuum spectrum and generalizes the free Schrodinger equation (recovered for l=12) in 1 + 1 dimension. The second equation (the “l-oscillator”) possesses a discrete, positive spectrum. It…

## 11 Citations

Invariant partial differential equations with two-dimensional exotic centrally extended conformal Galilei symmetry

- Mathematics, Physics
- 2015

Conformal Galilei algebras (CGAs) labeled by d, l (where d is the number of space dimensions and l denotes a spin-l representation w.r.t. the 𝔰𝔩(2) subalgebra) admit two types of central…

On dynamical realizations of l-conformal Galilei and Newton–Hooke algebras

- Physics, Mathematics
- 2015

In two recent papers (Aizawa et al., 2013 [15]) and (Aizawa et al., 2015 [16]), representation theory of the centrally extended l-conformal Galilei algebra with half-integer l has been applied so as…

Higher-Spin Symmetries and Deformed Schrödinger Algebra in Conformal Mechanics

- Physics, MathematicsAdvances in Mathematical Physics
- 2018

The dynamical symmetries of 1+1-dimensional Matrix Partial Differential Equations with a Calogero potential (with/without the presence of an extra oscillatorial de Alfaro-Fubini-Furlan, DFF, term)…

N ov 2 01 6 Z 2 × Z 2-graded Lie Symmetries of the Lévy-Leblond Equations

- 2016

The first-order differential Lévy-Leblond equations (LLE’s) are the non-relativistic analogs of the Dirac equation, being square roots of (1 + d)-dimensional Schrödinger or heat equations. Just like…

Centrally Extended Conformal Galilei Algebras and Invariant Nonlinear PDEs

- Physics, MathematicsSymmetry
- 2015

It is observed that the invariant PDEs have significant difference for \( \ell > \frac{1}{3}.

Invariant PDEs of Conformal Galilei Algebra as deformations: cryptohermiticity and contractions

- Physics, Mathematics
- 2015

We investigate the general class of second-order PDEs, invariant under the $d=1$ $\ell=\frac{1}{2}+{\mathbb N}_0$ centrally extended Conformal Galilei Algebras, pointing out that they are…

$\mathbb{Z}_2\times \mathbb{Z}_2$-graded Lie symmetries of the Lévy-Leblond equations

- Physics, Mathematics
- 2016

The first-order differential L\'evy-Leblond equations (LLE's) are the non-relativistic analogs of the Dirac equation, being square roots of ($1+d$)-dimensional Schr\"odinger or heat equations. Just…

Generalization of Superalgebras to Color Superalgebras and Their Representations

- Mathematics, Physics
- 2017

For a given Lie superalgebra, two ways of constructing color superalgebras are presented. One of them is based on the color superalgebraic nature of the Clifford algebras. The method is applicable to…

Generalized Niederer's transformation for quantum Pais–Uhlenbeck oscillator

- Physics
- 2015

Abstract We extend, to the quantum domain, the results obtained in [Nucl. Phys. B 885 (2014) 150] and [Phys. Lett. B 738 (2014) 405] concerning Niederer's transformation for the Pais–Uhlenbeck…

Symmetries of the Pais-Uhlenbeck oscillator on the Hamiltonian level

- Physics
- 2016

In this survey we report on the symmetry properties of the Pais-Uhlenbeck oscillator. Especially, basing on the paper [Nucl. Phys. B 889 (2014) 333] we describe the Hamiltonian formalisms and its…

## References

SHOWING 1-10 OF 109 REFERENCES

Four types of (super)conformal mechanics: D-module reps and invariant actions

- Physics, Mathematics
- 2014

(Super)conformal mechanics in one dimension is induced by parabolic or hyperbolic/trigonometric transformations, either homogeneous (for a scaling dimension λ) or inhomogeneous (at λ = 0, with ρ an…

The exotic conformal Galilei algebra and nonlinear partial differential equations

- Mathematics, Physics
- 2010

Abstract The conformal Galilei algebra ( cga ) and the exotic conformal Galilei algebra ( ecga ) are applied to construct partial differential equations (PDEs) and systems of PDEs, which admit these…

Dynamical realizations of l-conformal Newton–Hooke group

- Physics, Mathematics
- 2013

Abstract The method of nonlinear realizations and the technique previously developed in [A. Galajinsky, I. Masterov, Nucl. Phys. B 866 (2013) 212, arXiv:1208.1403 ] are used to construct a dynamical…

Intertwining operators for l-conformal Galilei algebras and hierarchy of invariant equations

- Mathematics, Physics
- 2013

l-Conformal Galilei algebra, denoted by g{l}{d}, is a non-semisimple Lie algebra specified by a pair of parameters (d,l). The algebra is regarded as a nonrelativistic analogue of the conformal…

Chiral and real N=2 supersymmetric ℓ-conformal Galilei algebras

- Physics, Mathematics
- 2013

Inequivalent N=2 supersymmetrizations of the l-conformal Galilei algebra in d-spatial dimensions are constructed from the chiral (2, 2) and the real (1, 2, 1) basic supermultiplets of the N=2…

On Schrödinger superalgebras

- Mathematics, Physics
- 1994

Using the supersymplectic framework of Berezin, Kostant, and others, two types of supersymmetric extensions of the Schrodinger algebra (itself a conformal extension of the Galilei algebra) were…

Algebraic structure of Galilean superconformal symmetries

- Physics, Mathematics
- 2011

The semisimple part of d-dimensional Galilean conformal algebra g^(d) is given by h^(d)=O(2,1)+O(d), which after adding via semidirect sum the 3d-dimensional Abelian algebra t^(d) of translations,…

Super-Galilean conformal algebra in AdS/CFT

- Physics
- 2010

Galilean conformal algebra (GCA) is an Inonu–Wigner (IW) contraction of a conformal algebra, while Newton–Hooke string algebra is an IW contraction of an Anti-de Sitter (AdS) algebra, which is the…

Symmetries of the Schrödinger equation and algebra/superalgebra duality

- Mathematics, Physics
- 2014

Some key features of the symmetries of the Schrodinger equation that are common to a much broader class of dynamical systems (some under construction) are illustrated. I discuss the…

Schrodinger equations for higher order nonrelativistic particles and N-Galilean conformal symmetry

- Physics, Mathematics
- 2012

Department of Physics, Toho University, Miyama, Funabashi, 274-8510, Japan(Received 26 September 2011; published 15 February 2012)We consider Schro¨dinger equations for a nonrelativistic particle…