, talk at the workshop "Quantengruppen und ihre Darstellungen", Oberflockenbach, 3.-7.4.1995
Abstract:
The talk is about the construction of coordinate rings for quantum monoids (and groups) which is due to L. Faddeev, N. Reshetikhin and L. Takhtadjian (FRT-construction). In the ordinary form this construction is applied in the case where the ground ring R is a field to an endomorphism of the natural module of the quantum group satisfying the quantum Yang-Baxter equation. If you take the ring of integer Laurent polynomials in the indeterminant q as ground ring R, the obtained coordinate rings of quantum monoids will have R-torsion in the symplectic and orthogonal cases. To avoid this the constructions must be applied to the above and a certain additional endomorphism, which is considered in the talk. Furthermore we look at the structure of the homogenous summands of the graded coordinate rings. Their definition is dual to the definition of centralizing algebras in a certain sense and they are projective as R-modules if and only if the corresponding centralizing algebra is stable under base changes. Finally, referring to work of T.Hayashi we explain how to obtain quantum versions of the groups of symplectic and orthogonal similitudes as open submonoids in the symplectic resp. orthogonal quantum monoid.
Remark:
The present German paper doesn't contain any proof. For a manuscript containing the proofs click here .