QUANTUM FUNCTION ALGEBRA AT ROOTS OF 1

We introduce a form of the quantum function algebra on a Drinfeld-Jimbo quantum group over the ring Z[q, q-1]. Specializing q to a root of 1, we show that over the cyclotomic field this algebra is a projective module over its central sub-algebra, which is the usual coordinate algebra of the group. We study the induced Poisson-Lie structure of the group. A bundle of algebras on a complex simply connected Lie group with hamiltonian flows in the bundle is constructed. Some representations of the quantum function algebra in a root of 1 are constructed as an application. An estimate of the dimension of an arbitrary representation is given. (C) 1994 Academic Press, Inc.