On the Geometry of Inhomogeneous Quantum Groups

Abstract

We first give a pedagogical introduction to the differential calculus on q-groups and analize the relation between differential calculus and q-Lie algebra. Equivalent definitions of bicovariant differential calculus are studied and their geometrical interpretation is explained. Vectorfields, contraction operator and Lie derivative are defined and their properties discussed. After a review of the geometry of the (multiparametric) linear q-group $GL_{q,r}(N)$ we construct the inhomogeneous q-group $IGL_{q,r}(N)$ as a projection from $GL_{q,r}(N+1)$, i.e. as a quotient of $GL_{q,r}(N+1)$ with respect to a suitable Hopf algebra ideal. The semidirect product structure of $IGL_{q,r}(N)$ given by the $GL_{q,r}(N)$ q-subgroup times translations is easily analized. A bicovariant calculus on $IGL_{q,r}(N)$ is explicitly obtained as a projection from the one on $GL_{q,r}(N+1)$. The universal enveloping algebra of $IGL_{q,r}(N)$ and its $R$-matrix formulation are constructed along the same lines. We proceed similarly in the orthogonal and symplectic case. We find the inhomogeneous multiparametric q-groups of the $B_n,C_n,D_n$ series via a projection from $B_{n+1}, C_{n+1},D_{n+1}$. We give an $R$-matrix formulation and discuss real forms. We study their universal enveloping algebras and differential calculi. In particular we obtain the bicovariant calculus on a dilatation-free minimal deformation of the Poincar\'e group $ISO_q(3,1)$. The projection procedure is also used to construct differential calculi on multiparametric q-orthogonal planes in any dimension $N$. Real forms are studied and in particular we obtain a q-Minkowski space and its q-deformed phase-space with hermitian operators $x^a$ and $p_a$.