Superintegrable models of Winternitz type

A general procedure is outlined which allows one to construct superintegrable models of Winternitz type. Some examples are presented.Comment: 6 pages, LaTeX; To appear in Phys. Lett.

The Quantum Galilei Group

The quantum Galilei group $G_{\varkappa}$ is defined. The bicrossproduct structure of $G_{\varkappa}$ and the corresponding Lie algebra is revealed. The projective representations for the two-dimensional quantum Galilei group are constructed.Comment: AMSTe

Superintegrable Calogero-type systems admit maximal number of Poisson structures

We present a general scheme for constructing the Poisson structure of super-integrable dynamical systems of which the rational Calogero-Moser system is one of the most interesting one. This dynamical system is $2N$ dimensional with $2N- 1$ first integrals and our construction yields $2N-1$ degenerate Poisson tensors that each admit $2(N-1)$ Casimirs. Our results are quite generally applicable to all super-integrable systems and form an alternative to the traditional bi-Hamiltonian approach

Global Space-Time Symmetries of Quantized Euclidean and Minkowski Superspaces

Starting with assumptions both simple and natural from "physical" point of view we present a direct construction of transformations preserving wide class of (anti)commutation relations which describe Euclidean/Minkowski superspace quantizations. These generalized transformations act on deformed superspaces as the ordinary ones do on undeformed spaces but they depend on non(anti)commuting parameters satisfying some consistent (anti)commutation relations. Once the coalgebraic structure compatible with the algebraic one is introduced in the set of transformations we deal with quantum symmetry supergroup. This is the case for intensively studied so called N=1/2 supersymmetry as well as its three parameter extension. The resulting symmetry transformations - supersymmetric extension of theta-Euclidean group can be regarded as global counterpart of appropriately twisted Euclidean superalgebra that has been shown to preserve N=1/2 supersymmetry.Comment: 34 pages; this is the final version accepted for publication in JHE