Effacement des dérivations et spectres premiers des algèbres quantiques

Given any (commutative) field k and any iterated Ore extension R=k[X1][X2;σ2,δ2] [XN;σN,δN] satisfying some suitable assumptions, we construct the so-called “Derivative-Elimination Algorithm.” It consists of a sequence of changes of variables inside the division ring F=Fract(R), starting with the indeterminates (X1,…,XN) and terminating with new variables (T1,…,TN). These new variables generate some quantum-affine space such that . This algorithm induces a natural embedding which satisfies the following property: . We study both the derivative-elimination algorithm and natural embedding and use them to produce, for the general case, a (common) proof of the “quantum Gelfand–Kirillov” property for the prime homomorphic images of the following quantum algebras: , (w W), Rq[G] (where G denotes any complex, semi-simple, connected, simply connected Lie group with associated Lie algebra and Weyl group W), quantum matrices algebras, quantum Weyl algebras and quantum Euclidean (respectively symplectic) spaces. Another application will be given in [G. Cauchon, J. Algebra, to appear]: In the general case, the prime spectrum of any quantum matrices algebra satisfies the normal separation property.