The Singularities of Quantum Groups

If CG] CH] is an extension of Hopf domains of degree d, thenH G is an étale map. Equivalently, the variety X^{C}H]of d-dimensional CH]-modules compatible with the trace mapof the extension, is a smooth GL_d-variety with quotient G. Ifwe replace CH] by a non-commutative Hopf algebra H, we constructsimilarly a GL^d-variety and quotient map : X_H G. The smoothlocus of H over CG] is the set of points g G such that X_His smooth along {^-1}(g). We relate this set to the separabilitylocus of H over CG] as well as to the (ordinary) smooth locusof the commutative extension CG] Z where Z is the centre ofH. In particular, we prove that the smooth locus coincides withthe separability locus whenever H is a reflexive Azumaya algebra.This implies that the quantum function algebras O(G) and quantisedenveloping algebras U{g} are as singular as possible. 1991 MathematicsSubject Classification: 16W30, 16R30.