Extensions of Noether's problem and Gelfand-Kirillov's conjecture to certain classes of noncommutative algebras

Abstract

Noether's Problem is a classical problem in the theory of commuative fields. J. Alev and F. Dumas introduced a noncommutative version of this problem, where the Weyl skew-fields substitute the role of the fields of rational functions. This topic was studied by the student for obtaining the Masters Degree, together with it's relation to another famous problem: Gelfand-Kirillov's Conjecture. During his studies, the student, together with his adivsor V. Futorny and Farkhod Eshmatov, obtained a new result, about invariants of the Weyl skew-fields under the action of pseudo-reflection groups. This PhD project seeks to continue the research developed in the process off obtaining the Master's degree. We search generalizations and applications of Noether's Problem and Gelfand-Kirillov's Conjecture for new classes of noncommutative algebras. We plan to study invariants under the action of Weyl groups on the ring of differential operators on the torus; to study possible applications of Noncommutative Noether's Problem to the theory of Galois Algebras, together with versions of Gelfand-Kirillov's Conjecture in this context. Applications and extensions of Noether's Problem and Gelfand-Kirillov's Conjecture will also be studied in the setting of Cherednik Algebras (in particular, the rational ones).Finally, we will search for generalizations of these results for a generalization of the concept of Galois Algebras, where the role of finite groups will be replaced by actions of Hopf algebras. (AU)