We apply our previous result [14] 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.
展开▼