Polynomial poisson algebras for two-dimensional classical superintegrable systems and polynomial associative algebras for quantum superintegrable systems

The integrals of motion of the classical two-dimensional superintegrable systems close in a restrained polynomial Poisson algebra, whose general form is discussed. Each classical superintegrable problem has a quantum counterpart, a quantum superintegrable system. The polynomial Poisson algebra is deformed to a polynomial associative algebra, the finite-dimensional representations of this algebra are calculated by using a deformed parafermion oscillator technique. It is conjectured that the finite-dimensional representations of the polynomial algebra are determined by the energy eigenvalues of the superintegrable system. The calculation of energy eigenvalues is reduced to the solution of algebraic equations, which are universal for a large number of two-dimensional superintegrable systems. Presented at the 9th Colloquium “Quantum Groups and Integrable Systems”, Prague, 22–24 June 2000.     

Polynomial associative algebras of quantum superintegrable systems

The integrals of motion of classical two-dimensional superintegrable systems, with polynomial integrals of motion, close in a restrained polynomial Poisson algebra; the general form of the quadratic case is investigated. The polynomial Poisson algebra of the classical system is deformed into a quantum associative algebra of the corresponding quantum system, and the finite-dimensional representations of this algebra are calculated by using a deformed parafermion oscillator technique. The finite-dimensional representations of the algebra are determined by the energy eigenvalues of the superintegrable system. The calculation of energy eigenvalues is reduced to the roots of algebraic equations in the quadratic case.     

Second-order superintegrable quantum systems

A classical (or quantum) superintegrable system on an n-dimensional Riemannian manifold is an integrable Hamiltonian system with potential that admits 2n ? 1 functionally independent constants of the motion that are polynomial in the momenta, the maximum number possible. If these constants of the motion are all quadratic, then the system is second-order superintegrable, the most tractable case and the one we study here. Such systems have remarkable properties: multi-integrability and separability, a quadratic algebra of symmetries whose representation theory yields spectral information about the Schrödinger operator, and deep connections with expansion formulas relating classes of special functions. For n = 2 and for conformally flat spaces when n = 3, we have worked out the structure of the classical systems and shown that the quadratic algebra always closes at order 6. Here, we describe the quantum analogs of these results. We show that, for nondegenerate potentials, each classical system has a unique quantum extension.     

Integrable quantum systems and classical Lie algebras

We have obtained six new infinite series of trigonometric solutions to triangle equations (quantumR-matrices) associated with the nonexceptional simple Lie algebras:sl(N),sp(N),o(N). TheR-matrices are given in two equivalent representations: in an additive one (as a sum of poles with matrix coefficients) and in a multiplicative one (as a ratio of entire matrix functions). TheseR-matrices provide an exact integrability of anisotropic generalizations ofsl(N),sp(N),o(N) invariant one-dimensional lattice magnetics and two-dimensional periodic Toda lattices associated with the above algebras.     

Quadratic algebras for three-dimensional superintegrable systems

The three-dimensional superintegrable systems with quadratic integrals of motion have five functionally independent integrals, one among them is the Hamiltonian. Kalnins, Kress, and Miller have proved that in the case of nondegenerate potentials with quadratic integrals of motion there is a sixth quadratic integral, which is linearly independent of the other integrals. The existence of this sixth integral implies that the integrals of motion form a ternary parafermionic-like quadratic Poisson algebra with five generators. In this contribution we investigate the structure of this algebra. We show that in all the nondegenerate cases there is at least one subalgebra of three integrals having a Poisson quadratic algebra structure, which is similar to the two-dimensional case.     

