On the quantum Poincare' group

TL;DR

This paper develops a method to derive bicovariant differential calculus on inhomogeneous quantum groups, including the quantum Poincare' group, by projecting from quantum general linear groups, thus extending quantum group geometry.

Contribution

It introduces a projection technique to obtain bicovariant calculus on inhomogeneous quantum groups, including the quantum Poincare' group, from known calculus on quantum general linear groups.

Findings

Derived bicovariant calculus on $IGL_q(n)$ and $IGL_q(n, ext{C})$

Recovered quantum Poincare' group and its geometry

Extended calculus methods to complex inhomogeneous quantum groups

Abstract

The inhomogeneous quantum groups $IGL_q(n)$ are obtained by means of a particular projection of $GL_q(n+1)$. The bicovariant differential calculus on $GL_q(n)$ is likewise projected into a consistent bicovariant calculus on $IGL_q(n)$. Applying the same method to $GL_q(n,\Cb)$ leads to a bicovariant calculus for the complex inhomogeneous quantum groups $IGL_q(n,\Cb)$. The quantum Poincare' group and its bicovariant geometry are recovered by specializing our results to $ISL_q(2,\Cb)$.