Determinants, integrality and Noether's theorem for quantum commutative algebras

The aim of this paper is to generalize Noether's theorem for finite groups acting on commutative algebras, to finite-dimensional triangular Hopf algebras acting on quantum commutative algebras. In the process we construct a non-commutative determinant function which yields an analogue of the Cayley-Hamilton theorem for certain endomorphisms.