Lie bialgebra contractions and quantum deformations of quasi-orthogonal algebras

TL;DR

This paper introduces and classifies Lie bialgebra contractions, applying them to pseudo-orthogonal algebras to generate new quantum deformations and non-semisimple quantum algebras like Euclidean, Poincaré, and Galilean.

Contribution

It provides a systematic classification of Lie bialgebra contractions and applies them to construct quantum deformations of quasi-orthogonal algebras.

Findings

Classified Lie bialgebra contractions for pseudo-orthogonal algebras.

Constructed quantum deformations for specific cases N=2,3,4.

Generated new non-semisimple quantum algebras from contractions.

Abstract

Lie bialgebra contractions are introduced and classified. A non-degenerate coboundary bialgebra structure is implemented into all pseudo-orthogonal algebras $so(p,q)$ starting from the one corresponding to $so(N+1)$. It allows to introduce a set of Lie bialgebra contractions which leads to Lie bialgebras of quasi-orthogonal algebras. This construction is explicitly given for the cases $N=2,3,4$. All Lie bialgebra contractions studied in this paper define Hopf algebra contractions for the Drinfel'd-Jimbo deformations $U_z so(p,q)$. They are explicitly used to generate new non-semisimple quantum algebras as it is the case for the Euclidean, Poincar\'e and Galilean algebras.