Introduction

L. Faddeev, N. Reshetikhin and L. Takhtadjian [RTF] introduced a construction to obtain quantum deformations of coordinate rings of classical groups. General considerations about this so called FRT-construction can be found for instance in [Ma],[Ta], [Ha], [Su] and also in many textbooks on quantum groups. Our approach differs from former ones in the following three aspects:

First, we focus attention to the graded matric bialgebra which arises in the first step of the construction. This means that we rather look at quantizations of appropriate closed monoids instead of classical groups. Especially we look at the homogenous summands of these graded bialgebras. These are coalgebras which can be defined in a dual way to centralizer algebras of subsets in an endomorphism ring. We therefore call them centralizer coalgebras and investigate their relationship to the corresponding centralizer algebras.

Further, we work over arbitrary noetherian integral domains as base rings. This makes sense since all known examples are already well defined over rings of integral Laurent polynomials, i.e. ${\mathbb{Z}}[X, X^{-1}]$ (in the indeterminate $X$). We will see that there are tremendous differences to the theory over fields, especially concerning the comparison of centralizer algebras and coalgebras. For instance the centralizer coalgebra may have $R$-torsion. We present the following criterion for $R$-projectivity: This property holds if and only if the centralizer algebra is stable under base changes. Furthermore, we will see that the latter property is always valid for centralizer coalgebras.

Finally, the FRT-construction in the ordinary form depends on exactly one endomorphism which usually is a quantum Yang-Baxter operator in the applications. Here we give another version of the FRT-construction which can be applied to sets of endomorphisms. This generalization is necessary to describe the coordinate rings of classical symplectic and orthogonal monoids by use of an FRT-type construction. We demonstrate this in the symplectic case, giving some improvements of results by S. Doty [Dt]. Furthermore, applying our results on centralizer coalgebras we obtain an integral form for the symplectic Schur algebra defined by S. Donkin in [Do2] without any use of the hyperalgebra or Kostant ${\mathbb{Z}}$-form. As an additional incredient we need a symplectic version of the straightening formula for bideterminants the proof of which covers all of the last two sections.