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

TL;DR

This paper introduces a novel non-standard quantum deformation of the so(2,2) algebra, explores its contractions and automorphisms, and links it to (2+1) kinematical and conformal algebras, revealing new algebraic structures.

Contribution

It presents a new family of non-standard quantum algebras derived from so(2,2), including a deformation of the Poincaré algebra and conformal algebras, with explicit realizations and properties.

Findings

New non-standard quantum algebra family introduced.

Quantum contractions of so(2,2) generalized via automorphisms.

Deformation of Poincaré and conformal algebras explicitly constructed.

Abstract

A new "non-standard" quantization of the universal enveloping algebra of the split (natural) real form $so(2,2)$ of $D_2$ is presented. Some (classical) graded contractions of $so(2,2)$ associated to a $Z_2 \times Z_2$ 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 Poincar\'e 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.