Dynamical symmetries for superintegrable quantum systems

We study the dynamical symmetries of a class of two-dimensional superintegrable systems on a 2-sphere, obtained by a procedure based on the Marsden-Weinstein reduction, by considering its shape-invariant intertwining operators. These are obtained by generalizing the techniques of factorization of one-dimensional systems. We firstly obtain a pair of noncommuting Lie algebras su(2) that originate the algebra so(4). By considering three spherical coordinate systems, we get the algebra u(3) that can be enlarged by “reflexions” to so(6). The bounded eigenstates of the Hamiltonian hierarchies are associated to the irreducible unitary representations of these dynamical algebras. The text was submitted by the authors in English.     

Superconformal algebras and their relation to integrable nonlinear systems

We relate the manifold of periodic functions on a circle with values in the Grassmann algebra to extended superconformal algebras. The graded Poisson brackets of these functions give the classical realization of the corresponding superconformal algebras and determine the hamiltonian structure for a class of integrable nonlinear equations. A super-generalization of the Korteweg-de Vries equation is found among these equations. In this way an important step in the program of the quantization of the Liouville equation is realized for the supersymmetric cases which are crucial in constructing a consistent quantum string theory. The construction of Miura transformations is outlined and the results for the N = 1,2 supersymmetric cases are presented.     

Isoperiodic classical systems and their quantum counterparts

One-dimensional isoperiodic classical systems have been first analyzed by Abel. Abel’s characterization can be extended for singular potentials and potentials which are not defined on the whole real line. The standard shear equivalence of isoperiodic potentials can also be extended by using reflection and inversion transformations. We provide a full characterization of isoperiodic rational potentials showing that they are connected by translations, reflections or Joukowski transformations. Upon quantization many of these isoperiodic systems fail to exhibit identical quantum energy spectra. This anomaly occurs at order O(?²) because semiclassical corrections of energy levels of order O(?) are identical for all isoperiodic systems. We analyze families of systems where this quantum anomaly occurs and some special systems where the spectral identity is preserved by quantization. Conversely, we point out the existence of isospectral quantum systems which do not correspond to isoperiodic classical systems.     

Loop algebras and their relation to the conformal structure of integrable systems

We use the theorem of Kostant, Adler and Symes to construct an infinite set of local polynomials in involution with respect to the Poisson bracket realisation of the Neveu-Schwartz sector of the N=1 superconformal algebra.     

Central extensions of classical and quantum q-Virasoro algebras

We investigate the central extensions of the q-deformed (classical and quantum) Virasoro algebras constructed from the elliptic quantum algebra A_q,p(sl(N))_c. After establishing the expressions of the cocycle conditions, we solve them, both in the classical and in the quantum case (for sl(2)). We find that the consistent central extensions are much more general that those found previously in the literature.     

Special tensors in the deformation theory of quadratic algebras for the classical Lie algebras

Using deformation theory, Braverman and Joseph constructed certain primitive ideals in the enveloping algebras of the simple Lie algebras. Except in the case sl(2,C)$\mathfrak{s}\mathfrak{l}(2,\mathbb{C})$, there is a special value of the deformation parameter giving an ideal of infinite codimension. For the classical Lie algebras, the uniqueness of the special value is equivalent to the existence of tensors with very particular properties. The existence of these tensors was concluded abstractly by Braverman and Joseph but here we present explicit formulae. This allows a rather direct computation of the special value of the deformation parameter.     

Complex quantum groups and their quantum enveloping algebras

We construct complexified versions of the quantum groups associated with the Lie algebras of typeA _n?1 ,B _n ,C _n , andD _n . Following the ideas of Faddeev, Reshetikhin and Takhtajan we obtain the Hopf algebras of regular functionals U_? on these complexified quantum groups. In the special exampleA ₁ we derive theq-deformed enveloping algebraU _q (sl(2, ?)). In the limitq→1 the undeformedU _q (sl(2, ?)) is recovered.     

