arXiv subjects
Publications and source records attributed to J. A. de Azcarraga.
New generalized Poisson structures are introduced by using suitable skew-symmetric contravariant tensors of even order. The corresponding `Jacobi identities' are provided by conditions on these tensors, which may be understood as cocycle conditions. As an example, we provide the linear generalized Poisson structures which can be constructed on the dual spaces of simple Lie algebras.
Using general but simple covariance arguments, we classify the `quantum' Minkowski spaces for dimensionless deformation parameters. This requires a previous analysis of the associated Lorentz groups, which reproduces a previous classification by Woronowicz and Zakrzewski. As a consequence of the unified analysis presented, we give the commutation properties, the deformed (and central) length element and the metric tensor for the different spacetime algebras.
The deformations of the Galilei algebra and their associated noncommutative Newtonian spacetimes are investigated. This is done by analyzing the possible nonrelativistic limits of an eleven generator (pseudo)extended \kap-Poincaré algebra $\tilde{\cal P}_κ$ and their implications for the existence of a first order differential calculus. The additional one-form needed to achieve a consistent calculus on \kap-Minkowski space is shown to be related to the additional central generator entering in the $\tilde{\cal P}_κ$ Hopf algebra. In the process, deformations of the extended Galilei and Galilei algebras are introduced which have, respectively, a cocycle and a bicrossproduct structure.
Some physical aspects of $q$-deformed spacetimes are discussed. It is pointed out that, under certain standard assumptions relating deformation and quantization, the classical limit (Poisson bracket description) of the dynamics is bound to contain unusual features. At the same time, it is argued that the formulation of an associated $q$-deformed field theory is fraught with serious difficulties.
An introduction to quantum groups and non-commutative differential calculus (Lecture at the III Workshop on Differential Geometry, Granada, September 1994)
Reflection and braid equations for rank two $q$-tensors are derived from the covariance properties of quantum vectors by using the $R$-matrix formalism.