R matrix and bicovariant calculus for the inhomogeneous quantum groups IGL_q(n)

TL;DR

This paper derives the R matrix for inhomogeneous quantum groups related to GL_q(n) and constructs a bicovariant differential calculus, with applications to the quantum Poincaré group in four dimensions.

Contribution

It provides the R matrix for inhomogeneous quantum groups and develops a bicovariant calculus applicable to quantum inhomogeneous groups like the quantum Poincaré group.

Findings

R matrix satisfies the quantum Yang-Baxter equation due to the Hecke relation.

Constructed a bicovariant differential calculus on IGL_q(n).

Applied calculus to the quantum Poincaré group in four dimensions.

Abstract

We find the R matrix for the inhomogeneous quantum groups whose homogeneous part is $GL_q(n)$, or its restrictions to $SL_q(n)$,$U_q(n)$ and $SU_q(n)$. The quantum Yang-Baxter equation for R holds because of the Hecke relation for the braiding matrix of the homogeneous subgroup. A bicovariant differential calculus on $IGL_q(n)$ is constructed, and its application to the $D=4$ Poincar\'e group $ISL_q(2,\Cb)$ is discussed.