Effacement des dérivations et spectres premiers des algèbres quantiques

Abstract

Given any (commutative) field k and any iterated Ore extension R = k [ X 1 ][ X 2 ; σ 2 , δ 2 ]⋯[ X N ; σ N , δ N ] 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 ( X 1 ,…, X N ) and terminating with new variables ( T 1 ,…, T N ). These new variables generate some quantum-affine space R such that F= Fract ( R ) . This algorithm induces a natural embedding ϕ : Spec (R)→ Spec ( R ) which satisfies the following property: If P is in Spec (R), then Fract (R/P)≃ Fract ( R /ϕ(P)) . 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: U q w ( g ) , B q w ( g ) ( w ∈ W ), R q [ G ] (where G denotes any complex, semi-simple, connected, simply connected Lie group with associated Lie algebra g 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.