Quantum Group Sheaf and Quantum Manifolds

Abstract

The problem of introducing a dependence of elements of quantum group on classical parameters is considered. It is suggested to interpret a homomorphism from the algebra of functions on quantum group to the algebra of sections of a sheaf of algebras on a classical manifold as describing such a dependence. It is argued that the functorial point of view of group schemes is more appropriate in quantum group field theory. A sheaf of the Hopf algebras over the manifold (quantum sheaf) is constructed by using bosonization formulas for the algebra of functions on the quantum group $SU_{q}(2)$ and the theory of repre- sentations of canonical commutation relations. A family of automorphisms of the Hopf algebra depending on classical variables is described. Quantum manifolds, i.e. manifolds with commutative and non-commutative coordinates are discussed as a generalization of supermanifolds. Quantum group chiral fields and relations with algebraic differential calculus are discussed.