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^a_b, 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).