Inhomogeneous quantum groups IGL_{q,r}(N): Universal enveloping algebra and differential calculus

TL;DR

This paper reviews multiparametric quantum groups, constructs the inhomogeneous quantum group IGL_qr(N), and develops its differential calculus, unifying quantum plane coordinates and group elements in a single framework.

Contribution

It introduces a method to derive the differential calculus on IGL_qr(N) from that on GL_qr(N+1), unifying quantum coordinates and group elements.

Findings

Explicit construction of IGL_qr(N) as a quotient of GL_qr(N+1)

Derivation of the bicovariant differential calculus on IGL_qr(N)

Illustration of the theory with the example of IGL_qr(2)

Abstract

A review of the multiparametric linear quantum group GL_qr(N), its real forms, its dual algebra U(gl_qr(N)) and its bicovariant differential calculus is given in the first part of the paper. We then construct the (multiparametric) linear inhomogeneous quantum group IGL_qr(N) as a projection from GL_qr(N+1), or equivalently, as a quotient of GL_qr(N+1) with respect to a suitable Hopf algebra ideal. A bicovariant differential calculus on IGL_qr(N) is explicitly obtained as a projection from the one on GL_qr(N+1). Our procedure unifies in a single structure the quantum plane coordinates and the q-group matrix elements T^a_b, and allows to deduce without effort the differential calculus on the q-plane IGL_qr(N) / GL_qr(N). The general theory is illustrated on the example of IGL_qr(2).