Quantum Q systems: from cluster algebras to quantum current algebras

This paper gives a new algebraic interpretation for the algebra generated by the quantum cluster variables of the \(A_r\) quantum Q-system (Di Francesco and Kedem in Int Math Res Not IMRN 10:2593–2642, 2014). We show that the algebra can be described as a quotient of the localization of the quantum algebra \(U_{\sqrt{q}}({\mathfrak {n}}[u,u^{-1}])\subset U_{\sqrt{q}}(\widehat{{\mathfrak {sl}}}_2)\), in the Drinfeld presentation. The generating current is made up of a subset of the cluster variables which satisfy the Q-system, which we call fundamental. The other cluster variables are given by a quantum determinant-type formula, and are polynomials in the fundamental generators. The conserved quantities of the discrete evolution (Di Francesco and Kedem in Adv Math 228(1):97–152, 2011) described by quantum Q-system generate the Cartan currents at level 0, in a non-standard polarization. The rest of the quantum affine algebra is also described in terms of cluster variables.     

Operator algebras related to quantum optical systems and integrations

We show that some quantum optical systems generate quantum algebras being the natural generalization of the Heisenberg-Weyl algebra. The importance of these algebras for the integration of the systems under consideration is discussed. Presented at the 11th Colloquium “Quantum Groups and Integrable Systems”, Prague, 20–22 June 2002. This work is supported in part by KBN grant 2 PO3 A 012 19.     

N-dimensional classical integrable systems from Hopf algebras

A systematic method to constructN-body integrable systems is introduced by means of phase space realizations of universal enveloping Hopf algebras. A particular realization for theso(2, 1) case (Gaudin system) is analysed and an integrable quantum deformation is constructed by using quantum algebras as Poisson-Hopf symmetries.Presented at the 5th International Colloquium on Quantum Groups: Quantum Groups and Integrable Systems, Prague, 20–22 June 1996.     

Quasigraded deformations of loop algebras and classical integrable systems

We construct a family of infinite-dimensional quasigraded Lie algebras, that could be viewed as deformation of the graded loop algebras. Using them we obtain new series of integrable Hamiltonian systems on semisimple Lie algebras and their extensions. Presented at the 11th Colloquium “Quantum Groups and Integrable Systems”, Prague, 20–22 June 2002.     

Nambu-Poisson dynamics of superintegrable systems

After an introduction to Nambu-Poisson dynamics (NPD), some applications of NPD in finite-dimensional (superintegrable) and infinite-dimensional (extended quantum mechanics and hydrodynamics) systems are considered. The text was submitted by the author in English.     

On the relation between classical and quantum observables

For systems with a finite number of degrees of freedom, the relation between classical and quantum observables is analysed. In particular, a precise statement of the correspondence limit is obtained.     

Factorization approach to superintegrable systems: Formalism and applications

The factorization technique for superintegrable Hamiltonian systems is revisited and applied in order to obtain additional (higher-order) constants of the motion. In particular, the factorization approach to the classical anisotropic oscillator on the Euclidean plane is reviewed, and new classical (super) integrable anisotropic oscillators on the sphere are constructed. The Tremblay–Turbiner–Winternitz system on the Euclidean plane is also studied from this viewpoint.

     

Pseudointegrable systems in classical and quantum mechanics

We study particles moving in planar polygonal enclosures with rational angles, and show by several methods that trajectories in the classical phase space explore two-dimensional invariant surfaces which are generically not tori as in integrable systems but instead have the topology of multiply-handled spheres. The quantum mechanics of one such ‘pseudointegrable system’ is studied in detail by computing energy levels using an exact formalism. This system consists of motion on a unit coordinate torus containing a square reflecting obstacle with side L. We find that neighbouring levels avoid degeneracies as L varies, and that the probability distribution for the spacing S of adjacent levels vanishes linearly as S→0 (‘level repulsion’). The Weyl area rule plus edge and corner corrections gives a very accurate approximation for the mean level density. Oscillatory corrections to the mean level density are given as a sum over closed classical paths; for pseudointegrable systems these closed paths form families covering part of the phase-space invariant surfaces.