Quantum deformations of certain prehomogeneous vector spaces III

摘要

We apply our previous result to the classical groups, and construct quantum analogues of the coordinate algebras of certain prehomogeneous vector spaces as non-commutative algebras equipped with actions of the quantized enveloping algebras. We also give explicit descriptions of the non-commutative counterparts for the generators of the defining ideals of the closures of orbits including basic relative invariants. In particular, quantum analogues of a quadratic form and the determinant of a symmetric matrix are naturally obtained.