arXiv subjects
Publications and source records attributed to S. Zakrzewski.
Quantum Lorentz groups H admitting quantum Minkowski space V are selected. Natural structure of a quantum space G = V x H is introduced, defining a quantum group structure on G only for triangular H (q=1). We show that it defines a braided quantum group structure on G for |q|=1.
The well known incompatibility between inhomogeneous quantum groups and the standard q-deformation is shown to disappear (at least in certain cases) when admitting the quantum group to be braided. Braided quantum ISO(p,N-p) containing SO_q(p,N-p) with |q|=1 are constructed for N=2p, 2p+1, 2p+2. Their Poisson analogues (obtained first) are presented as an introduction to the quantum case.
The existence of the theory of `twisted cotangent bundles' (symplectic groupoids) allows to study classical mechanical systems which are generalized in the sense that their configurations form a Poisson manifold. It is natural to study from this point of view first such systems which arise in the context of some basic physical symmetry (space-time, rotations, etc.). We review results obtained so far in this direction.
Poisson plane and sphere --- homogeneous spaces of Poisson groups E(2) and SU(2) (resp.) --- have phase spaces (corresponding symplectic groupoids), in which a free Hamiltonian is naturally defined. We solve the equations of motion and point out some unexpected features: free motion on the plane is bounded (periodic) and free trajectories on the sphere are all circles except the big ones.
The general framework of Legendre transformation is extended to the case of symplectic groupoids, using an appropriate generalization of the notion of generating function (of a Lagrangian submanifold).
SL(N,C) is the phase space of the Poisson SU(N). We calculate explicitly the symplectic structure of SL(N,C), define an analogue of the Hamiltonian of the free motion on SU(N) and solve the corresponding equations of motion. Velocity is related to the momentum by a non-linear Legendre transformation.
Natural conditions on a Poisson/quantum group G to implement Poisson/quantum gauge transformations on the lattice are investigated. In addition to our previous result that transformations on one lattice link require G to be coboundary, it is shown that for a sequence of links one needs a quasitriangular G.
We show that a Poisson Lie group $(G,π)$ is coboundary if and only if the natural action of $G\times G$ on $M=G$ is a Poisson action for an appropriate Poisson structure on $M$ (the structure turns out to be the well known $π_+$). We analyze the same condition in the context of Hopf algebras. Quantum analogue of the $π_+$ structure on SU(N) is described in terms of generators and relations as an example.
An introduction to inhomogeneous Poisson groups is given. Poisson inhomogeneous $O(p,q)$ are shown to be coboundary, the generalized classical Yang-Baxter equation having only one-dimensional right hand side. Normal forms of the classical $r$-matrices for the Poincaré group (inhomogeneous $O(1,3)$) are calculated.
Fundamental representations of real simple Poisson Lie groups are Poisson actions with a suitable choice of the Poisson structure on the underlying (real) vector space. We study these (mostly quadratic) Poisson structures and corresponding phase spaces (symplectic groupoids).
The dependence of velocity on momentum for the free massive particle obeying the $κ$-Poincaré Poisson symmetry is calculated in terms of intrinsic non-commuting space-time coordinates and shown to have a monotonic character, with upper limit of velocity equal to 1.
We present almost complete list of normal forms of classical $r$-matrices on the Poincaré group.
General framework for Poisson homogeneous spaces of Poisson groups is introduced. Poisson Minkowski spaces are discussed as a particular example.
Extended phase space of an elementary (relativistic) system is introduced in the spirit of the Souriau's definition of the `space of motions' for such system. Our formulation is generally applicable to any homogeneous space-time (e.g. de Sitter) and also to Poisson actions. Calculations concerning the Minkowski case for non-zero spin particles show an intriguing alternative: we should either accept two-dimensional trajectories or (Poisson) noncommuting space-time coordinates.