Contractions, deformations and curvature

The role of curvature in relation with Lie algebra contractions of the pseudo-ortogonal algebras so(p,q) is fully described by considering some associated symmetrical homogeneous spaces of constant curvature within a Cayley-Klein framework. We show that a given Lie algebra contraction can be interpreted geometrically as the zero-curvature limit of some underlying homogeneous space with constant curvature. In particular, we study in detail the contraction process for the three classical Riemannian spaces (spherical, Euclidean, hyperbolic), three non-relativistic (Newtonian) spacetimes and three relativistic ((anti-)de Sitter and Minkowskian) spacetimes. Next, from a different perspective, we make use of quantum deformations of Lie algebras in order to construct a family of spaces of non-constant curvature that can be interpreted as deformations of the above nine spaces. In this framework, the quantum deformation parameter is identified as the parameter that controls the curvature of such "quantum" spaces.Comment: 17 pages. Based on the talk given in the Oberwolfach workshop: Deformations and Contractions in Mathematics and Physics (Germany, january 2006) organized by M. de Montigny, A. Fialowski, S. Novikov and M. Schlichenmaie

Quantum superintegrability and exact solvability in N dimensions

A family of maximally superintegrable systems containing the Coulomb atom as a special case is constructed in N-dimensional Euclidean space. Two different sets of N commuting second order operators are found, overlapping in the Hamiltonian alone. The system is separable in several coordinate systems and is shown to be exactly solvable. It is solved in terms of classical orthogonal polynomials. The Hamiltonian and N further operators are shown to lie in the enveloping algebra of a hidden affine Lie algebra

Fourier and Gegenbauer expansions for a fundamental solution of the Laplacian in the hyperboloid model of hyperbolic geometry

Due to the isotropy $d$-dimensional hyperbolic space, there exist a spherically symmetric fundamental solution for its corresponding Laplace-Beltrami operator. On the $R$-radius hyperboloid model of $d$-dimensional hyperbolic geometry with $R>0$ and $d\ge 2$, we compute azimuthal Fourier expansions for a fundamental solution of Laplace's equation. For $d\ge 2$, we compute a Gegenbauer polynomial expansion in geodesic polar coordinates for a fundamental solution of Laplace's equation on this negative-constant sectional curvature Riemannian manifold. In three-dimensions, an addition theorem for the azimuthal Fourier coefficients of a fundamental solution for Laplace's equation is obtained through comparison with its corresponding Gegenbauer expansion.Comment: arXiv admin note: substantial text overlap with arXiv:1201.440

Superintegrability on N-dimensional spaces of constant curvature from so(N+1) and its contractions

The Lie-Poisson algebra so(N+1) and some of its contractions are used to construct a family of superintegrable Hamiltonians on the ND spherical, Euclidean, hyperbolic, Minkowskian and (anti-)de Sitter spaces. We firstly present a Hamiltonian which is a superposition of an arbitrary central potential with N arbitrary centrifugal terms. Such a system is quasi-maximally superintegrable since this is endowed with 2N-3 functionally independent constants of the motion (plus the Hamiltonian). Secondly, we identify two maximally superintegrable Hamiltonians by choosing a specific central potential and finding at the same time the remaining integral. The former is the generalization of the Smorodinsky-Winternitz system to the above six spaces, while the latter is a generalization of the Kepler-Coulomb potential, for which the Laplace-Runge-Lenz N-vector is also given. All the systems and constants of the motion are explicitly expressed in a unified form in terms of ambient and polar coordinates as they are parametrized by two contraction parameters (curvature and signature of the metric).Comment: 14 pages. Based on the contribution presented at the "XII International Conference on Symmetry Methods in Physics", Yerevan (Armenia), July 2006. To appear in Physics of Atomic Nucle

Synthesis, Structure and Stereochemistry of Dispirocompounds Based on Imidazothiazolotriazine and Pyrrolidineoxindole

Methods for the synthesis of two types of isomeric dispirocompounds based on imidazothiazolotriazine and pyrrolidineoxindole, differing in the structure of imidazothiazolotriazine fragment, namely, linear dispiro[imidazo[4,5-e]thiazolo[3,2-b][1,2,4]triazine-6,3′-pyrrolidine- 4′,3″-indolines] and angular dispiro[imidazo[4,5-e]thiazolo[2,3-c][1,2,4]triazine-7,3′-pyrrolidine-4′,3″-indolines] were proposed. The first method relies on a 1,3-dipolar cycloaddition of azomethine ylides generated in situ from paraformaldehyde and N-alkylglycine derivatives to the corresponding oxindolylidene derivatives of imidazothiazolotriazine. The cycloaddition leads to a mixture of two diastereomers resulted from anti- and syn-approaches of azomethine ylide in approximately a 1:1 ratio, which were separated by column chromatography. Another method consists in rearrangement of linear dispiro[imidazo[4,5-e]thiazolo[3,2-b][1,2,4]triazine-6,3′-pyrrolidine-4′,3″-indolines] into hitherto unavailable angular dispiro[imidazo[4,5-e]thiazolo[2,3-c]-[1,2,4]triazine-7,3′-pyrrolidine-4′,3″-indolines] upon treatment with KOH. It was found that the anti-diastereomer of linear type underwent rearrangement into the isomeric angular syn-diastereomer, while the rearrangement of the linear syn-diastereomer gave the angular anti-diastereomer

An effective one-pot access to polynuclear dispiroheterocyclic structures comprising pyrrolidinyloxindole and imidazothiazolotriazine moieties via a 1,3-dipolar cycloaddition strategy

An effective and highly regio- and diastereoselective one-pot method for the synthesis of new polynuclear dispiroheterocyclic systems with five stereogenic centers (dispiro[imidazo[4,5-e]thiazolo[3,2-b]-1,2,4-triazine-6,3′-pyrrolidine-2′,3′′-indoles]) comprising pyrrolidinyloxindole and imidazo[4,5-e]thiazolo[3,2-b]-1,2,4-triazine moieties has been developed. The method relies on a 1,3-dipolar cycloaddition of azomethine ylides generated in situ from isatin derivatives and sarcosine to 6-benzylideneimidazo[4,5-e]thiazolo[3,2-b]-1,2,4-triazine-2,7-diones