Non-standard quantum so(2,2) and beyond

A new “non-standard” quantization of the universal enveloping algebra of the split (natural) real form so(2, 2) of D2 is presented. Some (classical) graded contractions of so(2, 2) associated to a Z2 × Z2 grading are studied, and the automorphisms defining this grading are generalized to the quantum case, thus providing quantum contractions of this algebra. This produces a new family of “non-standard” quantum algebras. Some of these algebras can be realized as (2+1) kinematical algebras; we explicitly introduce a new deformation of Poincaré algebra, which is naturally linked to the null plane basis. Another realization of these quantum algebras as deformations of the conformal algebras for the two-dimensional Euclidean, Galilei and Minkowski spaces is given, and its new properties are emphasized.