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

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

1994-08-05

arxiv.org/abs/hep-th/9408031v1

Int.J.Mod.Phys.A11:1019-1056,1996

Physics

High Energy Physics

High Energy Physics - Theory

38 pages

Scientific paper

10.1142/S0217751X96000481

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).

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