Differential Calculus on Iso_q (N), Quantum Poincare' Algebra and q - Gravity

TL;DR

This paper develops a method to deform inhomogeneous algebras and constructs a quantum Poincaré algebra, enabling a $q$-deformation of gravity with a generalized Einstein-Cartan lagrangian.

Contribution

It introduces a projection-based method to deform inhomogeneous algebras and explicitly constructs a quantum Poincaré algebra suitable for $q$-gravity models.

Findings

Constructed bicovariant differential calculus for deformed algebras.

Derived explicit quantum Poincaré algebra with 10 generators.

Proposed a $q$-deformed gravity lagrangian extending Einstein-Cartan theory.

Abstract

We present a general method to deform the inhomogeneous algebras of the $B_n,C_n,D_n$ type, and find the corresponding bicovariant differential calculus. The method is based on a projection from $B_{n+1}, C_{n+1}, D_{n+1}$. For example we obtain the (bicovariant) inhomogeneous $q$-algebra $ISO_q(N)$ as a consistent projection of the (bicovariant) $q$-algebra $SO_q(N+2)$. This projection works for particular multiparametric deformations of $SO(N+2)$, the so-called ``minimal" deformations. The case of $ISO_q(4)$ is studied in detail: a real form corresponding to a Lorentz signature exists only for one of the minimal deformations, depending on one parameter $q$. The quantum Poincar\'e Lie algebra is given explicitly: it has 10 generators (no dilatations) and contains the {\sl classical} Lorentz algebra. Only the commutation relations involving the momenta depend on $q$. Finally, we discuss a $q$-deformation of gravity based on the ``gauging" of this $q$-Poincar\'e algebra: the lagrangian generalizes the usual Einstein-Cartan lagrangian.