The aim of this work is to describe the geometry underlying such algebras as the coordinate rings of quantum groups at roots of unity. Proofs of the results described here can be found in [Bull. Soc. Math. Fr. 122, No. 4, 443-485 (1994)]. These algebras are known to be finite modules over their centers. In fact, the center contains a copy of the coordinate algebra of the (classical) corresponding group. This embedding was discovered by Manin, Parshall-Wang in the \(GL_ n\) case and Lusztig in the general case. It was named after Frobenius. The present work gives a complete description of the center in the case where the parameter is a primitive \(l = p^ \alpha\)-th root of 1 (\(p\) is a prime, \(l\) is not dividing any entry of the Cartan matrix). In addition to its “Frobenius part”, the center contains elements which N. Andruskiewitsch and the author observed to be linked to the Poisson (i.e., prequantized) structure of the group. This structure is analogous to what was discovered by de Concini, Kac and Procesi in the dual case (analysis of the algebras \({\mathcal U}_ q {\mathbf g}\), \(q^ l = 1\)).