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

A review of the multiparametric linear quantum group GL(q,r)(N), its real Forms, its dual algebra U[gl(q,r)(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(q,r)(N) as a projection from GL(q,r)(N + 1) or, equivalently, as a quotient of GL(q,r)(N + 1) with respect to a suitable Hopf algebra ideal. A bicovariant differential calculus on IGL(q,r)(N) is explicitly obtained as a projection from that on GL(q,r)(N + 1). Our procedure unifies in a single structure the quantum plane coordinates and the q group matrix elements T-b(a), and allows one to deduce without effort the differential calculus on the q plane IGL(q,r)(N)/GL(q,r)(N). The general theory is illustrated on the example of IGL(q,r)(2).